# A doubly nonlinear evolution problem involving the fractional $p$-Laplacian

Timothy A. Collier<EMAIL_ADDRESS>Daniel Hauer11footnotemark: 1
<EMAIL_ADDRESS>The University of Sydney, School of Mathematics and
Statistics, NSW 2006, Australia

###### Abstract

In this article we focus on a doubly nonlinear nonlocal parabolic initial
boundary value problem driven by the fractional $p$-Laplacian equipped with
homogeneous Dirichlet boundary conditions on a domain in ${\mathbb{R}}^{d}$
and composed with a continuous, strictly increasing function. We establish
well-posedness in $L^{1}$ in the sense of mild solutions, a comparison
principle, and for restricted initial data we obtain that mild solutions of
the inhomogeneous evolution problem are strong. We obtain $L^{q}-L^{\infty}$
regularity estimates for mild solutions, implying decay estimates and
extending the property of strong solutions for more initial data. Moreover, we
prove local and global Hölder continuity results as well as a comparison
principle that yields extinction in finite time of mild solutions to the
homogeneous evolution equation.

###### keywords:

doubly nonlinear, porous media, fractional p-Laplacian, mild solutions,
accretive operators in $L^{1}$, nonlinear semigroups

###### MSC:

[2010] 35R11, 35B40, 35K55, 35B65, 47H06, 35K10

##  1 Introduction and main results

###  1.1 Introduction

Let $\Omega$ be an open set in $\mathbb{R}^{d}$, $d\geq 1$, $0<T<\infty$,
$1<p<\infty$ and $0<s<1$. Then our main focus in this article is the following
initial boundary value problem

$\begin{cases}\begin{aligned}
u_{t}(t)+(-\Delta_{p})^{s}\varphi(u(t))+f(\cdot,u(t))&=g(\cdot,t)\quad&&\text{in
$\Omega\times(0,T),$}\\\ u(t)&=0\quad&&\text{in
${\mathbb{R}}^{d}\setminus\Omega\times(0,T),$}\\\ u(0)&=u_{0}\quad&&\text{on
$\Omega$,}\end{aligned}\end{cases}$ (1.1)

for given $u_{0}\in L^{1}$ and $g\in L^{1}(0,T;L^{1})$, where we abbreviate
the Lebesgue space $L^{q}(\Omega)$ by $L^{q}$, $1\leq q\leq\infty$, and impose
the following conditions on $\varphi$ and $f$:

$\varphi\in C({\mathbb{R}})$ is a strictly increasing function satisfying
$\varphi(0)=0$, (1.2)

and $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ admits the following
properties:

$f$ is a Lipschitz-continuous Carathéodory function; that is, (1.3a)
$\displaystyle\left\\{\begin{array}[c]{l}\text{for every $u\in{\mathbb{R}}$,
$x\mapsto f(x,u)$ is measurable on $\Omega$, and there is}\\\ \text{an
$\omega>0$ such that}\\\\[5.0pt] \hskip
28.45274pt|f(x,u_{1})-f(x,u_{2})|\leq\omega|u_{1}-u_{2}|\qquad\text{for all
$u_{1},u_{2}\in\mathbb{R}$,}\\\\[5.0pt] \text{uniformly for
a.e.~{}$x\in\Omega$,}\end{array}\right.$ (1.7) and $f(x,0)=0$ for a.e.
$x\in\Omega$. (1.3b)

The doubly nonlinear nonlocal operator $(-\Delta_{p})^{s}\varphi$ in the
evolution problem (1.1) models a (singular or degenerated) nonlocal diffusion
and is the composition of the _(variational) Dirichlet fractional
$p$-Laplacian_ (see Section 2.3)

$\langle(-\Delta_{p})^{s}u,v\rangle:=\int_{{\mathbb{R}}^{2d}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{d+ps}}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$
(1.9)

for every $u$, $v\in W^{s,p}_{0}$ where we write $W^{s,p}_{0}$ for the
fractional Sobolev space $W^{s,p}_{0}(\Omega)$ (see Section 2.1) and a
function $\varphi$ satisfying (1.2).

In this paper, we present well-posedness of (1.1) in the sense of _mild_
solutions $u:[0,T]\to L^{1}$, that is, $u(t)$ is the limit in $L^{1}$ of step-
functions $u_{n}$ with coefficients solving the corresponding time discretized
problem (for a precise definition, we refer to Definition 2.4), and a
comparison result for such solutions (see Theorem 1.1 below). Moreover, we
establish global regularity estimates implying an immediate smoothing effect,
and finite time of extinction under minimal natural hypotheses on the given
initial data $u_{0}$ and the forcing term $g$. Further, we give sufficient
conditions implying that mild solutions $u$ of (1.1) are _strong
distributional solutions_ in $L^{1}$, that is, for a.e. $t\in(0,T)$, $u$ is
differentiable at $t$, $\varphi(u(t))\in W^{s,p}_{0}$ and satisfies
$(-\Delta_{p})^{s}\varphi(u(t))=g(\cdot,t)-f(\cdot,u(t))-u_{t}(t)$ in $L^{1}$
(see Definition 2.6).

We focus, in particular, on the case $\varphi(u)=|u|^{m-1}u$ (which we denote
by $u^{m}$) for $m>0$. Then the equation in (1.1) reduces to

$u_{t}(t)+(-\Delta_{p})^{s}u^{m}(t)+f(\cdot,u(t))=g(\cdot,t)\qquad\text{in
$\Omega\times(0,T)$.}$ (1.10)

For mild solutions of the initial boundary value problem associated to (1.10)
with $m\geq 1$, we prove global $L^{\ell}-L^{\infty}$ regularity estimates,
$1\leq\ell\leq m+1$, implying an immediate _smoothing effect_ (see Theorem
1.2). This also allows us to improve the integrability of the time derivative
of solutions as presented in Theorem 1.5. For this case, we establish in
Theorem 1.7 local Hölder continuity of the solutions to (1.10) provided $p\geq
2$ and $sp\geq d$ and global continuity in Theorem 1.8 for the case $m=1$ and
$1<p<\infty$. We prove these in Section 7. In Section 8, we provide a
comparison principle for the solutions of evolution problem (1.1) when the
homogeneous boundary data is replaced by time-dependent, inhomogeneous data.
With this tool, we can establish finite time of extinction of the solutions to
(1.1).

Nonlinear integro-differential operators such as the doubly nonlinear nonlocal
operator $(-\Delta_{p})^{s}\varphi$ have received recent interest for their
role in the mathematical analysis of anomalous diffusion, as well as their
applications in such fields as statistical mechanics, physics, finance, fluid
dynamics and image processing. We refer the interested reader, for example, to
[1, 2, 3, 4, 5].

In recent years, the first-order evolution problem for the fractional
$p$-Laplacian has been studied by many authors, including [6, 7, 8, 9, 10, 11,
12, 13]. This problem corresponds to that of (1.10) with $m=1$ so that the
equation in (1.1) reduces to

$u_{t}(t)+(-\Delta_{p})^{s}u(t)+f(\cdot,u(t))=g(\cdot,t)\qquad\text{in
$\Omega\times(0,T)$.}$ (1.11)

In particular, Mazón et al. [7] (see also, [8]) obtained existence and
uniqueness of strong solutions to equation (1.11) for $f\equiv g\equiv 0$
equipped with either homogeneous Dirichlet or Neumann boundary conditions.
Giacomoni and Tiwari [9] obtained well-posedness in $L^{\infty}$ of strong
solutions to equation (1.11) equipped with homogeneous Dirichlet boundary
data. Global $L^{q}-L^{\infty}$ regularity estimates, $1\leq q<\infty$, for
solutions to the parabolic equation (1.11) for $g\equiv 0$ and equipped with
homogeneous Dirichlet boundary conditions have been obtained in the monograph
[14] of the second author.

It is worth noting that the operator $(-\Delta_{p})^{s}\varphi$ is the
_nonlocal_ counterpart of the _local_ doubly nonlinear operator
$(-\Delta_{p})\varphi$ (see, for example [15, 16, 14]). In addition, the
fractional $p$-Laplacian $(-\Delta_{p})^{s}$ is a natural generalization of
the well-known linear _fractional Laplacian_ $(-\Delta)^{s}$. Hence, an
important special case of this doubly nonlinear nonlocal evolution problem
(1.1) is given by the celebrated _fractional porous medium equation_

$u_{t}(t)+(-\Delta)^{s}u^{m}(t)=g(\cdot,t)\qquad\text{in
$\Omega\times(0,T)$,}$ (1.12)

for $m>0$. For initial boundary-value problems associated with (1.12), Vázquez
and his collaborators [17, 18, 19] have developed a theory to obtain
existence, uniqueness, regularity, and decay estimates. Their methods were
partially based on classical nonlinear semigroup theory (cf. [20, 21, 22]),
the extension technique [23] (when $\Omega={\mathbb{R}}^{d}$) and other
classical PDE methods. Finite time of extinction of classical solutions to
(1.12) satisfying homogeneous Dirichlet boundary conditions was proved in
[24]. Here we refine the method in [24] and make it available for mild
solutions of the more general boundary value problem (1.1).

The parabolic boundary-value problem (1.1) studied in this paper was recently
studied by Giacomoni et al. [25] in the case $\varphi(r)=r^{m}$,
$r\in{\mathbb{R}}$, given by (1.10), for $\frac{1}{2p-1}\leq m<1$,
$p<\frac{d}{s}$, and $\Omega$ a bounded domain with boundary $\partial\Omega$
of class $C^{1,1}$. By using similar techniques as Bénilan and Gariepy [26],
they obtained existence and uniqueness of positive strong solutions under the
assumption that the forcing term $g\equiv 0$ and the lower-order perturbation
$f(\cdot,u)$ satisfies a specific growth condition related to $u^{q-1}$,
$1<q\leq p<\infty$. Our results presented here complement and improve those in
[25] in various directions as specified below.

###  1.2 Main results

We begin by stating our well-posedness result in the sense of _mild solutions_
in $L^{1}$ and a comparison principle. We denote the intersection space
$L^{1}\cap L^{\infty}$ by $L^{1\cap\infty}$ and restrict the fractional
p-Laplacian to $L^{1\cap\infty}$ with $(-\Delta_{p})^{s}_{|L^{1\cap\infty}}$
(see Section 2.2.4 and, in particular, Definition 2.13). The composition
operator $(-\Delta_{p})^{s}_{|L^{1\cap\infty}}\circ\varphi$ is defined in
$L^{1}$ (see Definition 2.15). Here we also use the notation $[u]^{+}$ to
denote $\max\\{u,0\\}$, the positive part of $u$, and $[u]^{1}=u$. We also use
the notation of $q$-brackets defined by Definition 2.2.

######  Theorem 1.1 (Well-posedness & comparison principle in $L^{1}$).

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Let
$1<p<\infty$, $0<s<1$, and suppose that $\varphi$, $f$ satisfy (1.2) and
(1.3a)-(1.3b), respectively. Then the following statements hold.

  1. 1.

If either $\varphi\in W^{1,\infty}_{loc}({\mathbb{R}})$ or $\varphi\in
W^{1,q}_{\operatorname{loc}}({\mathbb{R}})$ for $q>\frac{1}{1-s}$, then one
has that

$\overline{D((-\Delta_{p})^{s}_{|L^{1\cap\infty}}\varphi)}^{\mbox{}_{L^{1}}}=L^{1}.$

  2. 2.

For every initial value
$u_{0}\in\overline{D((-\Delta_{p})^{s}_{|L^{1\cap\infty}}\varphi)}^{\mbox{}_{L^{1}}}$
and $g\in L^{1}(0,T;L^{1})$, there is a unique mild solution $u\in
C([0,T];L^{1})$ of the initial boundary value problem (1.1). Moreover, for all
$1\leq q\leq\infty$, one has that

$\lVert u(t)\rVert_{q}\leq e^{\omega t}\lVert
u_{0}\rVert_{q}+\int_{0}^{t}e^{\omega(t-r)}\lVert
g(r)\rVert_{q}\mathop{}\\!\mathrm{d}r$ (1.13)

for every
$u_{0}\in\overline{D((-\Delta_{p})^{s}_{|L^{1\cap\infty}}\circ\varphi)}^{\mbox{}_{L^{1}}}\cap
L^{q}$ and $g\in L^{1}(0,T;L^{1}\cap L^{q})$.

  3. 3.

For every $y_{1}$, $y_{2}\in L^{1}$, $g_{1}$, $g_{2}\in L^{1}(0,T;L^{1})$, and
corresponding mild solutions $u_{1}$, $u_{2}$ of (1.1) with initial data
$y_{1}$, $y_{2}$ respectively, one has

$\begin{split}&\lVert[u_{1}(t)-u_{2}(t)]^{\nu}\rVert_{1}\leq e^{\omega
t}\lVert[y_{1}-y_{2}]^{\nu}\rVert_{1}\\\ &\hskip
99.58464pt+\int_{0}^{t}e^{\omega(t-r)}[u_{1}(r)-u_{2}(r),g_{1}(r)-g_{2}(r)]_{\nu}\mathop{}\\!\mathrm{d}r\end{split}$
(1.14)

for all $0\leq t\leq T$, and $\nu\in\\{+,1\\}$.

The statements of Theorem 1.1 follow as an application of the existence theory
developed by [26], in the monograph [14] by Coulhon and the second author, and
by classical nonlinear semigroup theory (cf. [21]). We give the details of the
proof in Section 3. Our well-posedness result presented here complements and
generalizes the recent result by Giacomoni et al. [25] by allowing general,
strictly increasing functions $\varphi$ and by considering solutions which may
change sign. In contrast to [25], Theorem 1.1 provides well-posedness for all
initial data $u_{0}$ in $L^{1}$.

Our next result is concerned with global $L^{\ell}-L^{\infty}$ regularity
estimates, $1\leq\ell<\infty$, for mild solutions $u$ of the initial boundary
value problem (1.1), implying an immediate smoothing effect. For these
regularity estimates, the Sobolev embedding (see, for instance, [27, 28])

$W^{s,p}\hookrightarrow L^{p_{s}}\quad\text{with}\quad
p_{s}=\begin{cases}\left(\frac{1}{p}-\frac{s}{d}\right)^{-1}&\text{ if
}p<\frac{d}{s},\\\ \tilde{p}&\text{ if }p=\frac{d}{s},\\\ \infty&\text{ if
}p>\frac{d}{s},\end{cases}$ (1.15)

and $\tilde{p}\in[p,\infty)$, is crucial, where we write $W^{s,p}$ for
$W^{s,p}(\Omega)$. Theorem 1.2 is a special case of Theorem 4.1 in Section 4
as illustrated in Section 5. Theorem 4.1 applies to abstract operators $A$
acting on $L^{q}$ and satisfying an abstract Sobolev inequality (as introduced
in [14]). In particular we apply a De Georgi iteration (cf. [29]) to obtain an
$L^{m+1}-L^{\infty}$ estimate which is then extrapolated to
$L^{\ell}-L^{\infty}$. Here we refine the methods in [29] by Caffarelli and
Vasseur, [30] by Porzio and [14] by Coulhon and the second author.

######  Theorem 1.2 (Global $L^{\ell}-L^{\infty}$ estimates).

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Let $p>1$,
$0<s<1$, and $m\geq 1$ such that

$m(p-1)+(m+1)\frac{sp}{d}>1.$ (1.16)

Further, let $\varphi(r)=r^{m}$ for $r\in{\mathbb{R}}$, and $f(\cdot,u)$
satisfy (1.3a)-(1.3b). Take $q_{s}=p_{s}$ if $p\neq\frac{d}{s}$ and
$q_{s}>\max(p,1+\frac{1}{m})$ if $p=\frac{d}{s}$. For $\rho\geq m+1$ and
$\psi>1$ satisfying

$\begin{cases}\frac{1}{\rho}<\left(1-\frac{1}{\psi}\right)p\left(\frac{1}{p}-\frac{1}{q_{s}}\right)&\text{if
}m(p-1)\geq 1,\\\
\frac{1}{\rho}\leq\left(1-\frac{1}{\psi}\right)p\left(\frac{m}{m+1}-\frac{1}{q_{s}}\right)&\text{if
}m(p-1)<1,\end{cases}$ (1.17)

and

$\rho\geq\frac{1-m(p-1)}{1-\frac{p}{q_{s}}},$ (1.18)

let $g\in L^{\psi}(0,T;L^{\rho})\cap L^{1}(0,T;L^{1}\cap L^{1+m+\varepsilon})$
for some $\varepsilon>0$. Let $1\leq\ell<m+1$ satisfy

$\ell>\frac{1-m(p-1)}{1-\frac{p}{q_{s}}}.$ (1.19)

Then for every $u_{0}\in L^{\ell}\cap L^{1}$ the mild solution $u$ of (1.1) in
$L^{1}$ satisfies

$\begin{split}\lVert u(t)\rVert_{\infty}&\leq
C\max\left(e^{\omega\beta_{1}t}\left(\tfrac{1}{t}+\omega\right)^{\alpha},\,e^{\omega\beta_{2}t}\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho})}^{\eta}\right)^{\frac{1}{\theta}}\left(1+N(t)^{\gamma}\right)\times\\\
&\hskip 85.35826pt\times\left(e^{\omega t}\lVert
u_{0}\rVert_{\ell}+\int_{0}^{t}e^{\omega(t-\tau)}\lVert
g(\tau)\rVert_{\ell}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{\ell\gamma}{(m+1)\theta}}\end{split}$
(1.20)

for all $t\in(0,T]$, where we set

$\begin{split}N(t)&=\sup\limits_{s\in(0,t]}\tfrac{M(\frac{s}{2})\lVert
g\rVert_{L^{1}(\frac{s}{2},s;L^{m+1})}+e^{\frac{\omega\beta_{2}s}{2\gamma}}\,\lVert
g\rVert_{L^{\psi}(0,\frac{s}{2};L^{\rho})}^{\frac{\eta}{\gamma}}}{M(s)^{\frac{1}{\theta}}\left(e^{\omega
s}\lVert u_{0}\rVert_{\ell}+\int_{0}^{s}e^{\omega(s-\tau)}\lVert
g(\tau)\rVert_{\ell}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{\ell}{(m+1)\theta}}},\\\
M(t)&=\max\left(e^{\omega\beta_{1}t}\left(\tfrac{1}{t}+\omega\right)^{\alpha},e^{\omega\beta_{2}t}\,\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho})}^{\eta}\right)^{\frac{1}{\gamma}},\end{split}$
(1.21)

for the constants given by

$\begin{split}\alpha&=\frac{1}{(m+1)p\left(\frac{m}{m+1}-\frac{1}{q_{s}}\right)},\quad\gamma=\frac{\frac{1}{p}-\frac{1}{q_{s}}}{\frac{m}{m+1}-\frac{1}{q_{s}}},\\\
\theta&=1-\gamma\left(1-\frac{\ell}{m+1}\right),\\\
\eta&=\frac{1}{1-\frac{m+1}{\rho}+mp\left(1-\frac{1}{\psi}\right)\left(1-\frac{m+1}{mq_{s}}\right)},\\\
\beta_{1}&=\begin{cases}\frac{\frac{1}{mp}-\frac{1}{m+1}}{\frac{1}{m+1}-\frac{1}{mq_{s}}}&\text{if
}m(p-1)<1,\\\ 0&\text{if }m(p-1)\geq 1,\end{cases}\\\
\beta_{2}&=\begin{cases}\eta(1-m(p-1))\left(1-\frac{1}{\psi}\right)&\text{if
}m(p-1)<1,\\\ 0&\text{if }m(p-1)\geq 1,\end{cases}\end{split}$ (1.22)

and where $C>0$ depends on $m$, $p$, $s$, $d$, $q_{s}$, $\ell$, $\rho$ and
$\psi$.

In the case of no forcing term, $g\equiv 0$, this simplifies to the following
$L^{\ell}-L^{\infty}$ estimate.

######  Corollary 1.3.

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Let $p>1$,
$0<s<1$, and $m\geq 1$ such that (1.16) holds. Let $q_{s}=p_{s}$ if
$p\neq\frac{d}{s}$ and $q_{s}>\max(p,1+\frac{1}{m})$ if $p=\frac{d}{s}$. Let
$\varphi(r)=r^{m}$ for $r\in{\mathbb{R}}$. Further, suppose that $f(\cdot,u)$
satisfies (1.3a)-(1.3b) and $g\equiv 0$. Let $1\leq\ell<m+1$ and suppose
$\ell$ satisfies (1.19). Then for every $u_{0}\in L^{\ell}\cap L^{1}$ the mild
solution $u$ of (1.1) in $L^{1}$ satisfies

$\begin{split}\lVert u(t)\rVert_{\infty}&\leq Ce^{\omega\beta
t}t^{-\alpha}\lVert u_{0}\rVert_{\ell}^{\gamma}\end{split}$ (1.23)

for all $t\in(0,T]$, where

$\begin{split}\alpha&=\frac{1}{m(p-1)-1+\ell(1-\frac{p}{q_{s}})},\quad\gamma=\frac{\ell(1-\frac{p}{q_{s}})}{m(p-1)-1+\ell(1-\frac{p}{q_{s}})},\\\
\beta&=\begin{cases}\frac{\frac{1}{p}-\frac{m}{m+1}}{\frac{m}{m+1}-\frac{1}{p}+\frac{\ell}{m+1}\left(\frac{1}{p}-\frac{1}{q_{s}}\right)}&\text{if
}m(p-1)<1,\\\ 0&\text{if }m(p-1)\geq 1,\end{cases}\end{split}$

and $C>0$ depends on $m$, $p$, $s$, $d$, $q_{s}$ and $\ell$.

Such regularity and decay estimates are common for these diffusion problems,
see for example [30] and the references therein. In particular, with $f\equiv
0$, $g\equiv 0$ we have Corollary 1.3 with $\omega=0$ and hence a decay in
time. Similar estimates have been found for related problems, including for a
class of local doubly nonlinear problems [30] related to a doubly nonlinear
$p$-Laplacian evolution equation. In [29], an $L^{1}-L^{\infty}$ estimate is
found corresponding to the fractional Laplacian with $s=\frac{1}{2}$ and in
[17] for the fractional porous medium equation with $s=\frac{1}{2}$. In the
case of the fractional porous medium equation on ${\mathbb{R}}^{d}$ ($d\geq
1$), the same authors in [18] find such an $L^{\ell}-L^{\infty}$ regularizing
effect for all $\ell\geq 1$. We can also compare this to the Barenblatt
solutions found for related evolution problems in [11, 12, 31], noting the
convergence in $L^{1}({\mathbb{R}}^{d})$ as $t\rightarrow\infty$. We emphasise
that in all cases the exponents given by (1.22) in Corollary 1.3 agree with
those found in these papers; in the case of [30], by taking $s=1$. We note
that compared to [25], we restrict to $m\geq 1$ rather than $0<m<1$ due to the
limitation of Lemma 5.1. Such regularizing effects require separate
consideration in this case.

