# Time-Fractional Porous Medium Type Equations.
Sharp Time Decay and Regularization

Matteo Bonforte${}^{\,a}$, Maria Gualdani${}^{\,b}$ and Peio
Ibarrondo${}^{\,a}$

###### Abstract

We consider a class of porous medium type of equations with Caputo time
derivative. The prototype problem reads as $D^{\alpha}_{t}u=-\mathcal{L}u^{m}$
and is posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^{N}$ with
smooth boundary and zero Dirichlet boundary conditions. The linear operator
$\mathcal{L}$ falls within a wide class of either local or nonlocal operators,
and the nonlinearity is allowed to be of degenerate or singular type, namely,
$0<m<1$ and $m>1$.

Using the fractional gradient flow theory, we show existence of unique
solution and prove new $L^{p}-L^{\infty}$ smoothing effects by employing the
Green function method. The comparison principle, which we provide in the most
general setting, serves as a crucial tool in the proof and provides a novel
monotonicity formula. To the best of our knowledge, this type of smoothing
effects are novel in nonlinear PDEs involving Caputo derivative. Consequently,
we establish that the regularizing effect due to the diffusion is stronger
than the memory effect introduced by the fractional time derivative. We also
prove that the solution satisfies the zero Dirichlet condition pointwise by
means of sharp boundary estimates.

Finally, we prove that our solution does not vanish in finite time if $0<m<1$,
unlike the case with the classical time derivative. Indeed, we provide a sharp
rate of decay for any $L^{p}$-norm of the solution for any $m>0$. Our findings
indicate that the difference between slow and fast diffusion is mitigated by
the memory effect, which dramatically slows the spatial diffusion.

Keywords:  porous medium equation; fast diffusion equation; nonlocal
operators; Caputo fractional time derivative; subdiffusion; comparison
principle; regularity estimates; long time behavior.

MSC2020 Classification. Primary: 34K37, 35K55, 35B51, 35B65, 35R11, 35B40.
Secondary: 35K61, 35K65, 35K67.

  * (a)

Universidad Autónoma de Madrid, Departamento de Matemáticas,
ICMAT - Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM
Campus de Cantoblanco, 28049 Madrid, Spain.

  * (b)

The University of Texas at Austin, Mathematics Department
PMA 8.100 2515 Speedway Stop C1200, Austin, TX 78712-1202, USA.

###### Contents

  1. 1 Introduction
    1. 1.1 Main theorems
    2. 1.2 Assumptions and functional setting of the operator $\mathcal{L}$
    3. 1.3 Examples of $\mathcal{L}$
  2. 2 Discretized problem
    1. 2.1 Existence and uniqueness for the discretized problem
    2. 2.2 Discrete Comparison Principle
  3. 3 Continuous problem
    1. 3.1 Existence and uniqueness
    2. 3.2 Comparison Principle
  4. 4 Smoothing effects
  5. 5 Sharp time decay of $L^{p}$-norms
  6. 6 Open questions
  7. A Appendix. Fractional ODEs

## Acknowledgments

M.B. and P.I. where partially supported by the Project PID2020-113596GB-I00
(Ministry of Science and Innovation, Spain) and by the E.U. H2020 MSCA
program, grant agreement 777822. M.B. acknowledges financial support from the
Spanish Ministry of Science and Innovation, through the “Severo Ochoa
Programme for Centres of Excellence in R&D” (CEX2019-000904-S). P.I. was
partially funded by the FPU-grant FPU19/04791 from the Spanish Ministry of
Universities.

This work started when P.I. was visiting M.G. at The University of Texas at
Austin in the year 2022, when an essential part of the work was done. P.I.
would like to thank both M.G. and the Mathematical Department of The
University of Texas at Austin for the kind hospitality, and the above MSCA
project for the financial support.

M.G. is partially supported by NSF grants DMS-2206677 and DMS-1514761. M.G.
also thanks the Isaac Newton Institute (Cambridge, UK) and ICMAT (Madrid,
Spain) for their kind hospitality.