Now, for $1<p<\infty$ and $0<s<1$, set

$\tilde{D}((-\Delta_{p})^{s}_{|_{L^{1\cap\infty}}})=\left\\{u\in
L^{1\cap\infty}\left|\begin{aligned} &\exists\,(u_{n},h_{n})_{n\geq
1}\subseteq(-\Delta_{p})^{s}_{|_{L^{1\cap\infty}}}\text{s.t.}\\\
&u_{n}\rightarrow u\text{ in }L^{1}\text{ and }\\\ &(u_{n},h_{n})_{n\geq
1}\text{ is bounded in }L^{\infty}\times L^{1}.\end{aligned}\right.\right\\}$

and for every continuous $\varphi$, let

$\hat{D}((-\Delta_{p})^{s}_{|L^{1\cap\infty}}\varphi)=\Big{\\{}u\in
L^{1}\,\Big{|}\,\varphi(u)\in\tilde{D}((-\Delta_{p})^{s}_{|_{L^{1\cap\infty}}})\Big{\\}}.$

Then, by taking advantage of [26, Theorem 4.1], we show in our next theorem
that for every $u_{0}\in\hat{D}((-\Delta_{p})^{s}_{|L^{1\cap\infty}}\varphi)$,
the corresponding mild solutions $u$ of the initial boundary value problem
(1.1) is strong and distributional. In this theorem, the term
$[\,\cdot\,]_{s,p}$ denotes the Gagliardo semi-norm (2.1). We prove this
result in Section 6.

######  Theorem 1.4 (Mild solutions are strong and distributional).

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$, of finite
Lebesgue measure. Let $1<p<\infty$, $0<s<1$, $\varphi\in C(\mathbb{R})$ be a
strictly increasing function such that $\varphi^{-1}\in
AC_{\operatorname{loc}}(\mathbb{R})$. Suppose $f(\cdot,u)$ satisfies
(1.3a)-(1.3b) and let $F$ be the Nemystkii operator of $f$. Further suppose
that $g\in BV(0,T;L^{1})\cap L^{1}(0,T;L^{\infty})$. Then for every
$u_{0}\in\hat{D}((-\Delta_{p})^{s}_{|L^{1\cap\infty}}\varphi)$, the mild
solution $u$ of (1.1) is a strong distributional solution in $L^{1}$ having
the regularity

$u\in W^{1,\infty}((0,T);L^{1})\cap L^{\infty}([0,T];L^{\infty})\cap
C([0,T];L^{q})$

for every $1\leq q<\infty$. Moreover, $\varphi(u)$ has a weak derivative given
by

$\frac{\mathrm{d}}{\mathrm{d}t}\varphi(u(t))=\varphi^{\prime}(u(t))\frac{\mathrm{d}u}{\mathrm{d}t}\qquad\text{
in }L^{2}\text{ for a.e.~{}}t>0$ (1.24)

and the function $t\mapsto\left[\varphi(u(t))\right]^{p}_{s,p}$ has derivative
given by

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}t}\left[\varphi(u(t))\right]^{p}_{s,p}&=-\lVert\sqrt{\varphi^{\prime}(u(t))}\frac{\mathrm{d}u}{\mathrm{d}t}(t)\rVert_{2}^{2}\\\
&\qquad-\braket{F(u(t))-g(t),\frac{\mathrm{d}u}{\mathrm{d}t}(t)\varphi^{\prime}(u(t))}\end{split}$
(1.25)

for a.e. $t>0$.

Strong solutions are naturally a standard aim of results in such a theory and
so have been investigated for many related problems. For the fractional
$p$-Laplacian evolution equation, these have been found in [7] and for the
fractional porous medium equation on ${\mathbb{R}}^{d}$, in [18]. Comparing
this with [25], we note that a different regime is studied with
$\frac{1}{2p-1}\leq m<1$ compared to $m>0$ for non-negative solutions.

In the case $\varphi(u)=u^{m}$, the operator is homogeneous and we have the
following Lipschitz estimate via the regularising effect of homogeneous
operators in [32]. Here we define, for $g\in
L^{1}_{\operatorname{loc}}(0,T;L^{1})$ and $0\leq t\leq T\leq\infty$,

$V(t,g)=\limsup_{\xi\rightarrow 0^{+}}\int_{0}^{t/(1+\xi)}\frac{\lVert
g(\tau(1+\xi))-g(\tau)\rVert_{1}}{\xi}\mathop{}\\!\mathrm{d}\tau.$ (1.26)

Note that $V(T,g)<\infty$ is equivalent to $t\rightarrow tg(t)$ having
(essentially) finite variation on $[0,T]$.

######  Theorem 1.5 (Derivative estimates for $\varphi(u)=u^{m}$).

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Let $p>1$,
$0<s<1$ and $f(\cdot,u)$ satisfy (1.3a)-(1.3b). Suppose $\varphi(r)=r^{m}$,
$r\in{\mathbb{R}}$ for $m>0$. Then we have the following regularity estimates.

  1. 1.

Suppose $m(p-1)\neq 1$, $g\in L^{1}(0,T;L^{1})$ and $V(T,g)<\infty$. Then for
every $u_{0}\in L^{1}$, the unique mild solution $u$ to (1.1) is Lipschitz
continuous on each compact subset of $(0,T]$ satisfying

$\begin{split}\limsup_{h\rightarrow 0^{+}}\frac{\lVert
u(t+h)-u(t)\rVert_{1}}{h}&\leq\frac{Ce^{2\omega t}}{t}\left(\lVert
u_{0}\rVert_{1}+\int_{0}^{t}\lVert
g(\tau)\rVert_{1}\mathop{}\\!\mathrm{d}\tau\right)\\\ &\hskip
85.35826pt+\frac{e^{\omega t}}{t}V(t,g)\end{split}$ (1.27)

where $C=\frac{m(p-1)+2}{|m(p-1)-1|}$.

  2. 2.

Let $g\in BV(0,T;L^{1})\cap L^{1}(0,T;L^{\infty})$. Further, suppose that
$m\geq 1$ satisfies

$m(p-1)+(m+1)\frac{sp}{d}>1.$ (1.16)

Take $q_{s}=p_{s}$ if $p\neq\frac{d}{s}$ and $q_{s}>\max(p,1+\frac{1}{m})$ if
$p=\frac{d}{s}$, and suppose that

$m\left(p-1\right)>\frac{p}{q_{s}}.$ (1.28)

If, in addition, for $\rho\geq m+1$ and $\psi>1$ satisfying

$\rho\geq\frac{m+1-mp}{1-\frac{p}{q_{s}}}\quad\text{ and
}\quad\frac{1}{\rho}<\left(1-\frac{1}{\psi}\right)\left(1-\frac{p}{q_{s}}\right),$

$g$ belongs to $L^{\psi}(0,T;L^{\rho})\cap L^{m+1}(0,T;L^{\infty})$, then for
every $u_{0}\in L^{1}$, the mild solution $u$ to (1.1) is a strong solution in
$L^{1}$ and satisfies

$\begin{split}\int_{0}^{t}s^{\tilde{\alpha}}&\int_{\Omega}u^{m-1}(s)\left|\frac{\mathrm{d}u}{\mathrm{d}s}\right|^{2}\mathop{}\\!\mathrm{d}\mu\mathop{}\\!\mathrm{d}s+t^{\tilde{\alpha}}\mathcal{E}(\varphi(u(t)))\\\
&\leq
Ct\left(1+t^{2}\right)\max\left(e^{\omega\beta_{1}t},t^{\alpha}e^{\omega\beta_{2}t}\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho})}^{\eta}\right)^{\frac{m}{\theta}}\times\\\
&\hskip 28.45274pt\left(e^{\omega t}\lVert
u_{0}\rVert_{1}+\int_{0}^{t}e^{\omega(t-\tau)}\lVert
g(\tau)\rVert_{1}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{\delta
m}{\theta(m+1)}+1}\left(1+F(t)^{\gamma}\right)\\\ &\hskip
85.35826pt+\int_{0}^{t}\left(\tilde{\alpha}+ms\right)s^{\tilde{\alpha}-1}\lVert
g(s)\rVert_{m+1}^{m+1}\mathop{}\\!\mathrm{d}s\end{split}$ (1.29)

for all $t\in(0,T]$ with $\tilde{\alpha}=\frac{\alpha m}{\theta}+2$, $F(s)$
given by (1.21), constants given by (1.22) with $\ell=1$ and where $C>0$
depends on $m$, $p$, $s$, $d$, $q_{s}$, $\psi$, $\rho$ and $\omega$.

In the case of no forcing term, $g\equiv 0$ and taking $\ell=1$, this
simplifies to the following $L^{1}-L^{\infty}$ estimate.

######  Corollary 1.6.

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Let $p>1$,
$0<s<1$, $f(\cdot,u)$ satisfy (1.3a)-(1.3b) and $g\equiv 0$. Suppose
$\varphi(r)=r^{m}$, $r\in{\mathbb{R}}$ for $m\geq 1$ satisfying

$m(p-1)+(m+1)\frac{sp}{d}>1.$ (1.16)

Take $q_{s}=p_{s}$ if $p\neq\frac{d}{s}$ and $q_{s}>\max(p,1+\frac{1}{m})$ if
$p=\frac{d}{s}$, and suppose that

$m\left(p-1\right)>\frac{p}{q_{s}}.$ (1.28)

Then for every $u_{0}\in L^{1}$, the mild solution $u$ to (1.1) satisfies

$\begin{split}\int_{0}^{t}s^{\tilde{\alpha}}\int_{\Omega}u^{m-1}(s)&\left|\frac{\mathrm{d}u}{\mathrm{d}s}\right|^{2}\mathop{}\\!\mathrm{d}\mu\mathop{}\\!\mathrm{d}s+t^{\tilde{\alpha}}\mathcal{E}(\varphi(u(t)))\\\
&\leq Ct\left(1+t^{2}\right)e^{\omega\beta t}\lVert
u_{0}\rVert_{1}^{\frac{\gamma m}{\theta(m+1)}+1}\end{split}$ (1.30)

for all $t\in(0,T]$ with

$\begin{split}\tilde{\alpha}&=\frac{\alpha m}{\theta}+2,\\\
\beta&=\frac{m\beta_{1}}{\theta}+\frac{\gamma m}{\theta(m+1)}+1,\end{split}$

constants given by (1.22) with $\ell=1$ and where $C>0$ depends on $m$, $p$,
$s$, $d$, $q_{s}$ and $\omega$.

In this case we also have local Hölder regularity. Here $V(T,g)$ is defined as
in (1.26).

######  Theorem 1.7 (Local Hölder continuity).

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Assume
$2\leq p<\infty$, $0<s<1$ such that $sp\geq d$ and $\varphi(u)=u^{m}$, $m>0$.
Suppose $f$ satisfies (1.3a)-(1.3b) with $F$ the Nemytskii operator of $f$,
$g\in L^{1}(0,T;L^{1})$ and $V(T,g)<\infty$. Let $u(t)$ be the mild solution
to (1.1) for $u_{0}\in L^{1}$.

Then $u^{m}(t)\in C_{\operatorname{loc}}^{\delta}(\Omega)$ for every
$0<\delta<\min(\frac{sp-d}{p-1},1)$ and a.e. $t\in(0,T)$. In particular, for
$m\geq 1$, $u(t)\in C_{\operatorname{loc}}^{\delta}(\Omega)$ for every
$0<\delta<\min(\frac{sp-d}{(p-1)m},1)$ and a.e. $t\in(0,T)$.

Local Hölder regularity has been established for the fractional porous medium
equation in [24] for $\frac{n-2s}{n+2s}<m<1$ via an oscillation lemma and in
[18] for the fractional porous medium equation on ${\mathbb{R}}^{d}$ for
$m\geq 1$. This result applies the elliptic local regularity proved in [33]
and extends the work in [34], which considers Hölder regularity in space and
time for a weak formulation of the fractional $p$-Laplacian evolution problem.

Furthermore, for $\varphi$ given by the identity, we have continuity in time
and global Hölder continuity in space for a bounded domain. Both Hölder
regularity results are proved in Section 7.

######  Theorem 1.8 (Global Hölder regularity for $\varphi(r)=r$).

Let $\Omega$ be a bounded
domain in ${\mathbb{R}}^{d}$, $d\geq 2$, with a boundary $\partial\Omega$ of
the class $C^{1,1}$, $1<p<\infty$, $0<s<1-\frac{1}{p}$, and suppose $f$
satisfies (1.3a)-(1.3b) with $F$ the Nemytskii operator of $f$ on
$C_{0}(\Omega)$. Then $-((-\Delta_{p})^{s}_{|_{C_{0}}}+F)$ generates a
strongly continuous semigroup of quasi contractions on $C_{0}(\Omega)$. In
particular, for $\varphi(r)=r$, $r\in{\mathbb{R}}$, $u_{0}\in C_{0}(\Omega)$
and $g\in L^{\infty}(0,T;L^{\infty})\cap BV(0,T;L^{\infty})$, let $u$ be the
unique mild solution of the initial boundary value problem (1.1). Then $u\in
C^{lip}([\delta,T];C_{0}(\Omega))$ for every $0<\delta<T$ and for some
$\alpha\in(0,s]$, $u(t)\in C^{\alpha}(\overline{\Omega})$ for all $t\in(0,T)$.

This result extends [9], wherein Giacomoni and Tiwari obtain continuity up to
the boundary in space, uniform in time. Here we refine the estimate, proving
that the resolvent is $m$-accretive in $C_{0}$. We use the global Hölder
regularity of the elliptic problem given by [35] with semigroup theory for the
operator restricted to the space of continuous functions. Hölder regularity
has been considered for the elliptic problem, for example, in [33].

By obtaining a comparison theorem for solutions with inhomogeneous boundary
data and considering the fundamental solution we can prove finite time of
extinction of solutions to (1.1). This is proved in Section 8.

######  Theorem 1.9 (Finite time of extinction).

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Let $u$ be
a strong distributional solution to (1.1) with
$u_{0}\in\tilde{D}((-\Delta_{p})^{s})$ where $\varphi$ is strictly increasing,
$\varphi(\mathbb{R})=\mathbb{R}$, $\varphi(0)=0$ and $\frac{1}{\varphi}\in
L^{1}(0,\lVert u_{0}\rVert_{\infty})$, $f\equiv 0$ and $g\equiv 0$. Then
$u(t,\cdot)=0$ for all $t\geq T^{*}$ where $T^{*}$ is given by

$T^{*}=\frac{1}{\tilde{C}R^{d-ps}}\int_{0}^{\lVert
u_{0}\rVert_{\infty}}\frac{1}{\varphi(s)}\mathop{}\\!\mathrm{d}s$ (1.31)

where $\tilde{C}$ is given by (8.6) and depends on $R$, $p$, $s$ and $d$.

We have the following Corollary in the case $\varphi(r)=r^{m}$, $r\in R$.

######  Corollary 1.10.

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Let $u$ be
a strong distributional solution to (1.1) with $u_{0}\in L^{\infty}$ where
$\varphi(r)=r^{m}$, $0<m<1$, $f\equiv 0$ and $g\equiv 0$. Then $u(t,\cdot)=0$
for all $t\geq T^{*}$ where $T^{*}$ is given by (1.31).

Finite time of extinction of solutions to (1.1) was also proved for the
fractional porous medium equation ($p=2$) in [24] in the Dirichlet case and
[18] for the Cauchy problem. See also [12, 13] for discussion of extinction
for the fractional $p$-Laplacian evolution problem on ${\mathbb{R}}^{d}$.

##  2 Preliminaries

In this section, we introduce some basic definitions and recall known results
used throughout this paper.

###  2.1 A brief primer on Gagliardo-Sobolev-Slobodeckiĭ spaces

In this section, we provide a short summary of Gagliardo-Sobolev-Slobodeckiĭ
spaces, which are necessary to study the intial boundary value problem (1.1)
with functional analytical tools. For a deeper understanding of this theory,
we refer the interested reader to [36, 27] or [37].

For $1<p<\infty$, $0<s<1$, and an open subset $\Omega$ of ${\mathbb{R}}^{d}$,
we write $W^{s,p}$ for the _Gagliardo-Sobolev-Slobodeckiĭ space_ of fractional
order $s$, also known as the _fractional Sobolev space_ $W^{s,p}(\Omega)$
given by

$W^{s,p}(\Omega)=\left\\{u\in L^{p}\Big{|}[u]_{s,p}<\infty\right\\}$

where

$[u]_{s,p}:=\left(\int_{\Omega}\int_{\Omega}\frac{\lvert
u(x)-u(y)\rvert^{p}}{\lvert
x-y\rvert^{d+sp}}\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x\right)^{1/p}$
(2.1)

denotes the _$s$ -Gagliardo semi-norm_. The space $W^{s,p}$ defines a Banach
space if it is equipped with the norm

$\lVert u\rVert_{W^{s,p}}:=\left(\lVert
u\rVert^{p}_{L^{p}}+[u]^{p}_{s,p}\right)^{1/p}.$

Further, let $W^{s,p}_{0}(\Omega)$, denoted by $W^{s,p}_{0}$, be the closure
in $W^{s,p}$ of the set $C^{\infty}_{c}(\Omega)$ of test functions. By [36,
Theorem 10.1.1], the space $W^{s,p}_{0}$ admits, for $1<p<\infty$, the
characterization

$W^{s,p}_{0}=\left\\{u\in
W^{s,p}({\mathbb{R}}^{d})\,\bigg{|}\begin{array}[c]{l}\exists\text{
}\overline{u}:{\mathbb{R}}^{d}\to{\mathbb{R}}\text{ s.t. }\overline{u}=u\text{
a.e.~{}on ${\mathbb{R}}^{d}$}\\\ \text{ and }\overline{u}=0\text{ quasi-
everywhere on }{\mathbb{R}}^{d}\setminus\Omega\end{array}\right\\},$

where $\overline{u}$ denotes a (quasi-continuous) representative of $u$.
Therefore, the space $W^{s,p}_{0}$ incorporates _homogeneous Dirichlet
boundary conditions_ in a weak sense.

###  2.2 Basics of nonlinear semigroup theory

We begin by reviewing some basic definitions and important results in
nonlinear semigroup theory from the standard literature [20, 22] and the
monograph [14].

####  2.2.1 The general framework

Let $(\Sigma,\mu)$ be a measure space with a positive $\sigma$-finite measure
$\mu$, and $M(\Sigma,\mu)$ be the set of $\mu$-a.e. equivalence classes of
measurable functions $u:\Sigma\to{\mathbb{R}}$. For $1\leq q\leq\infty$, we
denote by $L^{q}_{\mu}$ the classical _Lebesgue space_ equipped with the
standard $L^{q}$-norm

$\lVert u\rVert_{q}:=\begin{cases}\bigg{(}\displaystyle\int_{\Sigma}\lvert
u\rvert^{q}\mathop{}\\!\mathrm{d}\mu\bigg{)}^{1/q}&\quad\text{if
$q<\infty$,}\\\\[7.0pt] \inf\Big{\\{}k\in[0,\infty]\,\Big{|}\,\mu(\\{\lvert
u\rvert>k\\})=0\Big{\\}}&\quad\text{if $q=\infty$.}\end{cases}$

If $\Omega$ is an open subset of ${\mathbb{R}}^{d}$, $d\geq 1$, and $\mu$ is
the classical $d$-dimensional Lebesgue measure restricted to the
trace-$\sigma$-algebra $\mathcal{B}({\mathbb{R}}^{d})\cap\Omega$ then we write
$L^{q}$ instead of $L^{q}_{\mu}$.

Let $X\subseteq M(\Sigma,\mu)$ be a Banach space with norm
$\lVert\cdot\rVert_{X}$. The main object in this section is the abstract
Cauchy problem (in $X$)

$\begin{cases}\tfrac{\mathrm{d}u}{\mathop{}\\!\mathrm{d}t}(t)+Au(t)\ni
g(t)&\quad\text{for a.e.~{}$t\in(0,T)$,}\\\ u(0)=u_{0},&\end{cases}$ (2.2)

for given initial value $u_{0}\in\overline{D(A)}^{\mbox{}_{X}}$ and forcing
term $g\in L^{1}(0,T;X)$. In (2.2), $A$ denotes a (possibly) multi-valued
operator $A:D(A)\to 2^{X}$ on $X$ with _effective domain_ $D(A):=\\{u\in
X\,|\,Au\neq\emptyset\\}$, the closure of $D(A)$ in $X$ denoted by
$\overline{D(A)}^{\mbox{}_{X}}$, and _range_
$\operatorname{Rg}(A):=\bigcup_{u\in D(A)}Au$. In this setting, it is standard
to view an operator $A$ as a subset of $X\times X$, or _relation_ on $X$, and
to identify $A$ with its graph

$A:=\Big{\\{}(u,v)\in X\times X\,\Big{|}\;v\in Au\Big{\\}}.$

######  Definition 2.1.

An operator $A$ on $X$ is called _accretive_ (in $X$) if

$\lVert u-\hat{u}\rVert_{X}\leq\lVert u-\hat{u}+\lambda(v-\hat{v})\rVert_{X}$
(2.3)

for every $(u,v)$, $(\hat{u},\hat{v})\in A$ and every $\lambda>0$. Further, an
operator $A$ on $X$ is called _quasi accretive_ if there is an
$\omega\in{\mathbb{R}}$ such that $A+\omega I$ is accretive in $X$.

Clearly, if $A+\omega I$ is accretive in $X$ for some $\omega\in{\mathbb{R}}$
then $A+\tilde{\omega}I$ is accretive for every $\tilde{\omega}\geq\omega$.
Thus, there is no loss of generality in assuming that if $A$ is quasi
accretive then there is an $\omega\geq 0$ such that $A+\omega I$ is accretive
in $X$.

Equivalently, $A$ is accretive in $X$ if and only if, for every $\lambda>0$,
the _resolvent operator_ of $A$, defined by $J_{\lambda}:=(I+\lambda A)^{-1}$,
is a single-valued mapping from $\operatorname{Rg}(I+\lambda A)$ to $D(A)$
which is _contractive_ (also called _non-expansive_) with respect to the norm
of $X$. That is,

$\lVert J_{\lambda}u-J_{\lambda}\hat{u}\rVert_{X}\leq\lVert
u-\hat{u}\rVert_{X}$

for all $u$, $\hat{u}\in\operatorname{Rg}(I+\lambda A)$ and $\lambda>0$.

The next definition is taken from [14].

######  Definition 2.2.

For $1\leq q<\infty$, we define the _$q$ -bracket_ on $L^{q}_{\mu}$ to be the
mapping $[\cdot,\cdot]_{q}:L^{q}_{\mu}\times L^{q}_{\mu}\to{\mathbb{R}}$
defined by

$[u,v]_{q}:=\displaystyle{\lim_{\lambda\to 0+}\frac{\frac{1}{q}\lVert
u+\lambda v\rVert_{q}^{q}-\frac{1}{q}\lVert u\rVert_{q}^{q}}{\lambda}}$

for $u$, $v\in L^{q}_{\mu}$. We further define the bracket

$[u,v]_{+}:=\displaystyle{\lim_{\lambda\to 0+}\frac{\lVert[u+\lambda
v]^{+}\rVert_{1}-\lVert[u]^{+}\rVert_{1}}{\lambda}}$

for $u$, $v\in L^{1}_{\mu}$.