© 2024 by the authors. This paper may be reproduced, in its entirety, for non-
commercial purposes.

##  1 Introduction

We consider the Dirichlet problem for a general class of Porous Medium type of
equations with Caputo fractional time derivative of order $\alpha\in(0,1)$:

$D_{t}^{\alpha}u=-\mathcal{L}u^{m}\qquad m>0\,,$ (CPME)

with $u(x,t)$ defined on a smooth bounded domain
$\Omega\subset\mathbb{R}^{N}$. The linear operator $\mathcal{L}$ belongs to a
general class of densely defined operators both of local and nonlocal type,
which includes the classical Laplacian and fractional Laplacian operator. The
fractional time operator $D_{t}^{\alpha}$ is the Caputo derivative, introduced
in [23] and defined for $\alpha\in(0,1)$ by

$D^{\alpha}_{t}f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{\rm d}{\,{\rm
d}t}\int_{0}^{t}\frac{f(\tau)-f(0)}{(t-\tau)^{\alpha}}\,{\rm d}\tau\,.$ (1)

In the last years, subdiffusive partial differential equations – equations
that combine diffusion operators in the space variable with fractional time
derivatives – have attracted significant attention, as they are used to model
anomalous diffusion, viscoelastic phenomena and processes with memory [40, 41,
42]. The prototype subdiffusive PDE is the “Heat Equation with memory”,

$D^{\alpha}_{t}u=\Delta u\,.$ (2)

For the well-posedness and regularity of (2) in $\mathbb{R}^{N}$, see for
instance the works of Allen, Caffarelli and Vasseur in [3]; Kemppainen,
Siljander, Vergara and Zacher in [34]; and Zacher in [52]. Optimal asymptotic
decay for solutions of (2) are obtained by Cortazar, Quiros and Wolansky in
[24] for the Cauchy problem in $\mathbb{R}^{N}$. For the Dirichlet problem on
bounded domains, Vergara and Zacher [49] show long time decay estimates, also
allowing for general operators with variable coefficients in space and time.
Recently, Chan, Gomez-Castro and Vázquez studied in [25] global well-posedness
of (CPME) with $m=1$ for singular solutions on bounded domains.

From a nonlinear perspective, Vergara and Zacher in [49] consider several
fractional nonlinear models and obtain sharp decay estimates of $L^{q}$ norms
using fractional ODEs techniques. The Porous Medium with fractional pressure,
associated to the so-called Caffarelli-Vázquez model first introduced in [22],
has been studied in its Caputo derivative version, by Allen, Caffarelli and
Vasseur in [2] and by Daus, Xu, Zamponi, Zhang and the second author in [27].

Akagi [1], and Li and Liu [37] developed a theory of fractional gradient flows
in Hilbert spaces, which is the analogous of the Brezis-Komura theory with
fractional time derivative [20, 36]. This theory provides well-posedness for
both linear and for nonlinear problems, and it is the starting point of our
paper.

Beside global well-posedness, the main contributions of this manuscript are
threefold: (i) the study of comparison principle and time monotonicity
formula, (ii) $L^{p}-L^{\infty}$ smoothing effects, and (iii) optimal long
time behaviour for (CPME). A surprising fact established with our analysis is
the non-extinction in finite time for all solutions of (CPME) when
$m\in(0,1)$. This fact was previously observed in [49] for separate-variable
solutions. In this work, we prove this property for any general solution.

The Caputo derivative adds the extra difficulty of the “memory effect”.
Roughly speaking, this memory effect somehow destroys the semigroup structure,
essential in the De Giorgi-Nash-Moser theory. Surprisingly enough, a
nontrivial adaptation of the so-called Green function method, introduced in
[17], allows us to establish our main results, overcoming the memory effect.
This is achieved by employing novel time monotonicity estimates, a feature
that comes as a surprise in the context of the Caputo setting, as we shall
further comment below.