Then an operator $A$ on $L^{q}_{\mu}$ is accretive in $L^{q}_{\mu}$ if and
only if

$[u-\hat{u},v-\hat{v}]_{q}\geq 0\qquad\text{for all $(u,v)$,
$(\hat{u},\hat{v})\in A$.}$

The $q$-bracket $[\cdot,\cdot]_{q}$ is upper semicontinuous (respectively,
continuous if $1<q<\infty$) and if $1<q<\infty$,

$[u,v]_{q}=\int_{\Sigma}\lvert u\rvert^{q-2}u\,v\mathop{}\\!\mathrm{d}\mu$
(2.4)

for every $u$, $v\in L^{q}_{\mu}$. While for $q=1$, $[\cdot,\cdot]_{1}$
reduces to the classical _brackets_ $[\cdot,\cdot]$ on $L^{1}_{\mu}$ given by

$[u,v]_{1}=\int_{\\{u\neq
0\\}}\text{sign}_{0}(u)\,v\mathop{}\\!\mathrm{d}\mu+\int_{\\{u=0\\}}\lvert
v\rvert\mathop{}\\!\mathrm{d}\mu$ (2.5)

for $u$, $v\in L^{1}_{\mu}$, where the _restricted signum_
$\operatorname*{sign}_{0}$ is defined by

$\text{sign}_{0}(s)=\begin{cases}1&\text{if $s>0$,}\\\ 0&\text{if $s=0$,}\\\
-1&\text{if $s<0$,}\end{cases}$

for $s\in{\mathbb{R}}$ (cf. [21, Section 2.2 & Example (2.8)] or [22, pp
102]).

Next, we introduce the following class of operators.

######  Definition 2.3.

An operator $A$ on $X$ is called _$m$ -accretive in $X$_ if $A$ is accretive
in $X$ and satisfies the so-called _range condition_

$\operatorname{Rg}(I+\lambda A)=X\qquad\text{for some (or equivalently all)
$\lambda>0$,}$ (2.6)

and an operator $A$ on $X$ is called _quasi $m$-accretive in $X$_ if there is
an $\omega\geq 0$ such that $A+\omega I$ is $m$-accretive in $X$.

By the classical theory of nonlinear evolution problems (cf. [21], or
alternatively, [22, Corollary 4.1]), the condition _‘ $A$ is quasi
$m$-accretive in $X$’_ ensures that for given
$u_{0}\in\overline{D(A)}^{\mbox{}_{X}}$ and $g\in L^{1}(0,T;X)$, the Cauchy
problem (2.2) admits a unique _mild solution_ , which is continuously
dependent on $u_{0}$ and $g$.

######  Definition 2.4.

Suppose $g\in L^{1}(0,T;X)$ for some $T>0$. A _mild solution_ $u$ in $X$ of
Cauchy problem (2.2) is a function $u\in C([0,T];X)$ such that for every
$\varepsilon>0$ there is a partition $\sigma:0=t_{0}<\cdots<t_{N}\leq T$ of
the interval $[0,t_{N}]$ and a finite sequence $(g_{i})_{i=q}^{N}$ with the
following properties: $T-t_{N}<\varepsilon$, $t_{i}-t_{i-1}<\varepsilon$ for
every $i=1,\dots,N$,

$\sum_{i=1}^{N}\int_{t_{i-1}}^{t_{i}}\lVert g(s)-g_{i}\rVert_{X}<\varepsilon,$

there exists a step function $u_{\varepsilon,\sigma}:[0,T]\to X$ of the form

$u_{\varepsilon,\sigma}(t)=u_{0}\,\mathds{1}_{\\{t_{0}=0\\}}(t)+\sum_{i=1}^{N}u_{\varepsilon,\sigma}(t_{i})\;\mathds{1}_{(t_{i-1},t_{i}]}(t),$
(2.7)

where the values of $u_{\varepsilon,\sigma}$ on $(t_{i-1},t_{i}]$, denoted by
$u_{i}$, recursively solve the finite difference equation

$u_{i}+(t_{i}-t_{i-1})Au_{i}\ni(t_{i}-t_{i-1})g_{i}+u_{i-1}\qquad\text{for
every $i=1,\dots,N$}$

and

$\sup_{t\in[0,T]}\lVert
u(t)-u_{\varepsilon,\sigma}(t)\rVert_{X}\leq\varepsilon.$

In the homogeneous case $g\equiv 0$, if $A$ is quasi $m$-accretive in $X$,
then the Crandall-Liggett theorem (cf. [38, Theorem I]) states that for every
element $u_{0}$ of $\overline{D(A)}^{\mbox{}_{X}}$, there is a unique mild
solution $u$ of (2.2) in $X$ for every $T>0$ and this solution $u$ can be
given by the _exponential formula_

$u(t)=\lim_{n\to\infty}\left(I+\tfrac{t}{n}A\right)^{-n}u_{0}$ (2.8)

uniformly in $t$ on compact intervals. For every
$u_{0}\in\overline{D(A)}^{\mbox{}_{X}}$, setting

$T_{t}u_{0}:=u(t),\qquad\text{for every $t\geq 0$,}$ (2.9)

defines a (nonlinear) _strongly continuous semigroup_ $\\{T_{t}\\}_{t\geq 0}$
of _( $\omega$-)quasi contractions_
$T_{t}:\overline{D(A)}^{\mbox{}_{X}}\to\overline{D(A)}^{\mbox{}_{X}}$ with
$\omega\in{\mathbb{R}}$. More precisely, the family $\\{T_{t}\\}_{t\geq 0}$
satisfies the following three properties:

  * 1.

_semigroup property_

$T_{s+t}=T_{t}\circ T_{s}\qquad\text{for every $s$, $t\geq 0$,}$

  * 2.

_strong continuity_

$\lim_{t\to 0+}\lVert T_{t}u-u\rVert_{X}=0\qquad\text{for every
$u\in\overline{D(A)}^{\mbox{}_{X}}$,}$

  * 3.

_exponential growth property in $X$_ or _( $\omega$-)quasi contractivity in
$X$_

$\lVert T_{t}u-T_{t}v\rVert_{X}\leq e^{\omega\,t}\lVert
u-v\rVert_{X}\qquad\text{for all $u$, $v\in\overline{D(A)}^{\mbox{}_{X}}$,
$t\geq 0$.}$

For the family $\\{T_{t}\\}_{t\geq 0}$ on $\overline{D(A)}^{\mbox{}_{X}}$, the
operator

$A_{\circ}:=\Bigg{\\{}(u,v)\in X\times X\Bigg{|}\;\lim_{h\rightarrow
0^{+}}\frac{T_{h}u-u}{h}=v\text{ in $X$}\Bigg{\\}}$

is a well-defined mapping $A_{\circ}:D(A_{\circ})\to X$ with domain

$D(A_{\circ}):=\Big{\\{}u\in X\,\Big{|}\,\lim_{h\rightarrow
0^{+}}\frac{T_{h}u-u}{h}\text{ exists in }X\Big{\\}}$

called the _infinitesimal generator_ of $\\{T_{t}\\}_{t\geq 0}$. If
$\\{T_{t}\\}_{t\geq 0}$ is $\omega$-quasi contractive in $X$, then
$-A_{\circ}$ is $\omega$-quasi accretive in $X$.

Since mild solutions of Cauchy problem (2.2) are only the locally uniform (in
time) limit of step functions (2.7) with values in $X$, it is important to
know whether they are actually _strong solutions_ of (2.2) in $X$.

######  Definition 2.5.

Given $u_{0}\in X$ and $g\in L^{1}(0,T;X)$ for some $T>0$, a function $u\in
C([0,T];X)$ is called a _strong solution_ in $X$ of Cauchy problem (2.2) if
$u(0)=u_{0}$, $u$ belongs to $W^{1,1}_{loc}((0,T);X)$ and for a.e. $0<t<T$,
one has that $u(t)\in D(A)$ and
$g(t)-\tfrac{\mathrm{d}u}{\mathop{}\\!\mathrm{d}t}(t)\in Au(t)$.

Since we typically take $A$ to be the closure of $(-\Delta_{p})^{s}$ in
$L^{1}\times L^{1}$, we also want to consider solutions with further
regularity on $\varphi(u)$. Hence we consider _distributional solutions_ with
the following definition.

######  Definition 2.6.

For given $f$ satisfying (1.3a)-(1.3b) and $g\in
L^{1}_{loc}((0,\infty);L^{1}_{loc})$, a function $u\in C([0,\infty);L^{1})$ is
called a _distributional_ solution of initial value problem (1.1) if
$u(0)=u_{0}$ in $L^{1}$, $\varphi(u)\in L^{p}_{loc}((0,\infty);W^{s,p})$ and
for every test function $\xi\in
C_{c}^{\infty}([0,\infty)\times{\mathbb{R}}^{d})$, one has that

$\displaystyle-\int_{{\mathbb{R}}^{d}}u\,\xi\,\mathop{}\\!\mathrm{d}x\Bigg{|}_{t_{1}}^{t_{2}}-\int_{t_{1}}^{t_{2}}\int_{{\mathbb{R}}^{d}}u\,\xi_{t}\,\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t+\int_{t_{1}}^{t_{2}}\int_{{\mathbb{R}}^{d}}(f(x,u)-g)\xi\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}t$
$\displaystyle\hskip
56.9055pt+\int_{t_{1}}^{t_{2}}\int_{{\mathbb{R}}^{2d}}\frac{(\varphi(u)(t,x)-\varphi(u)(t,y))^{p-1}(\xi(t,x)-\xi(t,y))}{|x-y|^{d+sp}}\mathrm{d}(x,y)\mathop{}\\!\mathrm{d}t=0$

for all $0<t_{1}<t_{2}<\infty$.

Furthermore, if $u$ is also differentiable with $u_{t}(t)\in L^{1}$ for a.e.
$t>0$, then we call this a _strong distributional_ solution in $L^{1}$.

If, for example, $X=L^{q}_{\mu}$ for $1<q<\infty$ and $g\equiv 0$, then $X$ is
a uniformly convex Banach space and so the classical regularity theory of
nonlinear semigroups (cf. [22, Theorem 4.6]) applies: let $A$ be a quasi
$m$-accretive operator on $L^{q}_{\mu}$, then for every $u_{0}\in D(A)$ the
mild solution $u$ of Cauchy problem (2.2) is a strong solution of (2.2) and
$t\mapsto T_{t}u_{0}$ given by (2.9) satisfies

$\tfrac{\mathrm{d}}{\mathrm{d}t}_{+}T_{t}u_{0}=-A^{\circ}T_{t}u_{0}\qquad\text{for
every $t>0$,}$ (2.10)

where $A^{\circ}$ denotes the _minimal selection_ of $A$ given by the operator

$A^{\circ}:=\Big{\\{}(x,y)\in A\,\Big{|}\,\lVert
y\rVert=\min\limits_{\hat{y}\in Ax}\lVert\hat{y}\rVert\Big{\\}}.$

Under additional geometric conditions on the Banach space $X$, one has that
$-A_{\circ}\subseteq A^{\circ}$. Ignoring these details on $X$, we
nevertheless say that a strongly continuous semigroup $\\{T_{t}\\}_{t\geq 0}$
of quasi contractions on $\overline{D(A)}^{\mbox{}_{X}}$ is _generated by
$-A$_ if $A$ is quasi $m$-accretive in $X$ and $\\{T_{t}\\}_{t\geq 0}$ is the
family induced by (2.9).

If $X$ is a Hilbert space $H$ with inner product $(\cdot,\cdot)_{H}$, then an
important class of $m$-accretive operators in $H$ is given by the
_subdifferential operator_

$\partial\mathcal{E}:=\Big{\\{}(u,v)\in H\times
H\,\Big{|}\;(v,\xi-u)_{H}\leq\mathcal{E}(\xi)-\mathcal{E}(u)\text{ for all
}\xi\in H\Big{\\}}$ (2.11)

of a proper, convex, lower semicontinuous functional
$\mathcal{E}:H\to(-\infty,+\infty]$. In Hilbert spaces, accretivity is
equivalent to _monotonicity_ ; that is, an operator $A$ is monotone if

$(u-\hat{u},v-\hat{v})_{H}\geq 0\qquad\text{for all $(u,v)$,
$(\hat{u},\hat{v})\in A$}$

(cf. [20], and see also [39, 40]). For this class of operators
$A=\partial\mathcal{E}$, the Cauchy problem (2.2) has the smoothing effect
that every mild solution $u$ of (2.2) is strong. This result is due to Brezis
[41] (see also [40]).

It is well known that the _fractional $p$-Laplacian_ $(-\Delta_{p})^{s}$
defined in (1.9) equipped with homogeneous Dirichlet boundary conditions on an
open set $\Omega$ can be realized as such a subdifferential operator
$\partial\mathcal{E}$ in $L^{2}$. To be more precise, let
$\Omega\subseteq{\mathbb{R}}^{d}$, $d\geq 1$, be an open domain. Define the
energy functional $\mathcal{E}:L^{2}\to[0,\infty]$ for $1<p<\infty$ and
$s\in(0,1)$ by

$\mathcal{E}(u)=\begin{cases}\frac{1}{2p}[u]_{s,p}^{p}&\text{ if }u\in
W^{s,p}_{0}\cap L^{2},\\\ \infty&\text{ if }u\in L^{2}\setminus
W^{s,p}_{0},\end{cases}$ (2.12)

for every $u\in L^{2}$, where we have defined the $s$-Gagliardo semi-norm by
(2.1) on ${\mathbb{R}}^{N}$ as in the definition of $W^{s,p}_{0}$. It is
immediate that this energy functional is convex, lower semicontinuous and
proper with effecive domain $D(\mathcal{E})=W^{s,p}_{0}\cap L^{2}$ (cf. [7],
[14]). Then we have the following characterization of the subdifferential
$\partial\mathcal{E}$ (see [7]).

######  Proposition 2.7 (Characterization of $(-\Delta_{p})^{s}$).

For $1<p<\infty$ and $0<s<1$, let $\mathcal{E}$ be given by (2.12). Then for
every $u\in W^{s,p}_{0}\cap L^{2}$,

$\partial\mathcal{E}(u)=\left\\{h\in
L^{2}\Biggr{|}\scalebox{0.9}{$\begin{aligned}
&\int_{\Omega}h(x)v(x)\mathop{}\\!\mathrm{d}x=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{(u(x)-u(y))^{p-1}(v(x)-v(y))}{|x-y|^{d+sp}}\\\
&\hskip 99.58464pt\text{ for all }v\in W^{s,p}_{0}\cap
L^{2}\end{aligned}$}\right\\}.$

We note that the case $p=1$ can be characterized similarly with a formulation
presented in [7], although we do not consider $p=1$ in this article.

####  2.2.2 Completely accretive and $T$-accretive operators

The notion of _completely accretive operators_ was introduced in [42] by
Crandall and Bénilan and further developed in [14]. Following the same
notation as in these two references, $\mathcal{J}_{0}$ denotes the set of all
convex, lower semicontinuous functions $j:{\mathbb{R}}\to[0,+\infty]$
satisfying $j(0)=0$.

######  Definition 2.8.

A mapping $S:D(S)\to M(\Sigma,\mu)$ with domain $D(S)\subseteq M(\Sigma,\mu)$
is called a _complete contraction_ if

$\int_{\Sigma}j(Su-S\hat{u})\mathop{}\\!\mathrm{d}\mu\leq\int_{\Sigma}j(u-\hat{u})\mathop{}\\!\mathrm{d}\mu$

for all $j\in\mathcal{J}_{0}$ and every $u$, $\hat{u}\in D(S)$. An operator
$A$ on $M(\Sigma,\mu)$ is called _completely accretive_ if for every
$\lambda>0$, the resolvent operator $J_{\lambda}$ of $A$ is a _complete
contraction_.

It is well-known (see, e.g., [7, 14]) that the fractional $p$-Laplacian
$(-\Delta_{p})^{s}$ equipped with Dirichlet boundary conditions is
$m$-completely accretive in $L^{2}$. A more general class of operators is that
of _$T$ -accretive_ operators. In particular, choosing
$j(\cdot)=\lvert[\,\cdot\,]^{+}\rvert^{q}\in\mathcal{J}_{0}$ if $1\leq
q<\infty$ and $j(\cdot)=[[\,\cdot\,]^{+}-k]^{+}\in\mathcal{J}_{0}$ for $k\geq
0$ large enough if $q=\infty$ shows that a complete contraction $S$ satisfies
the following _$T$ -contractivity_ property in $L^{q}_{\mu}$ for every $1\leq
q\leq\infty$.

######  Definition 2.9.

A mapping $S:D(S)\to L^{q}_{\mu}$ with domain $D(S)\subseteq L^{q}_{\mu}$,
$1\leq q\leq\infty$, is called a _$T$ -contraction_ if

$\lVert[Su-S\hat{u}]^{+}\rVert_{q}\leq\lVert[u-\hat{u}]^{+}\rVert_{q}$

for every $u$, $\hat{u}\in D(S)$. We say that an operator $A$ on $L^{q}_{\mu}$
is _$T$ -accretive_ if, for every $\lambda>0$, the resolvent $J_{\lambda}$ of
$A$ defines a $T$-contraction with domain $D(J_{\lambda})=Rg(I+\lambda A)$.

By the previous remark, a completely accretive operator is $T$-accretive in
$L^{q}_{\mu}$ for all $1\leq q\leq\infty$. Furthermore, a $T$-accretive
operator in $L^{q}_{\mu}$, $1\leq q\leq\infty$, is accretive in $L^{q}_{\mu}$
and the resolvent is order-preserving in $L^{q}_{\mu}$. That is, denoting the
usual order relation on $L^{q}_{\mu}$ by $\leq$, if $S$ is $T$-contractive in
$L^{q}_{\mu}$, $1\leq q\leq\infty$, then $u\leq\hat{u}$ implies that $Su\leq
S\hat{u}$ for $u$, $\hat{u}\in L^{q}$ (see [21, Lemma 19.11] for further
properties).

We can naturally extend these definitions to quasi $m$-completely accretive
operators (and similarly for quasi $m$-$T$-accretive operators).

######  Definition 2.10.

An operator $A$ on $M_{\mu}$ is called _quasi completely accretive_ if there
is an $\omega>0$ such that for every $\lambda>0$, the resolvent operator
$J_{\lambda}$ of $A+\omega I$ is a complete contraction. Moreover, for $1\leq
q<\infty$, an operator $A$ on $L^{q}_{\mu}$ is said to be _quasi
$m$-completely accretive_ on $L^{q}_{\mu}$ if there is an $\omega>0$ such that
$A+\omega I$ is completely accretive and the range condition (2.6) holds with
$X=L^{q}_{\mu}$.

####  2.2.3 $T$-accretive operators in $L^{1}$ with complete resolvent

In this last part of Section 2.2, we introduce the class of operators $A$
which are merely $T$-accretive in $L^{1}_{\mu}$ but have a so-called _complete
resolvent_. This class of operators was introduced in [43] and further
elaborated in [14]. It’s worth mentioning that for a given completely
accretive operator $A$ in $L^{1}$, the composed operator $A\varphi$ becomes
$T$-accretive in $L^{1}$ with complete resolvent so long as $\varphi$ is a
strictly increasing, continuous function on ${\mathbb{R}}$ (see [14]). Typical
examples of this class of operators include the doubly-nonlinear operators
$-\Delta_{p}\varphi$ and $(-\Delta_{p})^{s}\varphi$. Hence we need to
introduce the notion _complete_ mappings.

######  Definition 2.11.

Let $D(S)$ be a subset of $M_{\mu}$. A mapping
$S:D(S)\to M(\Sigma,\mu)$ is called _complete_ if

$\int_{\Sigma}j(Su)\mathop{}\\!\mathrm{d}\mu\leq\int_{\Sigma}j(u)\mathop{}\\!\mathrm{d}\mu$
(2.13)

for every $j\in\mathcal{J}_{0}$ and $u\in D(S)$.

We now introduce the class of accretive operators in $L^{1}_{\mu}$ with
complete resolvent (similarly for $T$-accretive operators with complete
resolvent).

######  Definition 2.12.

An operator $A$ on $L^{1}_{\mu}$ is called _( $m$-)accretive in $L^{1}_{\mu}$
with complete resolvent_ if $A$ is ($m$-)accretive in $L^{1}_{\mu}$ and for
every $\lambda>0$, the resolvent operator $J_{\lambda}:Rg(I+\lambda A)\to
D(A)$ of $A$ is a complete mapping. For $\omega\in{\mathbb{R}}$, we call an
operator $A$ on $L^{1}_{\mu}$ _$\omega$ -quasi ($m$-)accretive in
$L^{1}_{\mu}$ with complete resolvent_ (or simply _quasi ( $m$-)accretive in
$L^{1}_{\mu}$ with complete resolvent_) if $A+\omega I$ is ($m$-)accretive in
$L^{1}_{\mu}$ with complete resolvent.

The condition (2.13) on the resolvent provides a growth estimate on $u(t)$,
allowing us to estimate $u(t)$ in $L^{p}_{\mu}$ by the norms of $u_{0}$ and
$g$ in $L^{p}_{\mu}$. In particular, we prove such an estimate in Lemma 4.3,
extending [43, Proposition 2.4].

####  2.2.4 The sub-differential operator in $X$

We rely on the $m$-accretivity of $(-\Delta_{p})^{s}$ in $L^{1}$ to obtain
mild and strong solutions to (1.1). We also find that it is sufficient to work
with the part in $L^{1\cap\infty}$ to establish existence of such solutions.
Hence we consider operators restricted to $L^{1\cap\infty}$ and introduce a
definition of subdifferential operators from [42] which we will apply in the
case $X=L^{1}$. We will also use this to apply the results of [26] and in
particular to obtain strong solutions. We include functionals on a (possibly
distinct) subspace $Y$ for completeness.

######  Definition 2.13.

Let $X$, $Y$ and $Z$ be linear subspaces of $M(\Sigma,\mu)$. Then for an
energy functional $\mathcal{E}:Y\to(-\infty,\infty]$ with effective domain
$D(\mathcal{E}):=\\{u\in Y\,|\,\mathcal{E}(u)<\infty\\}$, we define the _part
of $\mathcal{E}$ in $X$_ by

$\mathcal{E}_{|_{X}}(u)=\begin{cases}\mathcal{E}(u)&\text{for }u\in
D(\mathcal{E})\cap X,\\\ \infty&\text{otherwise.}\\\ \end{cases}$

Further, we define the operator $\partial_{X}\mathcal{E}$ in $X$ by

$\partial_{X}\mathcal{E}=\Bigg{\\{}(u,v)\in X\times
X\,\Bigg{|}\begin{array}[c]{l}u\in D(\mathcal{E})\;\text{ and
}\displaystyle\int_{\Sigma}v(w-u)\mathop{}\\!\mathrm{d}\mu\leq\mathcal{E}(w)-\mathcal{E}(u)\\\
\text{ for all }w\in X\text{ with }v(w-u)\in L^{1}_{\mu}\end{array}\Bigg{\\}}$

and the _part of $\partial_{X}\mathcal{E}$ in $Z$_ by

$(\partial_{X}\mathcal{E})_{|_{Z}}=\Big{\\{}(u,h)\in Z\times
Z\,\Big{|}(u,h)\in\partial_{X}\mathcal{E}\Big{\\}}.$

In the case $X=L^{2}_{\mu}$ and $Y$ a linear subspace of $M(\Sigma,\mu)$, this
coincides with the previous definition of the subdifferential
$\partial\mathcal{E}$ in $L^{2}_{\mu}$ given by (2.11). One sees that for a
functional $\mathcal{E}:Y\rightarrow(-\infty,\infty]$ and
$\tilde{\mathcal{E}}$ given by $\mathcal{E}_{|_{L^{2}_{\mu}}}$, that
$\partial_{L^{2}_{\mu}}\mathcal{E}=\partial\tilde{\mathcal{E}}$.

In the case $X=L^{1}_{\mu}$, $Y=L^{2}_{\mu}$, we have the inclusion
$\left(\partial\mathcal{E}\right)_{|_{L^{1\cap\infty}_{\mu}}}\subseteq\left(\partial_{L^{1}_{\mu}}\mathcal{E}\right)_{|_{L^{1\cap\infty}_{\mu}}}$.
However, since we are primarily interested in $m$-accretivity, they will be
largely interchangeable due to the $m$-accretivity properties of
$\left(\partial\mathcal{E}\right)_{|_{L^{1\cap\infty}_{\mu}}}$ proved in
Theorem 3.1. For the class of operators $\partial_{X}\mathcal{E}$, Bénilan and
Crandall [42] found sufficient conditions implying that the closure
$\overline{\partial_{X}\mathcal{E}}^{\mbox{}_{X}}$ of
$\partial_{X}\mathcal{E}$ in $X$ is $m$-completely accretive. We note that if
$\mathcal{E}$ is lower semicontinuous then
$\overline{\partial_{L^{2}_{\mu}}\mathcal{E}}^{\mbox{}_{L^{2}_{\mu}}}=\partial_{L^{2}_{\mu}}\mathcal{E}$.

######  Theorem 2.14 ([42, Lemma 7.1 & Theorem 7.4]).

Let $X$ be either $L^{r}_{\mu}$, $1\leq r\leq\infty$, or
$L^{1\cap\infty}_{\mu}$. Then the following statements hold.

  1. 1.

If a functional $\mathcal{E}:X\to(-\infty,\infty]$ satisfies

$\left\\{\begin{array}[c]{l}\hskip
28.45274pt\mathcal{E}(u+q(\hat{u}-u))+\mathcal{E}(\hat{u}+q(\hat{u}-u))\leq\mathcal{E}(u)+\mathcal{E}(\hat{u})\\\\[5.0pt]
\text{for all $u$, $\hat{u}\in X$ and $q\in C^{\infty}({\mathbb{R}})$ such
that $q(0)=0$ and}\\\\[5.0pt] \text{$q^{\prime}$ has compact support,
satisfying $0\leq q^{\prime}\leq 1$ on ${\mathbb{R}}$}\end{array}\right.$
(2.14)

then the operator $\partial_{X}\mathcal{E}$ is completely accretive.

  2. 2.

If $\mathcal{E}:X\to[0,\infty]$ satisfies (2.14),
$(0,0)\in\partial_{X}\mathcal{E}$, and if $\mathcal{E}$ is lower
semicontinuous for the topology of $X+L^{2}_{\mu}$, then the closure
$\overline{\partial_{X}\mathcal{E}}^{\mbox{}_{X}}$ of
$\partial_{X}\mathcal{E}$ in $X$ is $m$-completely accretive in $X$.

Note that if $\mathcal{E}(0)=0$ and $\mathcal{E}(u)\geq 0$ for all $u\in X$
with $\partial_{X}\mathcal{E}$ completely accretive, then the condition
$(0,0)\in\partial\mathcal{E}$ of Theorem 2.14 is satisfied.

###  2.3 The doubly nonlinear operator $(-\Delta_{p})^{s}\varphi$

Throughout this paper we focus on the composed operator
$(-\Delta_{p})^{s}\varphi=(-\Delta_{p})^{s}\circ\varphi$. Hence, we introduce
the composition operator on $L^{1}$ of the form $A\circ\varphi$ for $A$ an
operator on $L^{1}_{\mu}$ and $\varphi$ a function on ${\mathbb{R}}$.

######  Definition 2.15.

For an operator $A$ on $L^{1}_{\mu}$ and a function $\varphi$ on
${\mathbb{R}}$, we define the composed operator $A\circ\varphi$ in
$L^{1}_{\mu}$ as a graph by

$A\varphi=\\{(u,v)\in L^{1}_{\mu}\times L^{1}_{\mu}:(\varphi(u),v)\in A\\}.$

and interchangeably as a (possibly multi-valued) operator on $L^{1}_{\mu}$.

We may also extend the domain of an operator defined on
$L^{1\cap\infty}_{\mu}$ in the following manner introduced by [26].

######  Definition 2.16.

For an operator $A$ in $L^{1\cap\infty}_{\mu}\times L^{1\cap\infty}_{\mu}$, we
extend the domain $D(A)$ by the set

$\tilde{D}(A):=\left\\{u\in L^{1\cap\infty}_{\mu}\left|\begin{aligned}
&\exists\,(u_{n},h_{n})_{n\geq 1}\subseteq A\text{ such that}\\\
&u_{n}\rightarrow u\text{ in }L^{1}_{\mu}\text{ and }\\\ &(u_{n},h_{n})_{n\geq
1}\text{ is bounded in }L^{\infty}_{\mu}\times
L^{1}_{\mu}.\end{aligned}\right.\right\\}$

and for a strictly increasing $\varphi\in C({\mathbb{R}})$, we define

$\hat{D}(A\varphi):=\Big{\\{}u\in
L^{1}_{\mu}\,\Big{|}\,\varphi(u)\in\tilde{D}(A)\Big{\\}}.$

Then, for an operator $A$ in $L^{1\cap\infty}_{\mu}\times
L^{1\cap\infty}_{\mu}$, one has that

$D(A)\subseteq\tilde{D}(A)\subseteq L^{1}_{\mu}\qquad\text{and}\qquad
D(A\varphi)\subseteq\hat{D}(A\varphi).$

Now, for $\Omega\subset{\mathbb{R}}^{d}$ and
$\varphi(u_{0})\in\tilde{D}\left(\left(\partial_{L^{1}}\mathcal{E}\right)|_{L^{1\cap\infty}}\right)$
we have the following theorem for existence of strong solutions from [26].
Note that here $\beta$ corresponds to $\varphi^{-1}$ in our setting and $v$ to
$\varphi(u)$. The statement in [26] also uses a slightly different
implementation of the sub-differential, closer to
$\partial_{L^{\infty}}\mathcal{E}$ (giving potentially a larger operator),
however this does not affect the proof. A similar result for the homogeneous
evolution problem can be found in [14].

######  Theorem 2.17 ([26, Theorem 4.1], Existence of strong distributional
solutions).

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$, of finite
Lebesgue measure, and $T>0$. Suppose $\mathcal{E}:L^{2}\rightarrow[0,\infty]$
is a lower semicontinuous function satisfying $\mathcal{E}(0)=0$ and (2.14)
for $X=L^{2}$. Further, let $\beta\in AC_{\operatorname{loc}}(\mathbb{R})$ be
nondecreasing satisfying $\beta(\mathbb{R})=\mathbb{R}$. Then, for every
$v_{0}\in\tilde{D}(\left(\partial_{L^{1}}\mathcal{E}\right)_{|_{L^{1\cap\infty}}})$
and $f\in BV((0,T);L^{1})\cap L^{1}(0,T;L^{1})\cap L^{1}(0,T;L^{\infty})$,
there exists $v\in L^{\infty}((0,T)\times\Omega)$ such that $u:=\beta(v)\in
W^{1,\infty}(0,T;L^{1})$ is the unique strong distributional solution to

$\begin{cases}u^{\prime}(t)+\left(\partial_{L^{1}}\mathcal{E}\right)_{|_{L^{1\cap\infty}}}v(t)\in
f(t)&\text{ for a.e.~{}}t\in(0,T),\\\ u(0)=\beta(v_{0}).&\end{cases}$ (2.15)

We also require the following chain rule.

######  Theorem 2.18 ([26, Theorem 1.1]).

Let $(\Sigma,\mu)$ be a measure space.
If $w\in W^{1,1}((0,T);L^{1}_{\mu})$, $p\in
L_{\operatorname{loc}}^{1}({\mathbb{R}})$ and

$u=\int_{0}^{w}p(r)\mathop{}\\!\mathrm{d}r\in BV(0,T;L^{1}_{\mu})\cap
L^{1}((0,T);L^{1}_{\mu})$

then $u\in W^{1,1}((0,T);L^{1}_{\mu})$ and for a.e. $t\in(0,T)$

$\frac{\mathrm{d}u}{\mathrm{d}t}(t)=p(w(t))\frac{\mathrm{d}w}{\mathrm{d}t}(t)$

for $\mu$-a.e. $x\in\Sigma$.

With the above preliminaries we can now focus on proving our main results.

##  3 Well-posedness and a comparison principle

We now apply the nonlinear semigroup theory summarized in the previous section
to the doubly nonlinear operator $(-\Delta_{p})^{s}\varphi$ to prove Theorem
1.1. To do this we prove two key results for operators of the form
$\overline{(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}\varphi}^{\mbox{}_{L^{1}}}+F$,
which might be of independent interest.

The first result gives us $T$-accretivity with complete resolvent for
operators $A\varphi$ with a Lipschitz perturbation. This follows from [14,
Proposition 2.17 and Proposition 2.19]. A comparable result can also be found
for operators defined on finite measure spaces $(\Sigma,\mu)$ in [26, pg. 24]
(see also [42, Lemma 7.1]).

######  Theorem 3.1.

Suppose $(\Sigma,\mu)$ is a $\sigma$-finite measure space,
$\mathcal{E}:L^{2}_{\mu}\rightarrow[0,\infty]$ is convex, lower semicontinuous
with $\mathcal{E}(0)=0$ and satisfying (2.14). Let $\varphi\in
C({\mathbb{R}})$ be strictly increasing with $\varphi(0)=0$ and satisfying

$[\beta_{\lambda}(u),(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}u]_{1}\geq
0$ (3.1)

and

$[\beta_{\lambda}(u),(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}u]_{2}\geq
0$ (3.2)

for every $\lambda>0$ and $u\in
D((\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}})$, where
$\beta=\varphi^{-1}$ and $\beta_{\lambda}$ is the Yosida approximation of
$\beta$. Suppose $F:L^{1}_{\mu}\rightarrow L^{1}_{\mu}$ satisfies the
Lipschitz property

$|F(u)-F(\hat{u})|\leq\omega|u-\hat{u}|\quad\text{ on }\Sigma$ (3.3)

for all $u$, $\hat{u}\in L^{1}_{\mu}$ with constant $\omega\geq 0$ and
satisfies $F(0)=0$. Then
$\overline{(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}\varphi}^{\mbox{}_{L^{1}_{\mu}}}+F$
is $\omega$-quasi $m$-$T$-accretive in $L^{1}_{\mu}$ with complete resolvent.

###### Proof.

Since $\mathcal{E}$ is convex and attains its global minimum at $0$, we have
that $(0,0)\in\partial\mathcal{E}$. Then by the complete accretivity property,
$\partial\mathcal{E}$ has complete resolvent. Hence $\partial\mathcal{E}$ is
$m$-completely accretive in $L^{2}$ with complete resolvent so that
$(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}$ is also completely
accretive. Since $\varphi$ is injective, we have by [14, Proposition 2.17]
that $(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}\varphi$ is
$T$-accretive in $L^{1}_{\mu}$ with complete resolvent. So by [14, p.31] and
[14, Proposition 2.12], the operator
$(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}\varphi+F$ is $\omega$-quasi
$T$-accretive in $L^{1}$ with complete resolvent. Then we can apply [14,
Proposition 2.19] to $\partial\mathcal{E}$ to obtain the range condition for
the closure and hence $\omega$-quasi $m$-$T$-accretivity. ∎

We now apply the above theorem to the doubly nonlinear operator
$(-\Delta_{p})^{s}\varphi$.

######  Corollary 3.2.

Let $\Omega$ be an open domain in ${\mathbb{R}}^{d}$, $d\geq 1$. Suppose
$F:L^{1}\rightarrow L^{1}$ satisfies the Lipschitz property (3.3) for all $u$,
$\hat{u}\in L^{1}$ with constant $\omega\geq 0$ and satisfies $F(0)=0$. Let
$\mathcal{E}$ be the energy functional given by (2.12) associated to the
$s$-Gagliardo semi-norm. Then the following statements hold.

  1. 1.

The sub-differential operator $\partial\mathcal{E}$ in $L^{2}$ of
$\mathcal{E}$ is $m$-completely accretive on $L^{2}$.

  2. 2.

The part $(\partial\mathcal{E})_{|L^{1\cap\infty}}$ of $\partial\mathcal{E}$
in $L^{1\cap\infty}\times L^{1\cap\infty}$ satisfies (3.1) and (3.2).

  3. 3.

The operator
$\overline{(\partial\mathcal{E})_{|_{L^{1\cap\infty}}}\varphi}^{\mbox{}_{L^{1}}}+F$
is $\omega$-quasi $m$-$T$-accretive in $L^{1}$ with complete resolvent.

###### Proof.

We first show that $\mathcal{E}$ satisfies (2.14) and so, the sub-differential
operator $\partial\mathcal{E}$ of $\mathcal{E}$ on $L^{2}$ is $m$-completely
accretive on $L^{2}$ owing to Theorem 2.14. Obviously, it is sufficient to
show that $\mathcal{E}$ satisfies (2.14) for every $v$ and $\hat{v}\in
D(\mathcal{E})$. For given $x$, $y\in{\mathbb{R}}^{d}$, set $a=v(x)-v(y)$ and
$b=\hat{v}(x)-\hat{v}(y)$, and let $q\in C^{1}({\mathbb{R}})$ satisfying
$q(0)=0$ and $0\leq q^{\prime}\leq 1$. Since $0\leq q^{\prime}\leq 1$, there
is a $k\in[0,1]$ such that $q(b-a)=k(b-a)$. Then, by the convexity of
$|\cdot|^{p}$, one has that

$\left|ka+(1-k)b\right|^{p}+\left|(1-k)a+kb\right|^{p}\leq|a|^{p}+|b|^{p}$

holds and so, we have

$\left|b-q(b-a))\right|^{p}+\left|a+q(b-a)\right|^{p}\leq|a|^{p}+|b|^{p}$

or, equivalently,

$\displaystyle\left|\hat{v}(x)-\hat{v}(y)-q(\hat{v}(x)-\hat{v}(y)-v(x)-v(y)))\right|^{p}$
$\displaystyle\qquad\qquad+\left|v(x)-v(y)+q(\hat{v}(x)-\hat{v}(y)-v(x)-v(y))\right|^{p}$
$\displaystyle\qquad\leq|v(x)-v(y)|^{p}+|\hat{v}(x)-\hat{v}(y)|^{p}.$

Therefore (2.14) follows from integrating over ${\mathbb{R}}^{2d}$ with
respect to $|x-y|^{-d-sp}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y$.

Further, since for every $\lambda>0$ and $q\geq 1$,
$u\mapsto\left(\beta_{\lambda}(\cdot)\right)^{q-1}$ is monotone increasing,
differentiable with a bounded derivative, it follows from the characterization
of the sub-differential operator $\partial\mathcal{E}$ given in Proposition
2.7 that he part $(\partial\mathcal{E})_{|L^{1\cap\infty}}$ of
$\partial\mathcal{E}$ in $L^{1\cap\infty}\times L^{1\cap\infty}$ satisfies
(3.1) and (3.2). Hence we may apply Theorem 3.1, to conclude that the operator
$\overline{(\partial\mathcal{E})_{|_{L^{1\cap\infty}}}\varphi}^{\mbox{}_{L^{1}}}+F$
is $\omega$-quasi $m$-$T$-accretive in $L^{1}$ with complete resolvent. ∎

In order to apply our regularity results to all initial data in $L^{1}$, we
use the following density result for the composition operator
$(\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}\varphi$. This generalizes
the classic density result for sub-differential operators (cf. [20] or [22,
Proposition 1.6]) and in particular generalizes an idea from [22, p.48]. We
note that (3.1) and (3.2) are satisfied for the fractional $p$-Laplacian.

######  Theorem 3.3 (Density of
$D((\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}\varphi)$ in
$L^{1}_{\mu}$).

Let $(\Sigma,\mu)$ be a $\sigma$-finite measure space,
$\mathcal{E}:M(\Sigma,\mu)\rightarrow[0,\infty]$ a proper, convex functional
satisfying $\mathcal{E}(0)=0$, and suppose the restriction
$\mathcal{E}_{|L^{2}_{\mu}}$ of $\mathcal{E}$ on $L^{2}_{\mu}$ is lower
semicontinuous on $L^{2}_{\mu}$ and the restriction
$\mathcal{E}_{|L^{1\cap\infty}_{\mu}}$ of $\mathcal{E}$ on
$L^{1\cap\infty}_{\mu}$ satisfies (2.14). Further, let $\varphi\in
C({\mathbb{R}})$ be a strictly increasing function such that
$\varphi({\mathbb{R}})={\mathbb{R}}$, $\varphi(0)=0$ and the Yosida
approximation $\beta_{\lambda}$ of $\varphi^{-1}$ satisfies (3.1) and (3.2)
for every $\lambda>0$. Then the following statements hold.

  1. 1.

The domain $D(\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\varphi)$ of the
composed operator $\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\varphi$ is
dense in the closure
$\overline{D(\mathcal{E}_{|L^{1\cap\infty}_{\mu}}\varphi)}^{\mbox{}_{L^{1}_{\mu}}}$
of $D(\mathcal{E}_{|L^{1\cap\infty}_{\mu}}\varphi)$ with respect to the
$L^{1}_{\mu}$-norm topology.

  2. 2.

If the set $D(\mathcal{E}_{|L^{1\cap\infty}_{\mu}}\varphi)$ is dense in
$L^{1}_{\mu}$, then
$D((\partial\mathcal{E})_{|_{L^{1\cap\infty}_{\mu}}}\varphi)$ is dense in
$L^{1}_{\mu}$.

###### Proof of Theorem 3.3.

Under the hypotheses of this theorem, one can apply [14, Lemma A.3.1] to
$A=\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}$ and obtains that for every
$\lambda>0$, every $\varepsilon>0$ sufficiently small, and every $u\in
D(\mathcal{E}_{|L^{1\cap\infty}_{\mu}}\circ\varphi)$, there is a unique
$u_{\lambda}\in
D(\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\circ\varphi)$ satisfying

$u_{\lambda}+\lambda\left(\varepsilon\varphi(u_{\lambda})+\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\varphi(u_{\lambda})\right)\ni
u,$

or equivalently, there exists
$v_{\lambda}\in\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\varphi(u_{\lambda})$
such that

$u_{\lambda}+\lambda\left(\varepsilon\varphi(u_{\lambda})+v_{\lambda}\right)=u$
(3.4)

Multiplying (3.4) by $\varphi(u_{\lambda})-\varphi(u)$ gives

$\begin{split}\big{(}u_{\lambda}-u,\varphi(u_{\lambda})-\varphi(u)\big{)}_{L^{2}_{\mu}}=&-\lambda\varepsilon\big{(}\varphi(u_{\lambda}),\varphi(u_{\lambda})-\varphi(u)\big{)}_{L^{2}_{\mu}}\\\
&\hskip
56.9055pt-\lambda\big{(}v_{\lambda},\varphi(u_{\lambda})-\varphi(u)\big{)}_{L^{2}_{\mu}}.\end{split}$

The first term on the right-hand side can be estimates by

$\begin{split}-\big{(}\varphi(u_{\lambda}),\varphi(u_{\lambda})-\varphi(u)\big{)}_{L^{2}_{\mu}}&\leq\lVert\varphi(u_{\lambda})\rVert_{\infty}\lVert\varphi(u)\rVert_{1}\\\
&\leq\sup_{[-\lVert u\rVert_{\infty},\lVert
u\rVert_{\infty}]}\lVert\varphi\rVert_{\infty}\lVert\varphi(u)\rVert_{1},\end{split}$

and since
$v_{\lambda}\in\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\varphi(u_{\lambda})$
and noting that $\mathcal{E}(\varphi(u_{\lambda}))\geq 0$, it follows that

$\begin{split}-\big{(}v_{\lambda},\varphi(u_{\lambda})-\varphi(u)\big{)}_{L^{2}_{\mu}}&=-\big{(}(\partial\mathcal{E})\varphi(u_{\lambda}),\varphi(u_{\lambda})-\varphi(u)\big{)}_{L^{2}_{\mu}}\\\
&\leq-\left(\mathcal{E}(\varphi(u_{\lambda}))-\mathcal{E}(\varphi(u))\right)\\\
&\leq\mathcal{E}(\varphi(u))\end{split}$

for all $\lambda>0$. Thus and since $\varphi$ is increasing, we have shown
that

$0\leq(u_{\lambda}-u,\varphi(u_{\lambda})-\varphi(u))_{L^{2}_{\mu}}\leq\lambda\,\mathcal{E}(\varphi(u))+\lambda\varepsilon\sup_{[-\lVert
u\rVert_{\infty},\lVert
u\rVert_{\infty}]}\lVert\varphi\rVert_{\infty}\lVert\varphi(u)\rVert_{1}$

for all $\lambda>0$, from where we can conclude that

$\lim_{\lambda\rightarrow
0^{+}}\int_{\Sigma}(u_{\lambda}-u)(\varphi(u_{\lambda})-\varphi(u))\mathop{}\\!\mathrm{d}\mu=0.$

Since

$f_{\lambda}(x):=(u_{\lambda}(x)-u(x))(\varphi(u_{\lambda}(x)-\varphi(u(x)))\geq
0\quad\text{$\mu$-a.e.~{}on $\Sigma$,}$

the letter limit means that $f_{\lambda}\rightarrow 0$ in $L^{1}_{\mu}$. After
possibly passing to a subsequence, we know that

$\lim_{\lambda\rightarrow 0^{+}}f_{\lambda}(x)=0\qquad\text{for
$\mu$-a.e.~{}$x\in\Sigma$,}$

which due to the strict monotonicity of $\varphi$ implies that

$\lim_{\lambda\rightarrow 0^{+}}u_{\lambda}(x)=u(x)\qquad\text{for
$\mu$-a.e.~{}$x\in\Sigma$.}$ (3.5)

We recall from [14, Lemma 2.2.1] that if $u_{\lambda}\geq 0$ for all
$\lambda>0$, then the $\mu$-pointwise limit (3.5) together with the fact that
$u_{\lambda}$ satisfies

$\lVert u_{\lambda}\rVert_{1}\leq\lVert u\rVert_{1}\qquad\text{for all
$\lambda>0$,}$ (3.6)

implies that $u_{\lambda}\rightarrow u$ in $L^{1}_{\mu}$.