To provide further insights about our results, some numerology is in order:
our asymptotic estimates correspond to the known ones in the formal limit
$m\rightarrow 1$. On the other hand, this does not happen in the –formal–
limit $\alpha\rightarrow 1^{-}$. Indeed, neither the exponents of smoothing
effects nor the ones in the long time behaviour estimates, correspond to the
known ones when $\alpha=1$.

Next, we provide a more detailed exposition of our principal findings:

Comparison principle and time monotonicity. We first recall the results about
existence and uniqueness of solution for (CPME) from [1, 37]. One of the
approximations steps proposed in [37] is the starting point of our proof of
the comparison principle. The comparison principle here corresponds to “well-
ordering”: if initial data for (CPME) are ordered, then solutions will
preserve the same order for any positive time. In the present case, we have to
overcome two difficulties: we do not have the standard semigroup property
$S(t+h)f=S(h)S(t)f$ because of the Caputo derivative, and we lack regularity
to perform rigorously the computations. For these reasons, our proof involves
several approximation steps. This comparison principle of Theorem 1.2 lays the
cornerstone for several other results in this manuscript, most importantly for
a new time monotonicity estimates “a la Benilan-Crandall”. Namely, we show
that $t^{\frac{\alpha}{m-1}}u(t)$ is monotone in time for every $t>0$, cf.
Theorem 1.3. The unexpected nature of this result arises from the memory
effect, which complicates any form of time monotonicity.

Smoothing effects and boundary estimates. We show new $L^{p}-L^{\infty}$
smoothing effects, stated in Theorem 1.4. To the best of our knowledge, these
kind of $L^{p}-L^{\infty}$ regularizing effects have been proved by
Kemppainen, Siljander, Vergara and Zacher in [34] only for the linear Caputo
Heat equation (2) on the whole space, whenever the initial data is integrable
enough. Our regularizing estimates hold for $u_{0}\in L^{p}(\Omega)$ with
$p>\frac{N}{2s}$. There is a structural reason why one should not expect
similar results for $p<\frac{N}{2s}$: the fundamental solution of (2) in
$\mathbb{R}^{N}$ is unbounded and belongs to $L^{q}(\mathbb{R}^{N})$ with
$q<\frac{N}{N-2}$. Therefore, one should ask the initial datum to be in
$L^{p}(\mathbb{R}^{N})$ with $p>\frac{N}{2}$.

One question for the future is the proof of higher regularity, starting for
example with the work of Allen, Caffarelli and Vasseur in [3] for Hölder
continuity of the linear problem. We do not address this issue in the current
manuscript. The results present in the literature concerning nonlinear
diffusions with Caputo time derivative are mostly of propagation of
$L^{p}$-norms: for instance, in [51] the authors prove propagation of
$L^{\infty}$-norm for (CPME) in the case when $\mathcal{L}=(-\Delta)$, meaning
that solutions remains bounded whenever $u_{0}$ is bounded.

We also study the behaviour of solutions near the boundary and show sharp
estimates of the form $u(t,x)\lesssim{\rm
dist}^{\gamma/m}(x,\partial\Omega)t^{-\alpha/m}$ in Theorem 1.5. Indeed, for
“big initial data” we obtain lower estimates with matching powers, proving the
so-called Global Harnack Principle, that is $u(t,x)\asymp{\rm
dist}^{\gamma/m}(x,\partial\Omega)t^{-\alpha/m}$, see Theorem 1.6. These
results are optimal both in space and in time, the latter optimality is
explained in what follows.