Now, let
$u_{+,\lambda}:=J_{\lambda}^{\varepsilon\varphi_{1}+A\varphi}(u^{+})$, where
$u^{+}=u\vee 0$ is the positive part of $u$. Since
$\varepsilon\varphi_{1}+A\varphi$ is $T$-accretive in $L^{1}_{\mu}$ (cf. [14,
Proposition 2.3.6]), one has that $u_{+,\lambda}\geq 0$ for all $\lambda>0$.
Moreover, by the above argument, $u_{+,\lambda}$ satisfies (3.5) and (3.6).
Therefore one has that $u_{+,\lambda}\rightarrow u^{+}$ in $L^{1}_{\mu}$.
Next, let $u^{-}=(-u)\vee 0$ be the negative part of $u$ and set
$u_{-,\lambda}:=J_{\lambda}^{\varepsilon\varphi_{1}+A\varphi}(-u^{-})$. Then,
one also has that $u_{-,\lambda}$ satisfies (3.5) and so, in particular,
$-u_{-,\lambda}$ satisfies (3.5). Since $-u_{-,\lambda}$ is positive and
satisfies (3.6), it follows that $-u_{-,\lambda}\rightarrow-u^{-}$ in
$L^{1}_{\mu}$. Moreover, for
$u_{\lambda}:=J_{\lambda}^{\varepsilon\varphi_{1}+A\varphi}u$, one has that

$-u_{-,\lambda}\leq u_{\lambda}\leq u_{+,\lambda}\qquad\text{for every
$\lambda>0$.}$

From this sandwich inequality and since $u_{+,\lambda}\rightarrow u^{+}$ in
$L^{1}_{\mu}$ and $-u_{-,\lambda}\rightarrow-u^{-}$ in $L^{1}_{\mu}$, one can
extract from every zero sequence $(\lambda_{n})_{n\geq 0}$ a subsequence
$(\lambda_{k_{n}})_{n\geq 1}$ and finds a positive function $g\in L^{1}_{\mu}$
such that $\lvert u_{\lambda_{k_{n}}}\rvert\leq g$ $\mu$-a.e. on $\Sigma$ for
all $n\geq 1$. Thus, by Lebesgue’s dominated convergence theorem, it follows
that $u_{\lambda_{k_{n}}}\to u$ in $L^{1}_{\mu}$ as $n\to\infty$ and, thereby
we have shown that the domain
$D(\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\circ\varphi)$ lies dense in
the closure
$\overline{D(\mathcal{E}_{|L^{1\cap\infty}_{\mu}}\circ\varphi)}^{\mbox{}_{L^{1}_{\mu}}}$
of $D(\mathcal{E}_{|L^{1\cap\infty}_{\mu}}\circ\varphi)$ with respect to the
$L^{1}_{\mu}$-norm topology. Thus, if the domain
$D(\mathcal{E}_{|L^{1\cap\infty}_{\mu}}\circ\varphi)$ lies dense in
$L^{1}_{\mu}$ then
$D(\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\circ\varphi)$ lies dense in
$L^{1}_{\mu}$. ∎

We can now apply the previous results to the operator
$\partial\mathcal{E}_{|_{L^{1\cap\infty}_{\mu}}}\circ\varphi$ in the case of
the fractional $p$-Laplacian $\partial\mathcal{E}=(-\Delta_{p})^{s}$ equipped
with homogeneous Dirichlet boundary conditions, obtaining $m$-$T$-accretivity
in $L^{1}$ and well-posedness of mild solutions.

###### Proof of Theorem 1.1.

Let $\mathcal{E}$ be given by (2.12) and $\varphi$, $f$ satisfy (1.2) and
(1.3a)-(1.3b), respectively. Then, it follows from Corollary 3.2 that
$\partial_{L^{2}}\mathcal{E}$ is $m$-completely accretive and
$(\partial\mathcal{E})_{|L^{1\cap\infty}}$ satisfies (3.1) and (3.2) with
respect to the Yosida approximation $\beta_{\lambda}$ of $\varphi^{-1}$.
Further, under the hypothesis $\varphi\in W^{1,\infty}_{loc}({\mathbb{R}})$,
one has that

$[\varphi(\xi)]_{s,p}\leq\lVert\varphi^{\prime}\rVert_{L^{\infty}(-\lVert\xi\rVert_{\infty},\lVert\xi\rVert_{\infty})}\,[\xi]_{s,p}$

for every $\xi\in C^{\infty}_{c}(\Omega)$, and if $\phi\in
W^{1,q}_{\operatorname{loc}}({\mathbb{R}})$ for $q>1/(1-s)$, then Hölder’s
inequality yields that

$\displaystyle\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}\frac{|\phi(\xi(x))-\phi(\xi(y))|^{p}}{|x-y|^{d+sp}}\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x$
$\displaystyle\qquad=\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}\frac{|\int_{\xi(y)}^{\xi(x)}\phi^{\prime}(r)\mathop{}\\!\mathrm{d}r|^{p}}{|x-y|^{d+sp}}\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x$
$\displaystyle\qquad\leq\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}\frac{\left|\left(\int_{\xi(y)}^{\xi(x)}|\phi^{\prime}(r)|^{q}\mathop{}\\!\mathrm{d}r\right)^{\frac{p}{q}}\right||\xi(x)-\xi(y)|^{\frac{p}{q^{\prime}}}}{|x-y|^{d+sp}}\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x$
$\displaystyle\qquad\leq\lVert\phi^{\prime}\rVert_{L^{q}(-\lVert\xi\rVert_{\infty},\lVert\xi\rVert_{\infty})}^{p}\int_{{\mathbb{R}}^{d}}\int_{{\mathbb{R}}^{d}}\frac{|\xi(x)-\xi(y)|^{\frac{p}{q^{\prime}}}}{|x-y|^{d+sp}}\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x$
$\displaystyle\qquad=\lVert\phi^{\prime}\rVert_{L^{q}(-\lVert\xi\rVert_{\infty},\lVert\xi\rVert_{\infty})}^{p}\,[\xi]_{sq^{\prime},\frac{p}{q^{\prime}}}$

for every $\xi\in C^{\infty}_{c}(\Omega)$. In the last estimate, we note that
$q>1/(1-s)$ is equivalent to $0<sq^{\prime}<1$ and hence,
$[\xi]_{sq^{\prime},\frac{p}{q^{\prime}}}$ is finite. Thus, under both each
condition $\varphi\in W^{1,\infty}_{loc}({\mathbb{R}})$ or $\phi\in
W^{1,q}_{\operatorname{loc}}({\mathbb{R}})$ for $q>1/(1-s)$, one has that the
set $C^{\infty}_{c}(\Omega)$ is contained in
$D(\mathcal{E}_{|L^{1\cap\infty}}\varphi)$ and dense in $L^{1}$. Thus Theorem
3.3 implies that under those conditions on $\varphi$, the domain
$D((\partial\mathcal{E})_{|_{L^{1\cap\infty}}}\varphi)$ is dense in $L^{1}$.

Since by Corollary 3.2, the operator
$\overline{(\partial\mathcal{E})_{|_{L^{1\cap\infty}}}\varphi}^{\mbox{}_{L^{1}}}+F$
is $m$-$T$ accretive in $L^{1}$ with complete resolvent, it follows from
standard semigroup theory (e.g. [22, Corollary 4.2]) that for every
$u_{0}\in\overline{D(\mathcal{E}_{|L^{1\cap\infty}}\circ\varphi)}^{\mbox{}_{L^{1}}}$,
there exists a unique mild solution $u$ to problem (1.1). Moreover this mild
solutions satisfies growth estimate (1.13) (see also Lemma 4.3) and (1.14) for
$\nu=1$. The case $\nu=+$ follows in the same way, applying the
$T$-contractivity condition of the resolvent [21]. This completes the proof of
this theorem. ∎

##  4 Extrapolation toward $L^{\infty}$

As in the previous section, we again state these results for Lebesgue spaces
$L^{q}_{\mu}$ with $1\leq q\leq\infty$. We note that the following theorem
generalizes [30, Theorem 2.1] and [29, Theorem 1]. While these two theorems in
[30, 29] are restricted to derive $L^{q}_{\mu}-L^{\infty}_{\mu}$ regularity
estimates of solutions of parabolic diffusion problems with homogeneous
forcing terms $g\equiv 0$ and without Lipschitz perturbations (see also [44]),
the following results can also treat evolution problems involving Lipschitz
continuous nonlinearities and $L^{\infty}_{\mu}$-bounded forcing terms. In
particular we consider mild solutions $u$ to

$\begin{cases}\begin{aligned} u^{\prime}(t)+Au(t)&=g(t)\quad&&\text{for
a.e.~{}}t\in(0,T),\\\ u(0)&=u_{0},\quad&&\end{aligned}\end{cases}$ (4.1)

where $A$ is quasi $m$-accretive in $L^{q_{0}}_{\mu}$ with complete resolvent
for some $1\leq q_{0}<\infty$.

For $\lambda\geq 0$, we define the signed truncator $G_{\lambda}(s):=[\lvert
s\rvert-\lambda]^{+}\operatorname*{sign}(s)$ for every $s\in{\mathbb{R}}$ and
we set $0<T\leq\infty$. Note that $q-1\in[0,\infty)$ so we are using the
notation $\lVert\cdot\rVert_{q-1}$ to denote the usual Lebesgue integral even
when this is not a norm. When $q-1=0$ this is given by

$\lVert
u\rVert_{0}=\int_{\Sigma}\text{sign}_{0}\left(\left|u\right|\right)\mathop{}\\!\mathrm{d}\mu.$

The condition (4.2) for the forcing term can always be satisfied by choosing
$\rho=\infty$ and $\psi=\infty$.

######  Theorem 4.1.

For $1\leq q<r\leq\infty$, $1\leq\sigma<r$, $q\leq\rho\leq\infty$ and
$1<\psi\leq\infty$ satisfying

$\begin{cases}\frac{1}{\rho}<\left(1-\frac{1}{\psi}\right)\left(1-\frac{\sigma}{r}\right)&\text{if
}\sigma\geq q,\\\
\frac{1}{\rho}\leq\left(1-\frac{1}{\psi}\right)\left(\frac{\sigma}{q}-\frac{\sigma}{r}\right)&\text{if
}\sigma<q,\end{cases}$ (4.2)

and

$\frac{1}{\rho}\leq\frac{1}{q}-\frac{\sigma}{r}\left(1-\frac{1}{\psi}\right)$
(4.3)

let $g\in L^{\psi}(0,T;L^{\rho}_{\mu})\cap L^{1}(0,T;L^{q}_{\mu})$ and
$u_{0}\in L^{q}_{\mu}$. Suppose $u\in C([0,T];L^{q}_{\mu})$ satisfies
$u(0)=u_{0}$ and for some $L>0$, $\omega\geq 0$ and every $\lambda\geq 0$, the
“level set energy inequality”

$\begin{split}&\lVert G_{\lambda}(e^{-\omega
t_{2}}u(t_{2}))\rVert_{q}^{q}+L\int_{t_{1}}^{t_{2}}e^{\omega(\sigma-q)s}\lVert
G_{\lambda}(e^{-\omega s}u(s))\rVert_{r}^{\sigma}\mathop{}\\!\mathrm{d}s\\\
&\hskip 56.9055pt\leq\lVert G_{\lambda}(e^{-\omega
t_{1}}u(t_{1}))\rVert_{q}^{q}+q\lambda\,\omega\int_{t_{1}}^{t_{2}}\lVert
G_{\lambda}(e^{-\omega s}u(s))\rVert_{q-1}^{q-1}\mathop{}\\!\mathrm{d}s\\\
&\hskip 56.9055pt\quad+q\int_{t_{1}}^{t_{2}}e^{-\omega
s}\left|[G_{\lambda}(e^{-\omega
s}u),g(s)]_{q}\right|\mathop{}\\!\mathrm{d}s\end{split}$ (4.4)

holds for all $0\leq t_{1}<t_{2}\leq T$. Further, assume $t\mapsto
G_{\lambda}(e^{-\omega t}u(t))$ satisfies the following growth estimate in the
$L^{q}$-norm,

$\begin{split}\lVert G_{\lambda}(e^{-\omega t}u(t))\rVert_{q}\leq&\left\lVert
G_{\lambda}(e^{-\omega s}u(s))\right\rVert_{q}\\\
&+\int_{s}^{t}e^{-\omega\tau}\lVert
g(\tau)\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda}}\rVert_{q}\mathop{}\\!\mathrm{d}\tau\end{split}$
(4.5)

for all $0\leq s<t\leq T$. Then there exists $C>0$ depending on $\sigma$, $q$,
$r$, $\rho$, $\psi$ and $L$ such that

$\begin{split}\lVert u(t)\rVert_{\infty}&\leq
C\max\Biggr{(}e^{\omega\beta_{1}t}\left(\tfrac{1}{t}+\omega\right)^{\frac{1}{\sigma(1-\frac{q}{r})}}\left(\lVert
u_{0}\rVert_{q}+\lVert g\rVert_{L^{1}(0,t;L^{q})}\right)^{\gamma},\\\ &\hskip
45.52458pte^{\omega\beta_{2}t}\,\lVert
g\rVert_{L^{\psi}(0,T;L^{\rho})}^{\eta}\left(\lVert u_{0}\rVert_{q}+\lVert
g\rVert_{L^{1}(0,t;L^{q})}\right)^{\gamma_{\psi}}\Biggr{)}\end{split}$ (4.6)

for all $t\in(0,T]$ with the exponents

$\begin{split}\gamma&=\frac{\frac{1}{\sigma}-\frac{1}{r}}{\frac{1}{q}-\frac{1}{r}},\quad\gamma_{\psi}=\tfrac{\left(1-\frac{1}{\psi}\right)\left(\frac{1}{\sigma}-\frac{1}{r}\right)-\frac{1}{\rho\sigma}}{\left(1-\frac{1}{\psi}\right)\left(\frac{1}{q}-\frac{1}{r}\right)-\frac{1}{\rho\sigma}+\frac{1}{q\sigma}},\quad\eta=\tfrac{\frac{1}{q\sigma}}{\left(1-\frac{1}{\psi}\right)\left(\frac{1}{q}-\frac{1}{r}\right)-\frac{1}{\rho\sigma}+\frac{1}{q\sigma}},\quad\\\
\beta_{1}&=\begin{cases}\frac{\frac{1}{\sigma}-\frac{1}{q}}{\frac{1}{q}-\frac{1}{r}}&\text{if
}\sigma<q,\\\ 0&\text{if }\sigma\geq
q,\end{cases}\quad\beta_{2}=\begin{cases}\eta(q-\sigma)(1-\frac{1}{\psi})&\text{if
}\sigma<q,\\\ 0&\text{if }\sigma\geq q.\end{cases}\end{split}$ (4.7)

Our proof of Theorem 4.1 is based on a DeGiorgi iteration inspired by [30] and
[29]. For this, we modify [45, Chapter 2.5, Lemma 5.6] to prove convergence of
the following recurrence relation.

######  Lemma 4.2.

Let $b\geq 1$, $0<f<1$ and $M\in\mathbb{N}\setminus\set{0}$. Suppose a
sequence $(y_{k})_{k\geq 0}$ in $[0,\infty)$ satisfies the recursion relation

$y_{k+1}\leq b^{k}\,\sum_{i=1}^{M}c_{i}\,y_{k}^{1+\delta_{i}}\qquad\text{for
all $k\in\mathbb{N}$}$

where $c_{i}$, $\delta_{i}$ are positive constants for all
$i\in\set{1,...,M}$. Choose $\delta_{m}=\min_{i\in\set{1,...,M}}\delta_{i}$
and $C=\min_{i\in\set{1,...,M}}\left(c_{i}^{-\frac{1}{\delta_{i}}}\right)$. If

$y_{0}\leq\frac{C}{M}b^{-\frac{1}{\delta_{m}^{2}}},$

then

$y_{k}\leq\frac{C}{M}b^{-\frac{1}{\delta_{m}^{2}}}b^{-\frac{k}{\delta_{m}}}\qquad\text{for
all $k\in\mathbb{N}$.}$ (4.8)

In particular, if $b>1$ then $y_{k}\to 0$ as $k\to\infty$.

###### Proof.

Estimate (4.8) follows via induction with

$\begin{split}y_{k+1}&\leq b^{k}\,\sum_{i=1}^{M}c_{i}\,y_{k}^{1+\delta_{i}}\\\
&\leq\frac{Cb^{k}}{M}\left(b^{-\frac{1}{\delta_{m}^{2}}}b^{-\frac{k}{\delta_{m}}}\right)^{1+\delta_{m}}\,\sum_{i=1}^{M}c_{i}\,\left(\frac{C}{M}\right)^{\delta_{i}}\\\
&\leq\frac{C}{M}b^{-\frac{1}{\delta_{m}^{2}}}b^{-\frac{k+1}{\delta_{m}}}.\end{split}$

∎

###### Proof of Theorem 4.1.

By (4.4), one sees that $G_{\lambda}(u)\in L^{\sigma}(0,T;L^{r}_{\mu})$ for
every $\lambda\geq 0$. Let $\lambda\geq 0$ and $t\in(0,T]$ and for every
integer $k\geq 0$, set

$t_{k}=t(1-2^{-k}),\quad\lambda_{k}=\lambda(1-2^{-k}),\quad
G_{k}(\cdot)=G_{\lambda_{k}}(\cdot),$

and

$U_{k}=\sup_{\hat{s}\in[t_{k},t]}\lVert
G_{k}(e^{-\omega\hat{s}}u(\hat{s}))\rVert_{q}^{q}+L\int_{t_{k}}^{t}e^{\omega(\sigma-q)s}\lVert
G_{k}(e^{-\omega s}u(s))\rVert_{r}^{\sigma}\mathop{}\\!\mathrm{d}s.$

Then the aim is to choose $\lambda\geq 0$ such that $U_{k}\to 0$ as
$k\to\infty$. By the continuity of $t\mapsto\lVert G_{k}(e^{-\omega
t}\,u(t))\rVert_{q}^{q}$, there is an $s_{k}\in(t_{k-1},t_{k})$ satisfying

$\lVert G_{k}(e^{-\omega
s_{k}}\,u(s_{k}))\rVert_{q}^{q}=\tfrac{2^{k}}{t}\int_{t_{k-1}}^{t_{k}}\lVert
G_{k}(e^{-\omega s}\,u(s))\rVert_{q}^{q}\mathop{}\\!\mathrm{d}s.$ (4.9)

Further, note that

$\mathds{1}_{\\{\lvert e^{-\omega
s}u\rvert>\lambda_{k}\\}}\leq\mathds{1}_{\\{\lvert e^{-\omega
s}u\rvert>\lambda_{k-1}\\}}\left(\frac{2^{k}\,[\lvert e^{-\omega
s}u\rvert-\lambda_{k-1}]^{+}}{\lambda}\right)^{\ell}$

for every $\ell\geq 0$. We can then estimate

$|G_{k}(e^{-\omega
s}u(s))|^{q}\leq\left(\frac{2^{k}}{\lambda}\right)^{\ell}|G_{k-1}(e^{-\omega
s}u(s))|^{q+\ell}$ (4.10)

on $[t_{k-1},t]$ for $q\geq 0$ and $\ell\geq 0$. We now aim to obtain a
recurrence relation for $U_{k}$ of the form in Lemma 4.2. Taking a supremum
over $[t_{k},t]$ in (4.4) we can bound $U_{k}$ by

$\begin{split}U_{k}&\leq 2\lVert G_{k}(e^{-\omega
t_{k}}u(t_{k}))\rVert_{q}^{q}+2q\lambda_{k}\,\omega\int_{t_{k}}^{t}\lVert
G_{k}(e^{-\omega s}u(s))\rVert_{q-1}^{q-1}\mathop{}\\!\mathrm{d}s\\\
&\quad+2q\int_{t_{k}}^{t}e^{-\omega s}|[G_{k}(e^{-\omega
s}u)g(s)]_{q}|\mathop{}\\!\mathrm{d}s.\end{split}$ (4.11)

Estimating the first term by Lemma 4.3 and choosing $s_{k}$ according to
(4.9),

$\displaystyle\lVert G_{k}(e^{-\omega t_{k}}u(t_{k}))\rVert_{q}^{q}$
$\displaystyle\quad\leq\left(\lVert G_{k}(e^{-\omega
s_{k}}u(s_{k}))\rVert_{q}+\int_{s_{k}}^{t_{k}}e^{-\omega\tau}\lVert
g(\tau)\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda_{k}}}\rVert_{q}\mathop{}\\!\mathrm{d}\tau\right)^{q}$
$\displaystyle\quad\leq\tfrac{2^{k+q}}{t}\int_{t_{k-1}}^{t_{k}}\lVert
G_{k}(e^{-\omega s}\,u(s))\rVert_{q}^{q}\mathop{}\\!\mathrm{d}s$
$\displaystyle\qquad+2^{q}\left(\int_{s_{k}}^{t_{k}}e^{-\omega\tau}\lVert
g(\tau)\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda_{k}}}\rVert_{q}\mathop{}\\!\mathrm{d}\tau\right)^{q}.$

Separating the $g$ in the second term here by Hölder’s inequality, we have

$\begin{split}&\int_{s_{k}}^{t_{k}}e^{-\omega\tau}\lVert
g(\tau)\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda_{k}}}\rVert_{q}\mathop{}\\!\mathrm{d}\tau\\\
&\quad\leq\left(\int_{s_{k}}^{t_{k}}\lVert
e^{-\omega\tau}g(\tau)\rVert_{\rho}^{\psi}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{1}{\psi}}\left(\int_{s_{k}}^{t_{k}}\lVert\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda_{k}}}\rVert_{\rho_{q}^{\prime}}^{\psi^{\prime}}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{1}{\psi^{\prime}}}\end{split}$

where we choose $\frac{1}{\rho}+\frac{1}{\rho_{q}^{\prime}}=\frac{1}{q}$ and
$\frac{1}{\psi}+\frac{1}{\psi^{\prime}}=1$. We can then estimate $U_{k}$,
extending the time integrals $(t_{k-1},t)$, with

$\begin{split}U_{k}&\leq\tfrac{2^{k+q+1}}{t}\int_{t_{k-1}}^{t}\lVert
G_{k}(e^{-\omega s}\,u(s))\rVert_{q}^{q}\mathop{}\\!\mathrm{d}s\\\
&\quad+2^{q+1}\lVert
g\rVert_{L^{\psi}(t_{k-1},t;L^{\rho}_{\mu})}^{q}\left(\int_{t_{k-1}}^{t}\lVert\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda_{k}}}\rVert_{\rho_{q}^{\prime}}^{\psi^{\prime}}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{q}{\psi^{\prime}}}\\\
&\quad+2q\lambda_{k}\,\omega\int_{t_{k-1}}^{t}\lVert G_{k}(e^{-\omega
s}u(s))\rVert_{q-1}^{q-1}\mathop{}\\!\mathrm{d}s\\\ &\quad+2q\lVert
g\rVert_{L^{\psi}(t_{k-1},t;L^{\rho}_{\mu})}\left(\int_{t_{k-1}}^{t}\lVert
G_{k}(e^{-\omega
s}u(s))\rVert_{(q-1)\rho^{\prime}}^{(q-1)\psi^{\prime}}\mathop{}\\!\mathrm{d}s\right)^{\frac{1}{\psi^{\prime}}}.\end{split}$

where we choose $\frac{1}{\rho}+\frac{1}{\rho^{\prime}}=1$.