Optimal long time behaviour. Our last contribution concerns a sharp analysis
of the long time behaviour of the solution, cf. Theorem 1.7: in contrast to
the local-in-time results, in the fast diffusion regime $m\in(0,1)$, solutions
to (CPME) do not extinguish in finite time. This is an important side effect
of the “memory” intrinsic in the Caputo derivative. We compute the rates of
decay of all $L^{p}$ norms from above and below, with matching powers. This
phenomenon of non-extinction was first observed by Vergara and Zacher in [49]
for separate-variables solutions. Here, by means of differential inequalities
involving Caputo derivative111We devote Appendix A to recall the main results
used here about ODE with Caputo derivative., we show that the particular
behaviour of separate-variables solutions is enjoyed by all the other
solutions as well, and is optimal. We cover the whole range $m>0$ and a wide
class of operators $\mathcal{L}$, see Section 1.2 and 1.3 for more details
about the diffusion operators. This aspect reveals a remarkable discontinuity
in the parameter $\alpha$: indeed we show here that solutions decay as
$u(t)\asymp t^{-\alpha/m}$, while in the local-in-time case $\alpha=1$ they
are known to decay as $u(t)\asymp t^{-1/(m-1)}$ when $m>1$ and extinguish in
finite time $u(t)\asymp(T-t)_{+}^{1/(1-m)}$ when $m\in(0,1)$.

Related results. We conclude this section by recalling previous work on the
porous medium (PME) and fast diffusion equations (FDE) with the classical time
derivative. The literature on these models is extremely rich and spans several
decades. The few references we mention here are by no means exhaustive. Both
the porous medium and fast diffusion equations have been widely used to
describe different physical phenomena, including flow of gas through porous
media [46], plasma in reactors [8], etc. For this reason and for their own
mathematical interest, they have attracted a lot of attention since the early
50s, see for instance the lecture notes of Aronson [5] and the books of
Barenblatt [7], Vázquez [45, 46] and Daskalopulos and Kenig [26]. See also the
papers [44] ($m>1$) and [9] ($m<1$).

In the nonlocal case, we refer to the survey [47] for a complete review up to
2016, and we just mention the works strictly related to ours. When dealing
with the Cauchy problem for the nonlocal PME and FDE, namely,
$u_{t}=-(-\Delta)^{s}u^{m}$ on the whole space, a complete theory has been
developed by De Pablo, Quirós, Rodríguez and Vázquez in [28, 29, 30, 31]; see
also [16, 48]. As for the Dirichlet problem in the degenerate case (i.e.
$\alpha=1$ and $m>1$ in the present paper), Figalli, Ros-Oton, Sire, Vazquez
and the first author, performed a sharp analysis of the behaviour of the
solutions in a series of papers [17, 15, 10, 12, 11, 18]: existence and
uniqueness (in optimal classes), comparison, smoothing effects, infinite speed
of propagation (contrary to the local case $\mathcal{L}=-\Delta$), local and
global Harnack inequalities, sharp interior and boundary regularity estimates,
sharp asymptotic behaviour. When $m<1$ the theory is still under development:
existence, uniqueness, smoothing effects and boundary estimates have been
recently proven by Ispizua, the first and last authors in [13], while Harnack
and higher regularity estimates shall appear in [14]. The proofs rely on the
so-called Green function method, introduced in [17], which offers a valid
alternative to the more-classical approaches “a la” De Giorgi-Nash-Moser; see
[13, Section 3] for a comparison between the two methods.

###  1.1 Main theorems

We study the following Cauchy-Dirichlet problem with Caputo time derivative

$\left\\{\begin{array}[]{lll}D_{t}^{\alpha}u(t,x)=-\mathcal{L}u^{m}(t,x)&\qquad\mbox{on
}(0,+\infty)\times\Omega\,,\\\ u(t,\cdot)=0&\qquad\mbox{on the lateral
boundary, }\forall t>0\,,\\\ u(0,\cdot)=u_{0}&\qquad\mbox{in
}\Omega\,,\end{array}\right.$ (CPME)

where $m>0$ and $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain, with
boundary at least $C^{2,\alpha}$ to avoid delicate questions about boundary
regularity. We also assume $N>2s$. Note that Caputo derivative is defined as a
classical time derivative of a convolution with the kernel $k\in L^{1}_{\rm
loc}((0,+\infty))$:

$D^{\alpha}_{t}f(t)=\partial_{t}[k*(f-f_{0})](t)=\frac{\rm d}{\,{\rm
d}t}\int_{0}^{t}k(t-\tau)\,(f(\tau)-f_{0})\,{\rm d}\tau\,,$

where the kernel is given by

$k(t)=\frac{t^{-\alpha}}{\Gamma(1-\alpha)}\,.$

This kernel $k$ admits an inverse of the form

$\ell(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}\,,$

with respect to the convolution, i.e., $k*\ell=1$. It is worth mentioning that
there are different types pf kernels that define other types of fractional
derivative, and it would be interesting to analyze, see [1, 43, 51].

The operator $\mathcal{L}$ is a linear diffusion operator of local or nonlocal
type. We refer to Section 1.2 for a precise description, and to Section 1.3
for some examples. The solutions that we consider here are defined in the
Hilbert space $H^{*}(\Omega)$, which is the topological dual space of
$H(\Omega)$, the domain of the quadratic form associated to the operator
$\mathcal{L}$:

$H(\Omega)=\left\\{u\in L^{2}(\Omega)\,:\,\int_{\Omega}u\mathcal{L}u\,{\rm
d}x<+\infty\right\\}\,.$

The space $H^{*}(\Omega)$ is equipped with the following inner product and
associated norm:

$\langle u,v\rangle_{H^{*}(\Omega)}=\int_{\Omega}u\,\mathcal{L}^{-1}u\,{\rm
d}x\qquad\mbox{and}\qquad\|u\|^{2}_{H^{*}(\Omega)}=\int_{\Omega}u\,\mathcal{L}^{-1}u\,{\rm
d}x\,.$

Let us recall that when $\mathcal{L}=(-\Delta_{\Omega})$ then
$H(\Omega)=H^{1}_{0}(\Omega)$ and $H^{*}(\Omega)=H^{-1}(\Omega)$. We refer to
[13, 15] for more details about the space $H^{*}(\Omega)$ when $\mathcal{L}$
is a general nonlocal operator. Now, we introduce the class of solutions that
we will consider in this manuscript.

######  Definition 1.1.

($H^{*}$-solutions). Let $u\in L^{2}((0,T):H^{*}(\Omega))$, then $u$ is a
$H^{*}$-solution of (CPME) if the following hold:

  1. i)

We have

$k*(u-u_{0})\in
W^{1,2}((0,T):H^{*}(\Omega)),\qquad[k*(u-u_{0})](0)=0,\qquad|u|^{m-1}u\in
L^{2}((0,T):H(\Omega)).$

  2. ii)

For a.e. $t\in(0,T)$

$\int_{\Omega}D^{\alpha}_{t}u(t,x)\mathcal{L}^{-1}\varphi(x)\,{\rm
d}x=-\int_{\Omega}u^{m}(t,x)\varphi(x)\,{\rm d}x\qquad\quad\forall\varphi\in
H^{*}(\Omega).$ (3)

To ensure existence and uniqueness in this class of solutions, we use the
theory of fractional gradient flows in Hilbert spaces developed in [1, 37,
38]. The basic idea is that one would like to perform, in the more delicate
framework of Caputo time derivatives, the strategy typical of the Brezis-
Komura approach [20, 36]: gradient flow solutions are obtained by minimizing
at each time step a $\lambda$-convex energy functional, which in the case of
the Porous Medium Equation is

${\mathcal{E}}(v)=\frac{1}{1+m}\int_{\Omega}|v|^{1+m}\,.$

For the Caputo derivative, in [1] it is proved that the solutions of
fractional gradient flows associated to ${\mathcal{E}}$ are indeed
$H^{*}$-solutions according to the above definition.

Our first main result is a comparison principle for $H^{*}$-solutions, which
has its own interest, and it is also a key ingredient for proving the rest of
the main results presented in this paper. To the best of our knowledge, this
comparison result is new and a complete proof can be found in Section 3.2,
based on the discretized problem studied in Section 2. For the sake of
completeness, together with the comparison principle, we state existence and
uniqueness result proven by Akagi in [1] and by Li and Liu in [37].