We apply (4.10) to each $G_{k}$ term, as well as
$\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda_{k}}}$, with
$\ell=\varepsilon_{1}$, $1+\varepsilon_{1}$,
$q+\varepsilon_{2}-\rho_{q}^{\prime}$ and
$q+\varepsilon_{3}-(q-1)\rho^{\prime}$. The positive constants
$\varepsilon_{1}$, $\varepsilon_{2}$ and $\varepsilon_{3}$ will later be
chosen such that $\ell\geq 0$ in each case and an appropriate recurrence
relation may be obtained. Note that the requirement $\ell\geq 0$ will be
satisfied as a result of assumption (4.2). Then noting that
$\lambda_{k}<\lambda$, there exists $C>0$ depending on $q$ such that

$\begin{split}\frac{U_{k}}{C}&\leq\frac{2^{k(1+\varepsilon_{1})}}{\lambda^{\varepsilon_{1}}}\left(\tfrac{1}{t}+\omega\right)\int_{t_{k-1}}^{t}\lVert
G_{k-1}(e^{-\omega
s}u(s))\rVert^{q+\varepsilon_{1}}_{q+\varepsilon_{1}}\;\mathop{}\\!\mathrm{d}s\\\
&+\left(\tfrac{2^{k}}{\lambda}\right)^{\frac{q(q+\varepsilon_{2})}{\rho_{q}^{\prime}}}\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho_{q}}_{\mu})}^{q}\left(\int_{t_{k-1}}^{t}\lVert
G_{k-1}(e^{-\omega
s}u(s))\rVert_{q+\varepsilon_{2}}^{(q+\varepsilon_{2})\frac{\psi^{\prime}}{\rho_{q}^{\prime}}}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{q}{\psi^{\prime}}}\\\
&+\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho}_{\mu})}\left(\tfrac{2^{k}}{\lambda}\right)^{\frac{q+\varepsilon_{3}}{\rho^{\prime}}-(q-1)}\left(\int_{t_{k-1}}^{t}\lVert
G_{k-1}(e^{-\omega
s}u(s))\rVert_{q+\varepsilon_{3}}^{(q+\varepsilon_{3})\frac{\psi^{\prime}}{\rho^{\prime}}}\mathop{}\\!\mathrm{d}s\right)^{\frac{1}{\psi^{\prime}}}.\end{split}$
(4.12)

Now it remains to recover $U_{k-1}$ from integrals of the form

$\int_{t_{k-1}}^{t}\lVert G_{k-1}(e^{-\omega
s}u(s))\rVert_{q+\varepsilon}^{(q+\varepsilon)M}\mathop{}\\!\mathrm{d}s$
(4.13)

where $\varepsilon>0$ and $M>0$. In particular, we set
$q_{\varepsilon}:=q+\varepsilon$ and choose $\varepsilon$ as follows. To
obtain $U_{k-1}$ from (4.13) we will apply Hölder’s inequality, so choose
$\varepsilon>0$ and $\theta\in[0,1]$ such that

$\frac{1}{q_{\varepsilon}}=\frac{\theta}{q}+\frac{1-\theta}{r}\quad\text{and}\quad(1-\theta)q_{\varepsilon}M=\sigma.$

In particular, we choose

$\varepsilon=\begin{cases}\frac{\sigma}{M}\left(1-\frac{q}{r}\right)&\text{if
}r<\infty,\\\ \frac{\sigma}{M}&\text{if }r=\infty,\end{cases}$ (4.14)

and

$\theta=\begin{cases}1-\frac{1}{1+q\left(\frac{M}{\sigma}-\frac{1}{r}\right)}&\text{if
}r<\infty,\\\ \frac{Mq}{\sigma+Mq}&\text{if }r=\infty,\end{cases}$ (4.15)

satisfying $\theta<1$ and $\varepsilon>0$ given that $M>0$. The condition
$\theta\geq 0$ requires that $M\geq\frac{\sigma}{r}$. Since we take $M=1$,
$\frac{\psi^{\prime}}{\rho_{q}^{\prime}}$ and
$\frac{\psi^{\prime}}{\rho^{\prime}}$ in the case of (4.12), this is satisfied
by assumptions (4.2) and (4.3). Then applying standard $L^{p}$ interpolation
with $\theta$,

$\displaystyle\int_{t_{k-1}}^{t}\lVert G_{k-1}(e^{-\omega
s}u(s))\rVert_{q_{\varepsilon}}^{q_{\varepsilon}M}\;\mathop{}\\!\mathrm{d}s$
$\displaystyle\hskip
56.9055pt\leq\int_{t_{k-1}}^{t}e^{-\omega(\sigma-q)s}\left(\lVert
G_{k-1}(e^{-\omega s}u(s))\rVert^{q}_{q}\right)^{\frac{\theta
q_{\varepsilon}M}{q}}\times$ $\displaystyle\hskip
85.35826pte^{\omega(\sigma-q)s}\lVert G_{k-1}(e^{-\omega
s}u(s))\rVert^{(1-\theta)q_{\varepsilon}M}_{r}\;\mathop{}\\!\mathrm{d}s$
$\displaystyle\hskip
56.9055pt\leq\frac{1}{L}\sup_{\hat{s}\in[t_{k-1},t]}e^{-\omega(\sigma-q)\hat{s}}\left(\lVert
G_{k-1}(e^{-\omega\hat{s}}u(\hat{s}))\rVert_{q}^{q}\,\right)^{\frac{\theta
q_{\varepsilon}M}{q}}\times$ $\displaystyle\hskip
85.35826ptL\int_{t_{k-1}}^{t}e^{\omega(\sigma-q)s}\lVert G_{k-1}(e^{-\omega
s}u(s))\rVert^{\sigma}_{r}\;\mathop{}\\!\mathrm{d}s.$

We estimate $e^{-\omega(\sigma-q)s}$ on $[t_{k-1},t]$ according to the sign of
$\sigma-q$ so that

$\sup_{s\in[t_{k-1},t]}e^{-\omega(\sigma-q)s}=\begin{cases}e^{-\omega(\sigma-q)t}&\text{if
}\sigma\leq q,\\\ 1&\text{if }\sigma>q.\end{cases}$

Hence, applying Young’s inequality such that both terms have the same exponent
and evaluating, we have

$\int_{t_{k-1}}^{t}\lVert G_{k-1}(e^{-\omega
s}u(s))\rVert_{q_{\varepsilon}}^{q_{\varepsilon}M}\;\mathop{}\\!\mathrm{d}s\leq\frac{1}{L}e^{\omega
t(q-\sigma)^{+}}U_{k-1}^{M-\frac{\sigma}{r}+1}$

where $(q-\sigma)^{+}=\max(0,q-\sigma)$.

To apply Lemma 4.2, the exponents of $U_{k-1}$ corresponding to (4.12) must be
of the form $1+\delta$ with $\delta>0$. Hence we require

$\begin{split}&\frac{q}{\rho_{q}^{\prime}}+\frac{q}{\psi^{\prime}}\left(1-\frac{\sigma}{r}\right)>1\quad\text{
and
}\quad\frac{1}{\rho^{\prime}}+\frac{1}{\psi^{\prime}}\left(1-\frac{\sigma}{r}\right)>1\end{split}$

which follow from (4.2). Rewriting (4.12) as a recurrence relation for
$U_{k+1}$, we introduce the following constants

$\begin{split}c_{1}&=\left(\tfrac{1}{\lambda}\right)^{\varepsilon_{1}}\left(\tfrac{1}{t}+\omega\right)e^{\omega
t(q-\sigma)^{+}},\quad
c_{2}=\left(\tfrac{1}{\lambda}\right)^{\frac{q(q+\varepsilon_{2})}{\rho_{q}^{\prime}}}\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho_{q}}_{\mu})}^{q}e^{\frac{q\omega
t}{\psi^{\prime}}(q-\sigma)^{+}},\\\
c_{3}&=\left(\tfrac{1}{\lambda}\right)^{\frac{q+\varepsilon_{3}}{\rho^{\prime}}-q+1}\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho}_{\mu})}e^{\frac{\omega
t}{\psi^{\prime}}(q-\sigma)^{+}},\quad
b=\max\left(2^{1+\varepsilon_{1}},2^{\frac{q(q+\varepsilon_{2})}{\rho_{q}^{\prime}}},2^{\frac{q+\varepsilon_{3}}{\rho^{\prime}}-q+1}\right),\end{split}$

and exponents

$\delta_{1}=1-\frac{\sigma}{r},\quad\delta_{2}=\frac{q}{\psi^{\prime}}\left(1-\frac{\sigma}{r}\right)+\frac{q}{\rho_{q}^{\prime}}-1,\quad\delta_{3}=\frac{1}{\psi^{\prime}}\left(1-\frac{\sigma}{r}\right)+\frac{1}{\rho^{\prime}}-1.$

Then we obtain

$U_{k+1}\leq b^{k+1}\sum_{i=1}^{3}Cc_{i}\,U_{k}^{1+\delta_{i}}$

for some $C>0$ depending on $q$, $L$ and $\psi$. Then setting

$\delta_{m}:=\min\left(\delta_{1},\delta_{2},\delta_{3}\right)=\delta_{3},$

in order to apply Lemma 4.2, we require that

$U_{0}\leq\frac{1}{3\,b^{\frac{1}{\delta_{m}^{2}}}}\min_{i\in\set{1,2,3}}\frac{1}{(Cc_{i})^{\frac{1}{\delta_{i}}}}.$
(4.16)

We estimate $U_{0}$ by (4.11) and (4.5), so that

$\displaystyle U_{0}$ $\displaystyle\leq 2\left(\lVert
u_{0}\rVert_{q}^{q}+q\int_{0}^{t}\lVert e^{-\omega
s}u(s)\rVert_{q}^{q-1}e^{-\omega s}\lVert
g(s)\rVert_{q}\mathop{}\\!\mathrm{d}s\right)$ $\displaystyle\leq 2\left(\lVert
u_{0}\rVert_{q}^{q}+q\left(\lVert u_{0}\rVert_{q}+\int_{0}^{t}e^{-\omega
r}\lVert
g(r)\rVert_{q}\mathop{}\\!\mathrm{d}r\right)^{q-1}\int_{0}^{t}e^{-\omega
s}\lVert g(s)\rVert_{q}\mathop{}\\!\mathrm{d}s)\right)$ $\displaystyle\leq
2(1+q)\left(\lVert u_{0}\rVert_{q}+\int_{0}^{t}e^{-\omega s}\lVert
g(s)\rVert_{q}\mathop{}\\!\mathrm{d}s\right)^{q}.$

As the previous estimates were for arbitrary $\lambda\geq 0$, relabelling
$C>0$ to include $b$, we want to find $\lambda$ such that

$c_{i}\leq\frac{C}{\left(\lVert u_{0}\rVert_{q}+\lVert
g\rVert_{L^{1}(0,t;L^{q})}\right)^{q\delta_{i}}}$ (4.17)

for $i\in\set{1,2,3}$. Set

$\begin{split}\beta_{1}&=\begin{cases}\frac{q-\sigma}{\varepsilon_{1}}&\text{if
}\sigma\leq q,\\\ 0&\text{if }\sigma>q,\end{cases}\\\
\kappa_{1}&=\begin{cases}\frac{(q-\sigma)\rho_{q}^{\prime}}{\psi^{\prime}q(q+\varepsilon_{2})}&\text{if
}\sigma\leq q,\\\ 0&\text{if }\sigma>q,\end{cases}\\\
\kappa_{2}&=\begin{cases}\frac{q-\sigma}{\psi^{\prime}\left(\frac{q+\varepsilon_{3}}{\rho^{\prime}}-q+1\right)}&\text{if
}\sigma\leq q,\\\ 0&\text{if }\sigma>q.\end{cases}\end{split}$

Then (4.17) holds if

$\begin{split}\lambda&\geq
Ce^{\omega\beta_{1}t}\left(\left(\tfrac{2^{q}}{t}+q\omega\right)\left(\lVert
u_{0}\rVert_{q}+\int_{0}^{t}e^{-\omega s}\lVert
g(s)\rVert_{q}\mathop{}\\!\mathrm{d}s\right)^{q\delta_{1}}\right)^{\frac{1}{\varepsilon_{1}}},\\\
\lambda&\geq Ce^{\omega\kappa_{1}t}\left(\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho_{q}}_{\mu})}^{q}\left(\lVert
u_{0}\rVert_{q}+\int_{0}^{t}e^{-\omega s}\lVert
g(s)\rVert_{q}\mathop{}\\!\mathrm{d}s\right)^{q\delta_{2}}\right)^{\frac{\rho_{q}^{\prime}}{q(q+\varepsilon_{2})}},\text{
and}\\\ \lambda&\geq Ce^{\omega\kappa_{2}t}\left(\lVert
g\rVert_{L^{\psi}(0,t;L^{\rho}_{\mu})}\left(\lVert
u_{0}\rVert_{q}+\int_{0}^{t}e^{-\omega s}\lVert
g(s)\rVert_{q}\mathop{}\\!\mathrm{d}s\right)^{q\delta_{3}}\right)^{\frac{1}{\frac{q+\varepsilon_{3}}{\rho^{\prime}}-(q-1)}}.\end{split}$

for some $C>0$ depending on $q,\sigma,r,\rho,\psi$ and $L$. So taking
$\lambda$ as the maximum of these estimates, we have by Fatou’s Lemma,

$0=\liminf_{k\to\infty}U_{k}\geq\sup_{\hat{s}\in[t,t]}\lVert
G_{\lambda}(e^{-\omega\hat{s}}u(\hat{s}))\rVert_{q}^{q}\,+\int_{t}^{t}e^{\omega(\sigma-q)t}\lVert
G_{\lambda}(e^{-\omega s}u(s))\rVert_{r}^{\sigma}\mathop{}\\!\mathrm{d}s.$

Noting that $t$ was chosen arbitrarily, this implies that

$\lVert u(t)\rVert_{\infty}\leq\lambda\quad\text{for all }t\in(0,T].$

Evaluating constants and simplifying, we obtain (4.6). ∎

The following lemma shows that the growth condition on $G_{\lambda}(e^{-\omega
t}u(t))$ given by (4.5) holds for operators with complete resolvent. In the
case $\lambda=0$ this reduces to the standard growth estimate for accretive
operators with complete resolvent.

######  Lemma 4.3.

Let $1\leq q_{0}<\infty$ and suppose $A$ is $\omega$-quasi $m$-accretive in
$L^{q_{0}}_{\mu}$ with complete resolvent for some $\omega\geq 0$. Let $1\leq
q\leq\infty$ such that $g\in L^{1}(0,T;L^{q}_{\mu}\cap
L^{q+\varepsilon}_{\mu})$ for some $\varepsilon>0$ and
$u_{0}\in\overline{D(A)}^{\mbox{}_{L^{q_{0}}_{\mu}}}\cap L^{q}_{\mu}$. Denote
by $u(t)$ the mild solution to (4.1). Then we have the growth estimate

$\begin{split}\lVert G_{\lambda}(e^{-\omega t}u(t))\rVert_{q}\leq&\left\lVert
G_{\lambda}(e^{-\omega s}u(s))\right\rVert_{q}\\\
&+\int_{s}^{t}e^{-\omega\tau}\lVert
g(\tau)\mathds{1}_{\set{e^{-\omega\tau}}{u(\tau)|>\lambda}}\rVert_{q}\mathop{}\\!\mathrm{d}\tau\end{split}$
(4.18)

for all $0\leq s\leq t\leq T$ and $\lambda\geq 0$.

###### Proof.

For $u\in D(J_{h}^{A})$, we can rewrite the resolvent operator in the
following way,

$J_{\frac{h}{1-h\omega}}^{A+\omega I}u=(1-h\omega)J_{h}^{A}u.$

Then for $A+\omega I$ having complete resolvent, consider
$\alpha\in{\mathbb{R}}$ and take $j(\cdot)=|G_{\lambda}(\alpha\cdot)|^{q}$ in
the complete resolvent property (2.13) with the resolvent operator
$J_{\frac{h}{1-h\omega}}^{A+\omega I}$ to obtain the estimate

$\begin{split}\int_{\Sigma}\left|G_{\lambda}(\alpha
v)\right|^{q}\mathop{}\\!\mathrm{d}\mu&\geq\int_{\Sigma}\left|G_{\lambda}\left(\alpha
J_{\frac{h}{1-h\omega}}^{A+\omega
I}v\right)\right|^{q}\mathop{}\\!\mathrm{d}\mu\\\ &=\left\lVert
G_{\lambda}\left(\alpha(1-h\omega)J_{h}^{A}v\right)\right\rVert_{q}^{q}.\end{split}$
(4.19)

Given $s<t$, take a partition $(s_{n})_{n\in\set{0,1,...,N}}$ of $[s,t]$ given
by $s_{n}=s+\frac{n(t-s)}{N}$. Let

$g_{n}:=\frac{N}{t-s}\int_{s_{n}}^{s_{n+1}}g(\tau)\mathop{}\\!\mathrm{d}\tau.$
(4.20)

Then let $(v_{n})_{n\in\set{0,1,...,N}}$ be the solution to the discrete
problem

$\begin{cases}\begin{aligned}
v_{n}+\frac{t-s}{N}Av_{n}&=v_{n-1}+\frac{t-s}{N}g_{n-1}\quad\text{ for
}n=1,...,N,\\\ v_{0}&=u(s).\end{aligned}\end{cases}$

We can apply the resolvent estimate (4.19) to $v_{n}$, taking
$h=\frac{t-s}{N}$. Further, let $S_{n}=\\{x\in\Sigma:\frac{e^{-\omega
t}}{(1-h\omega)^{N-n}}\left|v_{n}+hg_{n}\right|>\lambda\\}$ so that we may
separate terms.

$\begin{split}&\left\lVert G_{\lambda}\left(\frac{e^{-\omega
t}}{(1-h\omega)^{N-n}}v_{n}\right)\right\rVert_{q}\leq\left\lVert
G_{\lambda}\left(\frac{e^{-\omega
t}}{(1-h\omega)^{N-n+1}}\left(v_{n-1}+hg_{n-1}\right)\right)\right\rVert_{q}\\\
&\hskip 17.07182pt\leq\left\lVert G_{\lambda}\left(\frac{e^{-\omega
t}}{(1-h\omega)^{N-n+1}}v_{n-1}\right)\right\rVert_{q}+\frac{e^{-\omega
t}h}{(1-h\omega)^{N-n+1}}\left\lVert
g_{n-1}\mathds{1}_{S_{n-1}}\right\rVert_{q}.\end{split}$

Repeating this, we have

$\begin{split}\lVert G_{\lambda}(e^{-\omega t}v_{N})\rVert_{q}&\leq\left\lVert
G_{\lambda}\left(\frac{e^{-\omega
t}}{\left(1-\tfrac{\omega(t-s)}{N}\right)^{N}}v_{0}\right)\right\rVert_{q}\\\
&\quad+\frac{t-s}{N}\sum_{n=0}^{N-1}\frac{e^{-\omega
t}}{\left(1-\tfrac{\omega(t-s)}{N}\right)^{N-n}}\left\lVert
g_{n}\mathds{1}_{S_{n}}\right\rVert_{q}\end{split}$

which converges to (4.18) as $N\rightarrow\infty$ by the definition of mild
solution and the projection (4.20). ∎

The following proposition introduces the pointwise estimate (4.21) for
operators with complete resolvent which we will use as the condition for
applying Theorem 4.1 to the doubly nonlinear problem (1.1) (see Section 5). In
particular, this provides (4.4).

######  Proposition 4.4.

For $1\leq q_{0}<\infty$ and $\omega\geq 0$, let $A$ be an $\omega$-quasi
$m$-accretive operator on $L^{q_{0}}_{\mu}$ with complete resolvent. Suppose
there are $q_{0}\leq q<r\leq\infty$, $1\leq\sigma<r$ and $C>0$ such that $A$
satisfies the one-parameter Sobolev type inequality

$\lVert G_{\lambda}(u)\rVert_{r}^{\sigma}\leq
C\;[G_{\lambda}(u),v+\omega(G_{\lambda}(u)+\lambda\mathds{1})]_{q}$ (4.21)

for every $(u,v)\in A$ and $\lambda\geq 0$. Let $g\in
L^{1}(0,T;L^{q_{0}}_{\mu})\cap L^{1}(0,T;L^{q+\varepsilon}_{\mu})$ for some
$\varepsilon>0$ and $u\in C([0,T];L^{q_{0}}_{\mu})\cap
L^{1}(0,T;L^{1\cap\infty}_{\mu})$ be the mild solution to (4.1) where
$u_{0}\in\overline{D(A)}^{\mbox{}_{L^{q_{0}}_{\mu}}}\cap
L^{1\cap\infty}_{\mu}$. Then for every $\lambda\geq 0$, $u$ satisfies the
“level set energy inequality”

$\begin{split}&\lVert G_{\lambda}(e^{-\omega
t_{2}}\,u(t_{2}))\rVert_{q}^{q}+\frac{q}{C}\int_{t_{1}}^{t_{2}}e^{\omega(\sigma-q)s}\lVert
G_{\lambda}(e^{-\omega s}\,u(s))\rVert_{r}^{\sigma}\mathop{}\\!\mathrm{d}s\\\
&\hskip 56.9055pt\leq\lVert G_{\lambda}(e^{-\omega
t_{1}}\,u(t_{1}))\rVert_{q}^{q}+\lambda\,\omega\int_{t_{1}}^{t_{2}}\lVert
G_{\lambda}(e^{-\omega s}\,u(s))\rVert_{q-1}^{q-1}\mathop{}\\!\mathrm{d}s\\\
&\hskip 56.9055pt\quad+q\int_{t_{1}}^{t_{2}}e^{-\omega
s}\left|[G_{\lambda}(e^{-\omega
s}u(s)),g]_{q}\right|\mathop{}\\!\mathrm{d}s\end{split}$ (4.22)

for all $0\leq t_{1}<t_{2}\leq T$.

###### Proof.

Let $\set{s_{n}}_{n\in\set{0,1,...,N}}$ be the discretization of the interval
$[t_{1},t_{2}]$ given by $s_{n}:=t_{1}+\frac{n(t_{2}-t_{1})}{N}$. Then for all
$n\in\set{0,...,N-1}$ set

$g_{N}(s):=\frac{N}{t_{2}-t_{1}}\int_{s_{n}}^{s_{n+1}}g(\tau)\mathop{}\\!\mathrm{d}\tau\quad\text{for
}s\in[s_{n},s_{n+1})$

which will converge to $g$ in $L^{1}(0,T;L^{q_{0}}_{\mu})\cap
L^{1}(0,T;L^{q}_{\mu})$ as $N\rightarrow\infty$. Let
$\set{v_{n}}_{n\in\set{0,1,...,N}}$ be the associated family of solutions to
the time discretized Cauchy problem satisfying

$v_{n+1}=J_{\frac{t_{2}-t_{1}}{N}}^{A}\left(v_{n}+\frac{t_{2}-t_{1}}{N}g_{N}(s_{n})\right)$
(4.23)

for all $n\in\set{0,...,N-1}$ with $v_{0}=u(t_{1})$. Note that by the complete
resolvent property of $A$ with $u_{0}\in L^{q}_{\mu}$ and $g\in
L^{1}(0,T;L^{q}_{\mu})$, $v_{n}\in L^{q}_{\mu}$ for all $n\in\set{0,...,N}$.
We first obtain a discrete version of the integral estimate (4.22) by
discretizing with a telescoping sum and applying a product rule. For $q\geq 1$
we use the following property of $q$-brackets,

$[u,v]_{q}\leq\frac{1}{q}\lVert u+v\rVert_{q}^{q}-\frac{1}{q}\lVert
u\rVert_{q}^{q}$ (4.24)

for every $u,v\in L^{q}_{\mu}$. Here we apply (4.24) to the following
telescoping sum, taking $u=G_{\lambda}(e^{-\omega s_{n}}v_{n})$ and
$v=G_{\lambda}(e^{-\omega s_{n-1}}v_{n-1})-G_{\lambda}(e^{-\omega
s_{n}}v_{n})$.