######  Theorem 1.2 (Existence, Uniqueness and Comparison Principle).

Assume (G1). Then, for each $u_{0}\in L^{1+m}(\Omega)\cap H^{*}(\Omega)$ with
$m>0$, there exists a unique $H^{*}$-solution for (CPME). Moreover, let $u$
and $v$ be two $H^{*}$-solutions with initial data $u_{0},v_{0}\in
L^{1+m}(\Omega)\cap H^{*}(\Omega)$, respectively. If assumption (K) holds and
$u_{0}\leq v_{0}$ a.e. in $\Omega$, then for every $t>0$

$u(t)\leq v(t)\qquad\mbox{ a.e. in }\,\Omega\,.$ (4)

In particular, if $u_{0}\geq 0$ a.e. in $\Omega$, then $u(t)\geq 0$ a.e. in
$\Omega$ for every $t>0$.

The proof of the comparison result is based on a suitable discretization in
the spirit of [37], together with a clever limiting process: we prove
comparison first for the Euler implicit scheme associated to the $H^{*}$
“gradient flow structure” of (CPME). We study the discretized problem in
Section 2, where we show existence, uniqueness of solutions and a
T-contractivity property in a weighted $L^{1}$ space, cf. Theorem 2.4. The
T-contractivity estimate at the discretized level, allows us to deduce a
discrete comparison principle, cf. Corollary 2.5. This last result goes all
the way through the limiting process and allows to prove the desired
comparison principle in the continuous framework, cf. Section 3.

A direct – and important – consequence of the comparison principle and of the
time-scaling of the equation are the Benilan-Crandall time monotonicity
formulae that we present below. Let us emphasize that achieving any time
monotonicity in PDEs involving the Caputo time derivative is considerably more
challenging than with the classical time derivative, due to the “memory
effect”: when $D^{\alpha}_{t}f(t)\leq 0$, we can only deduce a stability
estimate, $f(t)\leq f(0)$, rather than the “full” monotonicity, i.e. $f(t)\leq
f(t_{0})$ for every $t_{0}<t$.

######  Theorem 1.3 (Time monotonicity formulae).

Assume (G1) and (K), then for any $u$ $H^{*}$-solution of (CPME) with
nonnegative initial datum $u_{0}\in L^{1+m}(\Omega)\cap H^{*}(\Omega)$, it
holds that:

  1. i)

If $0<m<1$, then the function

$t\mapsto t^{\frac{-\alpha}{1-m}}u(t,x)$

is non increasing for any $t>0$ and a.e $x\in\Omega$.

  2. ii)

If $m>1$, then the function

$t\mapsto t^{\frac{\alpha}{m-1}}u(t,x)$

is non decreasing for any $t>0$ and a.e $x\in\Omega$.

Section 4 is devoted to the proof of the new $L^{p}-L^{\infty}$ smoothing
effects and boundary estimates. As already mentioned, the only
$L^{p}-L^{\infty}$ regularizing effects involving the Caputo time derivative
present in the literature have been established for the Heat equation in the
whole space $\mathbb{R}^{N}$, see [34, Theorem 3.3.]: the proof relies on the
representation formula and on the fundamental solution which is unbounded and
has an explicit form. In our setting the proof is much more involved, due to
the lack of a representation formula and the nonlinearity.

######  Theorem 1.4 ($L^{p}-L^{\infty}$ Smoothing effects).

Let $m>0$, $m\neq 1$, $N>2s$. Assume that (G1) and (K) hold. Given a
$H^{*}$-solution $u$ with initial datum $u_{0}\in L^{p}(\Omega)\cap
H^{*}(\Omega)$ with $p>1$ satisfiying

$\begin{cases}p\geq 1+m,&\mbox{if }1+m>\frac{N}{2s},\\\
p>\frac{N}{2s},&\mbox{if }\frac{N}{2s}\geq 1+m,\end{cases}$ (5)

then, $u$ satisfies

$\|u(t)\|_{\infty}\leq\overline{\kappa}\,\frac{\|u_{0}\|_{p}^{\frac{1}{m}}}{t^{\frac{\alpha}{m}}}\qquad\quad\forall
t>0\,,$ (6)

with $\overline{\kappa}>0$ depending only on $N,s,\alpha,p,m,\Omega$.