$\displaystyle\lVert G_{\lambda}(e^{-\omega t_{2}}v_{N})\rVert_{q}^{q}-\lVert
G_{\lambda}(e^{-\omega t_{1}}u(t_{1}))\rVert_{q}^{q}$ $\displaystyle\hskip
68.28644pt=\sum_{n=1}^{N}\lVert G_{\lambda}(e^{-\omega
s_{n}}v_{n})\rVert_{q}^{q}-\lVert G_{\lambda}(e^{-\omega
s_{n-1}}v_{n-1})\rVert_{q}^{q}$ $\displaystyle\hskip
68.28644pt\leq\sum_{n=1}^{N}q[G_{\lambda}(e^{-\omega
s_{n}}v_{n}),G_{\lambda}(e^{-\omega s_{n}}v_{n})-G_{\lambda}(e^{-\omega
s_{n-1}}v_{n-1})]_{q}.$

Noting that $G_{\lambda}$ is a Lipschitz continuous function, we can
differentiate almost everywhere on $\mathbb{R}$. Here we define

$G_{\lambda}^{\prime}(s)=\begin{cases}1&\text{ if }|s|>\lambda,\\\ 0&\text{ if
}|s|\leq\lambda,\end{cases}$

and

$c_{n}=\int_{0}^{1}G_{\lambda}^{\prime}(\theta e^{-\omega
s_{n}}v_{n}+(1-\theta)e^{-\omega s_{n-1}}v_{n-1})d\theta$

so that we can rewrite this difference as an integral of the derivative with

$G_{\lambda}(e^{-\omega s_{n}}v_{n})-G_{\lambda}(e^{-\omega
s_{n-1}}v_{n-1})=c_{n}\left(e^{-\omega s_{n}}v_{n}-e^{-\omega
s_{n-1}}v_{n-1}\right).$

Then returning to the estimate in the discrete setting,

$\displaystyle\lVert G_{\lambda}(e^{-\omega t_{2}}v_{N})\rVert_{q}^{q}-\lVert
G_{\lambda}(e^{-\omega t_{1}}u(t_{1}))\rVert_{q}^{q}$ $\displaystyle\hskip
31.2982pt\leq q\sum_{n=1}^{N}[G_{\lambda}(e^{-\omega
s_{n}}v_{n}),c_{n}\left(e^{-\omega s_{n}}v_{n}-e^{-\omega
s_{n-1}}v_{n-1}\right)]_{q}$ $\displaystyle\hskip 31.2982pt\leq
q\sum_{n=1}^{N}[G_{\lambda}(e^{-\omega s_{n}}v_{n}),c_{n}e^{-\omega
s_{n-1}}(v_{n}-v_{n-1})]_{q}$ $\displaystyle\hskip
31.2982pt\quad+q\sum_{n=1}^{N}[G_{\lambda}(e^{-\omega
s_{n}}v_{n}),c_{n}(e^{-\omega s_{n}}-e^{-\omega s_{n-1}})v_{n}]_{q}.$

Defining

$R_{n}=[G_{\lambda}(e^{-\omega s_{n}}v_{n}),(c_{n}-1)(v_{n}-v_{n-1})]_{q},$

we can rewrite the previous estimate as

$\begin{split}&\lVert G_{\lambda}(e^{-\omega t_{2}}v_{N})\rVert_{q}^{q}-\lVert
G_{\lambda}(e^{-\omega t_{1}}u(t_{1}))\rVert_{q}^{q}\\\ &\hskip
19.91684pt\leq\frac{q(t_{2}-t_{1})}{N}\sum_{n=1}^{N}e^{-\omega
s_{n-1}}[G_{\lambda}(e^{-\omega s_{n}}v_{n}),-Av_{n}+g_{N}(s_{n-1})]_{q}\\\
&\hskip 19.91684pt\quad+q\sum_{n=1}^{N}e^{-\omega
s_{n-1}}R_{n}+q\sum_{n=1}^{N}[G_{\lambda}(e^{-\omega s_{n}}v_{n}),e^{-\omega
s_{n}}v_{n}c_{n}]_{q}\left(1-e^{\frac{\omega(t_{2}-t_{1})}{N}}\right).\end{split}$

We now consider the values of $v_{n}$ and $v_{n-1}$ for almost every
$x\in\Sigma$ to show that $R_{n}$ is non-positive. Note that $c_{n}\leq 1$ and
in particular,

$\displaystyle c_{n}(x)\in\begin{cases}\set{0}&\text{ if }|\theta e^{-\omega
s_{n}}v_{n}+(1-\theta)e^{-\omega s_{n-1}}v_{n-1}|\leq\lambda\text{ for
a.e.~{}}\theta\in(0,1),\\\ \set{1}&\text{ if }|\theta e^{-\omega
s_{n}}v_{n}+(1-\theta)e^{-\omega s_{n-1}}v_{n-1}|>\lambda\text{ for
a.e.~{}}\theta\in(0,1),\\\ (0,1)&\text{ otherwise}.\end{cases}$

Since $|e^{-\omega s_{n}}v_{n}(x)|\leq\lambda$ implies that
$G_{\lambda}(e^{-\omega s_{n}}v_{n}(x))=0$ and so does not contribute to
$R_{n}$, we consider only $x\in\Sigma$ such that $|e^{-\omega
s_{n}}v_{n}(x)|>\lambda$. Then there will be some subinterval of
$\theta\in(0,1)$ such that

$|\theta e^{-\omega s_{n}}v_{n}(x)+(1-\theta)e^{-\omega
s_{n-1}}v_{n-1}(x)|>\lambda$

implying that $c_{n}(x)>0$. If $c_{n}=1$ then $c_{n}(x)-1=0$ and so this will
not contribute to $R_{n}$. Hence we only consider $x$ such that
$c_{n}(x)\in(0,1)$. Since $|e^{-\omega s_{n}}v_{n}(x)|>\lambda$, this implies
that either

$|e^{-\omega s_{n-1}}v_{n-1}(x)|\leq\lambda$

or

$\operatorname*{sign}(v_{n-1}(x))=-\operatorname*{sign}(v_{n}(x)).$

For the first case, $G_{\lambda}(e^{-\omega s_{n-1}}v_{n-1}(x))=0$ so

$\displaystyle(G_{\lambda}(e^{-\omega
s_{n}}v_{n}(x)))^{q-1}(v_{n}(x)-v_{n-1}(x))$
$\displaystyle\quad=\left((G_{\lambda}(e^{-\omega
s_{n}}v_{n}(x)))^{q-1}-(G_{\lambda}(e^{-\omega
s_{n-1}}v_{n-1}(x)))^{q-1}\right)(v_{n}(x)-v_{n-1}(x))$
$\displaystyle\quad\geq 0.$

Note that for $q=1$,
$(G_{\lambda}(v_{n}(x)))^{q-1}=\operatorname*{sign}(G_{\lambda}(v_{n}(x)))$.
For the second case,
$\operatorname*{sign}(v_{n}(x)-v_{n-1}(x))=\operatorname*{sign}(v_{n}(x))$ so

$(G_{\lambda}(e^{-\omega s_{n}}v_{n}(x)))^{q-1}(v_{n}(x)-v_{n-1}(x))\geq 0.$

Putting this together we have that $R_{n}\leq 0$. Returning to the discrete
estimate,

$\begin{split}\lVert G_{\lambda}(e^{-\omega t_{2}}v_{N})\rVert_{q}^{q}&-\lVert
G_{\lambda}(e^{-\omega t_{1}}u(t_{1}))\rVert_{q}^{q}\\\
&\leq\frac{q(t_{2}-t_{1})}{N}\sum_{n=1}^{N}e^{-\omega
s_{n-1}}[G_{\lambda}(e^{-\omega s_{n}}v_{n}),-Av_{n}+g_{N}(s_{n})]_{q}\\\
&\quad+\sum_{n=1}^{N}\lVert G_{\lambda}(e^{-\omega
s_{n}}v_{n})\rVert_{q}^{q}\left(1-e^{\frac{q\omega(t_{2}-t_{1})}{N}}\right).\end{split}$

Note that by (4.21),

$\displaystyle e^{-\omega s_{n-1}}$ $\displaystyle[G_{\lambda}(e^{-\omega
s_{n}}v_{n}),Av_{n}]_{q}=e^{-\omega s_{n-1}}e^{-(q-1)\omega s_{n}}[G_{\lambda
e^{\omega s_{n}}}(v_{n}),Av_{n}]_{q}$
$\displaystyle=e^{\frac{\omega(t_{2}-t_{1})}{N}}e^{-q\omega s_{n}}[G_{\lambda
e^{\omega s_{n}}}(v_{n}),Av_{n}]_{q}$ $\displaystyle\geq
e^{\frac{\omega(t_{2}-t_{1})}{N}}e^{-q\omega s_{n}}\frac{1}{C}\lVert
G_{\lambda e^{\omega s_{n}}}(v_{n})\rVert_{r}^{\sigma}$
$\displaystyle\quad-e^{\frac{\omega(t_{2}-t_{1})}{N}}e^{-q\omega
s_{n}}\omega\left(\lVert G_{\lambda e^{\omega
s_{n}}}(v_{n})\rVert_{q}^{q}+\lambda e^{\omega s_{n}}\lVert G_{\lambda
e^{\omega s_{n}}}(v_{n})\rVert_{q-1}^{q-1}\right)$
$\displaystyle=e^{\frac{\omega(t_{2}-t_{1})}{N}}\frac{e^{\omega
s_{n}(\sigma-q)}}{C}\lVert G_{\lambda}(e^{-\omega
s_{n}}v_{n})\rVert_{r}^{\sigma}$
$\displaystyle\quad-e^{\frac{\omega(t_{2}-t_{1})}{N}}\omega\left(\lVert
G_{\lambda}(e^{-\omega s_{n}}v_{n})\rVert_{q}^{q}+\lambda\lVert
G_{\lambda}(e^{-\omega s_{n}}v_{n})\rVert_{q-1}^{q-1}\right).$

Hence we have

$\begin{split}e^{-\omega s_{n-1}}&[G_{\lambda}(e^{-\omega
s_{n}}v_{n}),Av_{n}]_{q}\\\ &\geq\frac{e^{\omega s_{n}(\sigma-q)}}{C}\lVert
G_{\lambda}(e^{-\omega s_{n}}v_{n})\rVert_{r}^{\sigma}\\\
&\quad-e^{\frac{\omega(t_{2}-t_{1})}{N}}\omega\left(\lVert
G_{\lambda}(e^{-\omega s_{n}}v_{n})\rVert_{q}^{q}+\lambda\lVert
G_{\lambda}(e^{-\omega s_{n}}v_{n})\rVert_{q-1}^{q-1}\right).\end{split}$

We now aim to take $N\rightarrow\infty$, first converting the discrete sums to
integrals. Let $U_{N}$ be a stepwise solution to (4.23) such that

$U_{N}(s)=v_{0}\mathds{1}_{\set{0}}(s)+\sum_{n=1}^{N}v_{n}\mathds{1}_{(s_{n-1},s_{n}]}(s)$

for every $s\in[t_{1},t_{2}]$. We have

$\displaystyle\lVert G_{\lambda}(e^{-\omega t_{2}}v_{N})\rVert_{q}^{q}-\lVert
G_{\lambda}(e^{-\omega t_{1}}u(t_{1}))\rVert_{q}^{q}$
$\displaystyle\quad\leq-\frac{q(t_{2}-t_{1})}{N}\sum_{n=1}^{N}\left(\frac{e^{\omega
s_{n}(\sigma-q)}}{C}\lVert G_{\lambda}(e^{-\omega
s_{n}}v_{n})\rVert_{r}^{\sigma}\right.$
$\displaystyle\qquad+e^{\frac{\omega(t_{2}-t_{1})}{N}}\omega[G_{\lambda}(e^{-\omega
s_{n}}v_{n}),G_{\lambda}(e^{-\omega
s_{n}}v_{n})-\lambda\operatorname*{sign}(G_{\lambda}(e^{-\omega
s_{n}}v_{n}))]_{q}$ $\displaystyle\qquad+e^{-\omega
s_{n-1}}[G_{\lambda}(e^{-\omega s_{n}}v_{n}),g_{N}(s_{n})]_{q}\Biggr{)}$
$\displaystyle\qquad+\sum_{n=1}^{N}\lVert G_{\lambda}(e^{-\omega
s_{n}}v_{n})\rVert_{q}^{q}\left(1-e^{\frac{q\omega(t_{2}-t_{1})}{N}}\right)$
$\displaystyle\quad\leq-\frac{q}{C}\int_{t_{1}}^{t_{2}}e^{\omega(\sigma-q)\left(s-\operatorname*{sign}(\sigma-q)\frac{t_{2}-t_{1}}{N}\right)}\left\lVert
G_{\lambda}(e^{-\omega(s+\frac{t_{2}-t_{1}}{N})}U_{N})\right\rVert_{r}^{\sigma}\mathop{}\\!\mathrm{d}s$
$\displaystyle\qquad+\left|\frac{e^{\frac{q\omega(t_{2}-t_{1})}{N}}-1}{\frac{t_{2}-t_{1}}{N}}-q\omega
e^{\frac{\omega(t_{2}-t_{1})}{N}}\right|\int_{t_{1}}^{t_{2}}\lVert
G_{\lambda}(e^{-\omega s}U_{N})\rVert_{q}^{q}\mathop{}\\!\mathrm{d}s$
$\displaystyle\qquad+q\omega\lambda
e^{\frac{\omega(t_{2}-t_{1})}{N}}\int_{t_{1}}^{t_{2}}\lVert
G_{\lambda}(e^{-\omega s}U_{N})\rVert_{q-1}^{q-1}\mathop{}\\!\mathrm{d}s$
$\displaystyle\qquad+q\int_{t_{1}}^{t_{2}}\int_{\Sigma}|G_{\lambda}(e^{-\omega
s}U_{N})|^{q-1}|g_{N}|\mathop{}\\!\mathrm{d}\mu\mathop{}\\!\mathrm{d}s.$

We now prove convergence of each term in the estimate in order to obtain the
continuous version (4.22). Noting that $U_{N}(s)\rightarrow u(s)$ in
$C([0,T];L^{q_{0}}_{\mu})$ and since $\lVert\cdot\rVert_{q}^{q}$ and
$\lVert\cdot\rVert_{r}^{\sigma}$ are lower semicontinuous on
$L^{q_{0}}_{\mu}$, we have that

$\liminf\limits_{N\rightarrow\infty}\lVert G_{\lambda}(e^{-\omega
t_{2}}v_{N})\rVert_{q}^{q}\geq\lVert G_{\lambda}(e^{-\omega
t_{2}}u(t_{2}))\rVert_{q}^{q}$

and applying Fatou’s lemma,

$\begin{split}\liminf_{N\rightarrow\infty}\int_{t_{1}}^{t_{2}}&e^{\omega(\sigma-q)s}\left\lVert
G_{\lambda}(e^{-\omega(s+\frac{t_{2}-t_{1}}{N})}U_{N}(s))\right\rVert_{r}^{\sigma}\mathop{}\\!\mathrm{d}s\\\
&\quad\geq\int_{t_{1}}^{t_{2}}e^{\omega(\sigma-q)s}\lVert
G_{\lambda}(e^{-\omega
s}u(s))\rVert_{r}^{\sigma}\mathop{}\\!\mathrm{d}s.\end{split}$

Next, note that by the complete resolvent property of $A$ and Lemma 4.3 we can
estimate $U_{N}$ and $u$ in $L^{q}_{\mu}$ uniformly on $[0,T]$. Hence let $M$
bound both $\lVert U_{N}\rVert_{q}$ and $\lVert u\rVert_{q}$.

$\begin{split}\left|\frac{1-e^{-\frac{q\omega
t}{N}}}{\frac{t}{N}}-q\omega\right|\int_{t_{1}}^{t_{2}}\lVert
G_{\lambda}(e^{-\omega
s}U_{N}(s))\rVert_{q}^{q}\mathop{}\\!\mathrm{d}s&\leq\left|\frac{1-e^{-\frac{q\omega
t}{N}}}{\frac{t}{N}}-q\omega\right|\int_{t_{1}}^{t_{2}}M^{q}\mathop{}\\!\mathrm{d}s\\\
&\rightarrow 0\end{split}$

as $N\rightarrow 0$. For the next term we prove uniform convergence of
$G_{\lambda}(e^{-\omega s}U_{N})$ to $G_{\lambda}(e^{-\omega s}u)$ in
$C([0,T];L^{q-1}_{\mu})$ when $\lambda>0$. For this fix $s\in[0,T]$ and let

$f_{N}=G_{\lambda}(e^{-\omega s}U_{N}(s))-G_{\lambda}(e^{-\omega s}u(s))$

so that $f_{N}\rightarrow 0$ in $L^{q_{0}}$. We note that

$\begin{split}\lVert f_{N}\rVert_{q}&\leq 2M\end{split}$

and by Chebyshev’s inequality

$\begin{split}\mu(\\{x\in\Sigma:|f_{N}|>0\\})&\leq\mu(\\{x\in\Sigma:e^{-\omega
s}|U_{N}|\geq\lambda\text{ or }e^{-\omega s}|u|\geq\lambda\\})\\\
&\leq\frac{1}{\lambda^{q}}\left(\lVert e^{-\omega s}U_{N}(s)\rVert_{q}+\lVert
e^{-\omega s}u(s)\rVert_{q}\right)\\\ &\leq\frac{2M}{\lambda^{q}}.\end{split}$

Here we consider cases for $q$. For $q-1\geq q_{0}$, apply Hölder’s inequality
with $\theta$ chosen to satisfy

$\frac{\theta}{q_{0}}+\frac{1-\theta}{q}=\frac{1}{q-1}$

to obtain

$\displaystyle\lim\limits_{N\rightarrow\infty}\lVert f_{N}\rVert_{q-1}$
$\displaystyle\leq\lim\limits_{N\rightarrow\infty}\left(\lVert
f_{N}\rVert_{q_{0}}^{\theta}\lVert f_{N}\rVert_{q}^{1-\theta}\right)=0.$

For $q-1<q_{0}$, we apply Jensen’s inequality noting that
$|\cdot|^{\frac{q_{0}}{q-1}}$ is convex. Let

$\Sigma_{N}=\\{x\in\Sigma:|f_{N}|>0\\}$

then we can again estimate $f_{N}$ with

$\lVert f_{N}\rVert_{q-1}\leq\mu(\Sigma_{N})^{\frac{q_{0}}{q-1}-1}\lVert
f_{N}\rVert_{q_{0}}.$

Note that for $q-1<1$, we have

$\displaystyle\left|\lVert G_{\lambda}(e^{-\omega
s}U_{N})\rVert_{q-1}^{q-1}-\lVert G_{\lambda}(e^{-\omega
s}u)\rVert_{q-1}^{q-1}\right|$
$\displaystyle\qquad\leq\int_{\Sigma}\left|\left|G_{\lambda}(e^{-\omega
s}U_{N})\right|^{q-1}-\left|G_{\lambda}(e^{-\omega
s}u)\right|^{q-1}\right|\mathop{}\\!\mathrm{d}\mu$
$\displaystyle\qquad\leq\int_{\Sigma}\left|f_{N}\right|^{q-1}\mathop{}\\!\mathrm{d}\mu$

so that

$\lim\limits_{N\rightarrow\infty}\lVert G_{\lambda}(e^{-\omega
s}U_{N}(s))\rVert_{q-1}^{q-1}=\lVert G_{\lambda}(e^{-\omega
s}u(s))\rVert_{q-1}^{q-1}.$

For the last term, note that $g_{N}\rightarrow g$ in $L^{1}(0,T;L^{q}_{\mu})$
as $N\rightarrow\infty$. So by a corollary of Riesz-Fischer, there exists a
subsequence $N_{k}$ and a function $h\in L^{1}(0,T;L^{q}_{\mu})$ such that
$|g_{N_{k}}(x)|\leq h(x)$ for all $k$ and a.e. $x\in\Sigma$. Similarly,
interpolating between $q_{0}$ and $q+\varepsilon$,

$\begin{split}\lVert G_{\lambda}(e^{-\omega s}U_{N}(s))-G_{\lambda}(e^{-\omega
s}u(s))\rVert_{q}&\leq\lVert G_{\lambda}(e^{-\omega
s}U_{N}(s))-G_{\lambda}(e^{-\omega s}u(s))\rVert_{q_{0}}^{\theta}\times\\\
&\qquad\lVert G_{\lambda}(e^{-\omega s}U_{N}(s))-G_{\lambda}(e^{-\omega
s}u(s))\rVert_{q+\varepsilon}^{1-\theta}\end{split}$

for some $\theta\in(0,1]$. Then with $U_{N}$ and $u$ bounded in
$L^{q+\varepsilon}_{\mu}$, $G_{\lambda}(e^{-\omega s}U_{N}(s))\rightarrow
G_{\lambda}(e^{-\omega s}u(s))$ in $L^{q_{0}}_{\mu}$ as $N\rightarrow\infty$
uniformly for all $t\in[0,T]$. Hence taking another subsequence we have a
dominant $H_{\lambda}\in L^{\infty}(0,T;L^{q}_{\mu})$. So we can estimate the
integrand pointwise

$\left|G_{\lambda}(e^{-\omega s}U_{N_{k}}(s))^{q-1}g_{N}\right|\leq
H_{\lambda}^{q-1}h\quad\text{a.e.~{}on }\Sigma\times[0,T).$

Moreover this dominant is in $L^{1}(0,T;L^{1}_{\mu})$ with

$\int_{t_{1}}^{t_{2}}\int_{\Sigma}H_{\lambda}^{q-1}h\mathop{}\\!\mathrm{d}\mu\mathop{}\\!\mathrm{d}s\leq\lVert
H_{\lambda}\rVert_{L^{\infty}(0,T;L^{q}_{\mu})}\lVert
h\rVert_{L^{1}(0,T;L^{q}_{\mu})}.$

Hence we apply dominated convergence to obtain the continuous estimate
(4.22). ∎

We now show that this pointwise estimate (4.21) for Proposition 4.4 implies a
similar estimate when adding a Lipschitz perturbation.

######  Lemma 4.5.

For $1\leq q<\infty$, let $A$ be an operator on $L^{q}_{\mu}$ and suppose
there are $1\leq r\leq\infty$, $\sigma>0$, $\omega\in\mathbb{R}$, $\lambda\geq
0$ and $C>0$ such that (4.21) is satisfied for all $(u,v)\in A$. Let
$F:L^{q}_{\mu}\to L^{q}_{\mu}$ be Lipschitz continuous with Lipschitz constant
$\omega^{\prime}\geq 0$ and satisfying $F(0)=0$. Then, the operator $A+F$ in
$L^{q}_{\mu}$ satisfies

$\lVert G_{\lambda}(u)\rVert_{r}^{\sigma}\leq
C\;[G_{\lambda}(u),v+(\omega+\omega^{\prime})(G_{\lambda}(u)+\lambda\operatorname*{sign}(u))]_{q}.$
(4.25)

###### Proof.

Let $\hat{v}=v+F(u)$. Then, since $[\cdot,\cdot]_{q}$ is linear in the second
term,

$\begin{split}[G_{\lambda}(u),\hat{v}+&(\omega+\omega^{\prime})(G_{\lambda}(u)+\lambda\operatorname*{sign}(u))]_{q}\\\
&=[G_{\lambda}(u),v+\omega(G_{\lambda}(u)+\lambda\operatorname*{sign}(u))]_{q}\\\
&\quad+[G_{\lambda}(u),F(u)+\omega^{\prime}(G_{\lambda}(u)+\lambda\operatorname*{sign}(u))]_{q}.\end{split}$
(4.26)

By the Lipschitz condition,

$-\omega^{\prime}|u|\leq F(u)\leq\omega^{\prime}|u|.$

Hence

$\begin{split}[G_{\lambda}(u),F(u)]_{q}&\geq-\omega^{\prime}[G_{\lambda}(u),u]_{q}\\\
&=-\omega^{\prime}[G_{\lambda}(u),G_{\lambda}(u)+\lambda\operatorname*{sign}(u)]_{q}.\end{split}$

So applying this and the initial assumption to (4.26), we have (4.25). ∎

We now extend the $L^{q}_{\mu}-L^{\infty}_{\mu}$ regularity of Theorem 4.1 to
obtain $L^{\ell}_{\mu}-L^{\infty}_{\mu}$ regularity as in Theorem 1.2. For
this we consider (4.6) applied to $\tilde{u}_{0}=u(\tfrac{t}{2})$ and
$\tilde{g}(s)=g(s+\tfrac{t}{2})$. The following theorem is an extension of
[14, Chapter 6] and in particular with $r=\infty$ gives the desired regularity
result.

######  Theorem 4.6.

For $1\leq\ell<q<r\leq\infty$ and $T>0$, let $g\in
L^{1}(0,T;L^{\ell}_{\mu}\cap L^{q}_{\mu})$ and $u\in
L^{\infty}(0,T;L^{\ell}_{\mu}\cap L^{r}_{\mu})$ satisfying the exponential
growth property (4.18) for some $\omega\geq 0$. Suppose there exist increasing
functions $c_{1}(t)$, $c_{2}(t)$ with $c_{1}(t)>0$ and $c_{2}(t)\geq 0$ for
all $t\in[0,T]$ and exponents $\alpha\geq 0$, $0<\gamma^{*}\leq\gamma<\infty$
such that

$\begin{split}\lVert
u(t)\rVert_{r}&\leq\max\left\\{c_{1}(\tfrac{t}{2})\left(\tfrac{2}{t}+\omega\right)^{\alpha}\left(\lVert
u(\tfrac{t}{2})\rVert_{q}+\lVert
g\rVert_{L^{1}(\frac{t}{2},t;L^{q}_{\mu})}\right)^{\gamma},\right.\\\ &\hskip
99.58464pt\left.c_{2}(\tfrac{t}{2})\left(\lVert
u(\tfrac{t}{2})\rVert_{q}+\lVert
g\rVert_{L^{1}(\frac{t}{2},t;L^{q}_{\mu})}\right)^{\gamma^{*}}\right\\}\end{split}$
(4.27)

for every $t\in(0,T]$. Define

$\theta:=1-\gamma\left(\frac{\frac{1}{\ell}-\frac{1}{q}}{\frac{1}{\ell}-\frac{1}{r}}\right)$

and suppose that $\theta>0$. Then one has the $L^{\ell}_{\mu}-L^{r}_{\mu}$
estimate

$\begin{split}\lVert u(t)\rVert_{r}&\leq
2^{\gamma}\left(2^{\frac{\alpha}{\gamma\theta}}+\sup_{s\in(0,t]}N(s)^{\theta}\right)^{\frac{\gamma}{\theta}}\max\left\\{c_{1}(t)^{\frac{1}{\theta}}\left(\tfrac{1}{t}+\omega\right)^{\frac{\alpha}{\theta}},c_{2}(t)^{\frac{1}{\theta}}\right\\}\times\\\
&\hskip 85.35826pt\times\left(e^{\omega t}\lVert
u_{0}\rVert_{\ell}+\int_{0}^{t}e^{\omega(t-\tau)}\lVert
g(\tau)\rVert_{\ell}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{\theta_{\ell}\gamma}{\theta}}\end{split}$
(4.28)

for all $t\in(0,T]$ where

$\begin{split}N(t)&:=\sup\limits_{s\in(0,t]}\frac{M(\tfrac{s}{2})\lVert
g\rVert_{L^{1}(\frac{s}{2},s;L^{q}_{\mu})}+c_{2}(\tfrac{s}{2})^{\frac{1}{\gamma}}}{M(s)^{\frac{1}{\theta}}\left(e^{\omega
s}\lVert u_{0}\rVert_{\ell}+\int_{0}^{s}e^{\omega(s-\tau)}\lVert
g(\tau)\rVert_{\ell}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{\theta_{\ell}}{\theta}}},\\\
M(t)&:=\max\left\\{c_{1}(t)^{\frac{1}{\gamma}}\left(\tfrac{1}{t}+\omega\right)^{\frac{\alpha}{\gamma}},c_{2}(t)^{\frac{1}{\gamma}}\right\\},\end{split}$

and

$\theta_{\ell}:=\frac{\frac{1}{q}-\frac{1}{r}}{\frac{1}{\ell}-\frac{1}{r}}.$

###### Proof.

We first note that since $\gamma^{*}\leq\gamma$, we can estimate

$\begin{split}\left(\lVert u(\tfrac{t}{2})\rVert_{q}+\lVert
g\rVert_{L^{1}(\frac{t}{2},t;L^{q}_{\mu})}\right)^{\gamma^{*}}&\leq\max\left\\{1,\left(\lVert
u(\tfrac{t}{2})\rVert_{q}+\lVert
g\rVert_{L^{1}(\frac{t}{2},t;L^{q}_{\mu})}\right)^{\gamma}\right\\}\end{split}$

for $t\in[0,T]$. Hence, taking (4.27) to the power $\frac{1}{\gamma}$, we have

$\begin{split}\lVert
u(t)\rVert_{r}^{\frac{1}{\gamma}}&\leq\max\left\\{c_{1}(\tfrac{t}{2})^{\frac{1}{\gamma}}\left(\tfrac{2}{t}+\omega\right)^{\frac{\alpha}{\gamma}}\left(\lVert
u(\tfrac{t}{2})\rVert_{q}+\lVert
g\rVert_{L^{1}(\frac{t}{2},t;L^{q}_{\mu})}\right),\right.\\\ &\hskip
71.13188pt\left.c_{2}(\tfrac{t}{2})^{\frac{1}{\gamma}}\left(\lVert
u(\tfrac{t}{2})\rVert_{q}+\lVert
g\rVert_{L^{1}(\frac{t}{2},t;L^{q}_{\mu})}\right),c_{2}(\tfrac{t}{2})^{\frac{1}{\gamma}}\right\\}.\end{split}$

We then apply standard interpolation on the $L^{q}_{\mu}$ norm with exponent
$\theta_{\ell}$ and the growth estimate (4.18) on $\lVert
u(\tfrac{t}{2})\rVert_{\ell}$ to obtain

$\begin{split}\lVert u(t)\rVert_{q}&\leq\left(e^{\omega t}\lVert
u_{0}\rVert_{\ell}+\int_{0}^{t}e^{\omega(t-\tau)}\lVert
g(\tau)\rVert_{\ell}\mathrm{d}\tau\right)^{\theta_{\ell}}\lVert
u(t)\rVert_{r}^{1-\theta_{\ell}}\end{split}$

for all $t\in[0,T]$. By choice of $\theta$, $\theta_{\ell}$ we have the
relation

$\gamma(1-\theta_{\ell})=1-\theta.$

Then for all $t\in(0,T]$,

$\begin{split}\lVert u(t)\rVert_{r}^{\frac{1}{\gamma}}&\leq
M(\tfrac{t}{2})\left(e^{\frac{\omega t}{2}}\lVert
u_{0}\rVert_{\ell}+\int_{0}^{\tfrac{t}{2}}e^{\omega(\tfrac{t}{2}-\tau)}\lVert
g(\tau)\rVert_{\ell}\right)^{\theta_{\ell}}\lVert
u(\tfrac{t}{2})\rVert_{r}^{\frac{1-\theta}{\gamma}}\\\
&\quad+M(\tfrac{t}{2})\lVert
g\rVert_{L^{1}(\frac{t}{2},t;L^{q}_{\mu})}+c_{2}(\tfrac{t}{2})^{\frac{1}{\gamma}}.\end{split}$
(4.29)

We aim to produce comparable terms on either side of this equation. Since
$c_{1}$ and $c_{2}$ are increasing, $M(\tfrac{t}{2})\leq
2^{\frac{\alpha}{\gamma}}M(t)$. Furthermore,

$e^{\frac{\omega t}{2}}\lVert
u_{0}\rVert_{\ell}+\int_{0}^{\tfrac{t}{2}}e^{\omega(\tfrac{t}{2}-\tau)}\lVert
g(\tau)\rVert_{\ell}\leq e^{-\frac{\omega t}{2}}\left(e^{\omega t}\lVert
u_{0}\rVert_{\ell}+\int_{0}^{t}e^{\omega(t-\tau)}\lVert
g(\tau)\rVert_{\ell}\right).$

Hence we define

$K_{u}(t):=\frac{M(t)^{-\frac{1}{\theta}}\lVert
u(t)\rVert_{r}^{\frac{1}{\gamma}}}{\left(e^{\omega t}\lVert
u_{0}\rVert_{\ell}+\int_{0}^{t}e^{\omega(t-\tau)}\lVert
g(\tau)\rVert_{\ell}\mathop{}\\!\mathrm{d}\tau\right)^{\frac{\theta_{\ell}}{\theta}}}\quad\text{for
}t\in[0,T]$ (4.30)

in order to estimate and rearrange (4.29) into a relation involving $K_{u}$
and $N$. Note that if the denominator of (4.30) is zero, we can add some
arbitrarily small $\varepsilon>0$ to the denominator of both $K_{u}$ and $F$,
taking $\varepsilon\rightarrow 0$ in the final estimate. Now we fix
$t\in(0,T]$ so that after rearranging we may take a supremum over $(0,t]$ to
obtain

$\sup_{s\in(0,t]}K_{u}(s)\leq
2^{\frac{\alpha}{\gamma\theta}}\sup_{s\in(0,\frac{t}{2}]}K_{u}(s)^{1-\theta}+N(t).$

We now aim to split the forcing term $N(t)$ so as to incorporate this into
each $K_{u}$ term. For this we define
$D(t)\geq(2^{\frac{\alpha}{\gamma\theta}})^{\frac{1}{\theta}}$ for $t\in[0,T]$
such that

$D(t)-2^{\frac{\alpha}{\gamma\theta}}D(t)^{1-\theta}=N(t).$ (4.31)

Noting that $\theta\in(0,1)$, this is possible as the function
$f(x)=x-cx^{\alpha}$ for $c\geq 0$ and $\alpha\in(0,1)$ is continuous,
satisfies $f(c^{\frac{1}{1-\alpha}})=0$ and is strictly increasing for
$x>(\alpha c)^{\frac{1}{1-\alpha}}$ (in particular for $x\geq
c^{\frac{1}{1-\alpha}}$). Further, we can estimate (4.31) by

$\displaystyle N(t)$
$\displaystyle=\left(D(t)^{\theta}\right)^{\frac{1-\theta}{\theta}}\left(D(t)^{\theta}-2^{\frac{\alpha}{\gamma\theta}}\right)$
$\displaystyle\geq\left(D(t)^{\theta}-2^{\frac{\alpha}{\gamma\theta}}\right)^{\frac{1}{\theta}}$

so that

$D(t)\leq\left(2^{\frac{\alpha}{\gamma\theta}}+N(t)^{\theta}\right)^{\frac{1}{\theta}}.$
(4.32)

Then,

$\sup_{s\in(0,t]}K_{u}(s)-D(t)\leq
2^{\frac{\alpha}{\gamma\theta}}\left(\left(\sup_{s\in\left(0,\tfrac{t}{2}\right]}K_{u}(s)\right)^{1-\theta}-D(t)^{1-\theta}\right).$

Noting that $K_{u}(s)$ is bounded for all $s\in[0,T]$, either

$\sup_{s\in(0,t]}K_{u}(s)\leq D(t)$

or we can extend to a supremum over $(0,t]$ and combine terms, obtaining

$\begin{split}\sup_{s\in(0,t]}K_{u}(s)-D(t)&\leq
2^{\frac{\alpha}{\gamma\theta}}\left(\sup_{s\in(0,t]}K_{u}(s)-D(t)\right)^{1-\theta}\\\
\left(\sup_{s\in(0,t]}K_{u}(s)-D(t)\right)^{\theta}&\leq
2^{\frac{\alpha}{\gamma\theta}}.\end{split}$

In either case, we have the uniform bound,

$K_{u}(s)\leq\left(2^{\frac{\alpha}{\gamma\theta}}\right)^{\frac{1}{\theta}}+D(t).$

for all $s\in(0,t]$. Applying (4.32),

$\displaystyle K_{u}(t)$
$\displaystyle\leq\left(2^{\frac{\alpha}{\gamma\theta}}\right)^{\frac{1}{\theta}}+\left(2^{\frac{\alpha}{\gamma\theta}}+N(t)^{\theta}\right)^{\frac{1}{\theta}}$
$\displaystyle\leq
2\left(2^{\frac{\alpha}{\gamma\theta}}+N(t)^{\theta}\right)^{\frac{1}{\theta}}.$

Rewriting this as an estimate on $\lVert u(t)\rVert_{r}$ we obtain (4.28). ∎

##  5 $L^{1}-L^{\infty}$ regularization

By Theorem 3.1 and the proof of Theorem 1.1 we know that
$\overline{(-\Delta_{p})^{s}\varphi}^{\mbox{}_{L^{1}}}+F$ is m-accretive in
$L^{1}$ with complete resolvent where $F$ is the Nemytskii operator of
$f(\cdot,u)$ satisfying (1.3a)-(1.3b). Hence to apply Theorem 4.1 we first
prove the pointwise estimate (4.21) for the operator
$\overline{(-\Delta_{p})^{s}\varphi}^{\mbox{}_{L^{1}}}$ in $L^{1}$, giving
Proposition 4.4 and thereby outlining the proof of the $L^{m+1}-L^{\infty}$
estimate of Theorem 5.3. We then apply Theorem 4.6, proving the
$L^{\ell}-L^{\infty}$ estimate of Theorem 1.2.

The following lemma allows us to estimate the $q$-bracket
$[G_{\lambda}(u),(-\Delta_{p})^{s}(u^{m})]_{m+1}$. In particular, the
restriction $m\geq 1$ in this lemma results in the same restriction in Theorem
1.2. Recall that we use the notation $r^{m}=|r|^{m-1}r$ for powers and that
$G_{\lambda}(r)=[\lvert r\rvert-\lambda]^{+}\operatorname*{sign}(r)$ for
$r\in{\mathbb{R}}$.

######  Lemma 5.1.

Let $1<p<\infty$, $m\geq 1$ and $\lambda\geq 0$. Given $a$, $b\in\mathbb{R}$
define $a_{\lambda}=G_{\lambda}(a)$, $b_{\lambda}=G_{\lambda}(b)$. Then,

$\displaystyle(a^{m}-b^{m})^{p-1}(a_{\lambda}^{m}-b_{\lambda}^{m})\geq|a_{\lambda}^{m}-b_{\lambda}^{m}|^{p}.$

###### Proof.

If $a_{\lambda}-b_{\lambda}=0$, both sides are $0$. For
$a_{\lambda}-b_{\lambda}\neq 0$,

$\displaystyle\operatorname*{sign}(a_{\lambda}^{m}-b_{\lambda}^{m})$
$\displaystyle=\operatorname*{sign}(a^{m}-b^{m})$
$\displaystyle|a^{m}-b^{m}|^{p-2}(a^{m}-b^{m})(a_{\lambda}^{m}-b_{\lambda}^{m})$
$\displaystyle=|a^{m}-b^{m}|^{p-1}|a_{\lambda}^{m}-b_{\lambda}^{m}|$

so we only need to prove that
$|a_{\lambda}^{m}-b_{\lambda}^{m}|\leq|a^{m}-b^{m}|$. We take cases, assuming
without loss of generality that $|a|\geq|b|$. First suppose that
$|a|\leq\lambda$ or $|b|\leq\lambda$ so that $a_{\lambda}=b_{\lambda}=0$ and
the inequality is clear. Next, if $|b|\leq\lambda$ and $|a|>\lambda$, then

$|a_{\lambda}^{m}-b_{\lambda}^{m}|=(|a|-\lambda)^{m}\leq(|a|-|b|)^{m}\leq|a|^{m}-|b|^{m}\leq|a^{m}-b^{m}|.$

Hence we consider cases for $a$, $b$ corresponding to $|a|>\lambda$ and
$|b|>\lambda$. Suppose that $a>\lambda$ and $b>\lambda$. Then noting that for
$m\geq 1$, $|x|^{m-1}$ is non-decreasing on the set $[0,\infty)$,

$\displaystyle a^{m}-b^{m}$
$\displaystyle=\int_{b}^{a}\frac{\mathrm{d}}{\mathrm{d}x}x^{m}\mathop{}\\!\mathrm{d}x$
$\displaystyle\geq\int_{b}^{a}m|x-\lambda|^{m-1}\mathop{}\\!\mathrm{d}x$
$\displaystyle=(a-\lambda)^{m}-(b-\lambda)^{m}.$

Similarly for $a<-\lambda$, $b<-\lambda$, noting that $|x|^{m-1}$ is non-
increasing on $(-\infty,0]$,

$a^{m}-b^{m}\geq(a+\lambda)^{m}-(b+\lambda)^{m}.$

Finally, suppose that $a>\lambda$ and $b<-\lambda$ (similarly for $a<-\lambda$
and $b>\lambda$). Then

$\displaystyle|a_{\lambda}^{m}-b_{\lambda}^{m}|$
$\displaystyle=(a-\lambda)^{m}-(b+\lambda)^{m}$ $\displaystyle\leq
a^{m}-b^{m}$ $\displaystyle=|a^{m}-b^{m}|.$

∎

We can now derive (4.21), the pointwise estimate for Proposition 4.4, giving
the Sobolev-type inequality required for Theorem 4.1 in the case of (1.1).
Recall the notation of $q$-brackets from Section 2.2. In this case with
$q=m+1$ for $m\geq 1$, the $q$-bracket is given by

$[u,v]_{m+1}=\int_{\Omega}\lvert u\rvert^{m-1}u\,v\mathop{}\\!\mathrm{d}\mu$

for every $u$, $v\in L^{m+1}$.

######  Lemma 5.2.

Let $\Omega$ be an open set in $\mathbb{R}^{d}$, $d\geq 1$. For $1<p<\infty$,
$0<s<1$, define $p_{\mbox{}_{\\!s}}$ according to the Sobolev embedding (1.15)
and let $\varphi(r)=r^{m}$ for $r\in\mathbb{R}$ where $m\geq 1$. Then
$(-\Delta_{p})^{s}\varphi$ satisfies the one-parameter Sobolev type inequality

$\lVert G_{\lambda}(u)\rVert_{mp_{\mbox{}_{\\!s}}}^{mp}\leq
C_{d}\;[G_{\lambda}(u),(-\Delta_{p})^{s}\varphi(u)]_{m+1}$ (5.1)

for all $\varphi(u)\in W^{s,p}_{0}$ and $\lambda\geq 0$. In particular, this
is (4.21) with $q=m+1$, $\omega=0$, $\sigma=mp$, $r=mp_{s}$ and $C=C_{d}$.

###### Proof.

Let $u^{m}\in W^{s,p}_{0}$. By Lemma 5.1 one sees that

$\begin{split}&[G_{\lambda}u,(-\Delta_{p})^{s}(u^{m})]_{m+1}\\\
&\qquad=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{\lvert((u(t,x))^{m}-(u(t,y))^{m}\rvert^{p-2}((u(t,x))^{m}-(u(t,y))^{m})}{|x-y|^{d+sp}}\times\\\
&\hskip
142.26378pt((G_{\lambda}u(t,x))^{m}-(G_{\lambda}u(t,y))^{m})\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\\\
&\qquad\geq\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{|(G_{\lambda}u(t,x))^{m}-(G_{\lambda}u(t,y))^{m}|^{p}}{|x-y|^{d+sp}}\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}y\end{split}$

Hence we can estimate by a semi-norm,

$\begin{split}[G_{\lambda}u,(-\Delta_{p})^{s}(u^{m})]_{m+1}&\geq[(G_{\lambda}u(t,x))^{m}]_{s,p}^{p}\\\
&\geq\frac{1}{C_{d}}\,\lVert(G_{\lambda}(u))^{m}\rVert_{p_{\mbox{}_{\\!s}}}^{p}\\\
&=\frac{1}{C_{d}}\,\lVert
G_{\lambda}(u)\rVert_{mp_{\mbox{}_{\\!s}}}^{mp}\end{split}$

by the classical Sobolev inequality for Gagliardo semi-norms (cf. [27])

$\lVert u\rVert_{p_{\mbox{}_{s}}}\leq C_{d}\,[u]_{s,p}$

where $p_{\mbox{}_{s}}$ is given by (1.15). ∎

With these preliminaries we can now apply Proposition 4.4 and Theorem 4.1 to
prove the $L^{m+1}-L^{\infty}$ regularisation effect for the doubly nonlinear
nonlocal problem (1.1).

######  Theorem 5.3.

Let $\Omega$ be an open set in $\mathbb{R}^{d}$, $d\geq 1$. Suppose $p>1$,
$0<s<1$, and $m\geq 1$ such that

$m(p-1)+(m+1)\frac{sp}{d}>1.$ (1.16)

Let $q_{s}=p_{s}$ if $p\neq\frac{d}{s}$ and $q_{s}>\max(p,1+\frac{1}{m})$ if
$p=\frac{d}{s}$. Let $T>0$ and $g\in L^{1}(0,T;L^{1})\cap
L^{\psi}(0,T;L^{\rho})\cap L^{1}(0,T;L^{m+1+\varepsilon})$ for some
$\varepsilon>0$ where $\rho\geq m+1$ and $\psi>1$ satisfy

$\begin{cases}\frac{1}{\rho}<\left(1-\frac{1}{\psi}\right)\left(1-\frac{p}{q_{s}}\right)&\text{if
}m(p-1)\geq 1,\\\
\frac{1}{\rho}\leq\left(1-\frac{1}{\psi}\right)p\left(\frac{m}{m+1}-\frac{1}{q_{s}}\right)&\text{if
}m(p-1)<1.\end{cases}$ (1.17)