Surprisingly, the decay rate $t^{-\frac{\alpha}{m}}$ of the $L^{\infty}$-norm
for $\alpha\in(0,1)$ presented in (6) is significantly different than in the
case $\alpha=1$. For the porous medium equation with classical time
derivative, the existence of the Friendly Giant provides that the decay of the
$L^{\infty}$-norm matches with the time scaling:

$\|u(t)\|_{\infty}\leq Kt^{-\frac{1}{m-1}}\qquad\forall t>1,\quad m>1\,,$ (7)

with a universal constant $K>0$, see [15, 17]. In addition, the smoothing
effects for small times provided in [13, 17] for the porous medium reads as

$\|u(t)\|_{\infty}\leq\overline{\kappa}\,\|u_{0}\|_{p}^{2sp\vartheta_{p}}t^{-N\vartheta_{p}}\qquad\forall
t>0,\quad\alpha=1,\quad m>0\,,$ (8)

where $u_{0}\in L^{p}(\Omega)$ with $p>\frac{N(1-m)}{2s}$ and
$\vartheta_{p}=(2sp-N(1-m))^{-1}$. However, for $\alpha\in(0,1)$, the time
scaling of (CPME) would suggest a $L^{\infty}$ decay of the form
$t^{-\frac{\alpha}{m-1}}$, but this is not the case, as (6) shows.

The condition $p\geq 1+m$ in Theorem 1.4 ensures that solutions are in the
energy space needed to define fractional gradient flows, see [1, 37]. On the
other hand, the condition $p>\frac{N}{2s}$ seems to be a natural threshold for
smoothing effects of (CPME). The key point is that the “memory” effect makes
$u$ behave like the solution of a suitable elliptic problem which always
remembers $u_{0}$,

$u^{m}(t,x_{0})\sim\int_{\Omega}\mathbb{G}_{\Omega}(x,x_{0})u_{0}(x)\,{\rm
d}x\,,$

where $\mathbb{G}_{\Omega}$ represents the Green function of the operator
$\mathcal{L}$, see for instance (40). Thus, the boundedness of $u$ depends
directly on the integrability of $\mathbb{G}_{\Omega}$, which by (G1) is
$L^{q}(\Omega)$ with $q<\frac{N}{N-2s}$. This result is also motivated by the
explicit solution of the Caputo Heat Equation in $\mathbb{R}^{N}$, where
bounded solutions require $u_{0}\in L^{p}(\Omega)$ with $p>\frac{N}{2s}$ due
to the unboundedness of the fundamental solution, see [24, 34]. This shows a
significant discrepancy between the smoothing effects of Caputo PME and
classical PME. We recall that for the latter we require $u_{0}\in
L^{p}(\Omega)$ with $p>\frac{N(1-m)}{2s}$, which implies the relaxation of the
integrability condition for $u_{0}$ when $m>0$ gets bigger. See the picture
below.

Classical PME
$\mathbf{0}$p$\mathbf{1}$${m}$$\mathbf{p=\frac{N(1-m)}{2s}}$$\frac{N}{2s}$--$\mathbf{1}$$\frac{N-2s}{N}$$L^{p}\to
L^{\infty}$$L^{1}\to L^{\infty}$ Caputo PME
$\mathbf{0}$p${m}$$\frac{N}{2s}$--$\mathbf{1}$$\frac{N}{2s}-1$$\mathbf{1}$$L^{p}\to
L^{\infty}$$\mathbf{p=1+m}$ Figure 1: On the left side, the smoothing effects
(8) for the PME with classical time derivative.
On the right side, the smoothing effects (6) for the PME with Caputo time
derivative.

The proof of Theorem 1.4 is based on the so-called Green function method,
introduced in [17] for $m>1$ and in [13] for $m<1$. The key tool of the proof
is the pointwise bound shown in Proposition 4.4, which plays the role of an
“almost representation formula”. This pointwise bound is a direct consequence
of the dual formulation

$\mathcal{L}^{-1}D^{\alpha}_{t}u=-u^{m}\,,$

and time monotonicity formula of Theorem 1.3. This approach replaces the
classical Moser iteration, which seems more challenging due to the memory
effect of the time derivative.

Thanks to the Green function method, we are able to show stronger boundary
estimates, and in particular we ensure that solutions satisfy the zero
Dirichlet boundary condition in a precise and quantitative way. The estimate
we provide involves a parameter $\gamma>0$, which characterizes the behaviour
of the first eigenfunction of $\mathcal{L}$. The value of $\gamma$ depends on
the diffusion operator under consideration, see Section 1.2 for further
explanations.

######  Theorem 1.5 (Upper boundary behaviour).

Let $m>0$, $m\neq 1$, $N>2s>\gamma$. Assume that (G2) and (K) hold. Given a
$H^{*}$-solution $u$ with initial datum $u_{0}\in L^{p}(\Omega)\cap
H^{*}(\Omega)$ with $p$ satisfying

$\begin{cases}p\geq 1+m,&\mbox{if }1+m>\frac{N}{2s-\gamma},\\\
p>\frac{N}{2s-\gamma},&\mbox{if }\frac{N}{2s-\gamma}\geq 1+m,\end{cases}$

then, we have

$\left\|\frac{u^{m}(t)}{\mbox{\rm
dist}(\cdot,\partial\Omega)^{\gamma}}\right\|_{\infty}\leq\overline{\kappa}\frac{\|u_{0}\|_{p}}{t^{\alpha}},\qquad\quad\forall
t>0\,,$ (9)

with $\overline{\kappa}$ depending only on $N,s,\alpha,p,m,\Omega$.

The upper bound (9) is sharp, since at least for “big initial data” we provide
matching lower bounds:

######  Theorem 1.6 (Global Harnack Principle for “big” initial data).

Under the assumptions of Theorem 1.5, assume moreover that
$m>\frac{N-2s}{N+2s}$ and that the initial data also satisfies
$u_{0}(x)\geq\mathsf{c}\,\mbox{\rm dist}(x,\partial\Omega)^{\frac{\gamma}{m}}$
for a.e. $x\in\Omega$ and some $\mathsf{c}>0$. Then, we have the following
Global Harnack Principle

$\frac{c_{0}}{1+t^{\alpha}}\leq\frac{u^{m}(t,x)}{\mbox{\rm
dist}(x,\partial\Omega)^{\gamma}}\leq\frac{c_{1}}{t^{\alpha}}\qquad\mbox{ for
a.e. }x\in\Omega\quad\forall t>0\,.$ (10)

with $c_{0},c_{1}>0$ depending only on $\mathsf{c},N,s,\alpha,p,m,\Omega$ and
$\|u_{0}\|_{p}$.

The above Global Harnack Principle (10), follows by the comparison principle
and from the form of the solution(s) of separation of variable. The proof of
the above results is contained in Section 4.

The next natural question is whether the asymptotic behaviour of the solution
is similar to the one with the classical time derivative. Surprisingly, the
answer is negative. We have already commented on the discontinuity in $\alpha$
of the time behaviour, we now state the precise result: all the $L^{p}$-norms
decay in time with the same rate as $t$ goes to infinity, for all $m>0$. In
particular, we would like to focus on the fast diffusion regime $m\in(0,1)$,
and compare the cases $\alpha=1$ and $0<\alpha<1$. In the latter case,
solutions do not extinguish in finite time, contrary to the case $\alpha=1$,
cf. [13], and this is one of the biggest differences between the classical and
the Caputo time derivatives.

######  Theorem 1.7 (Sharp Time Decay of $L^{p}$-norms).

Assume (G1) and (K). Let $u$ be a $H^{*}$-solution of (CPME) with nonnegative
initial datum $u_{0}\in L^{p_{0}}(\Omega)\cap H^{*}(\Omega)$ with $p_{0}>1$
such that

$\begin{cases}p_{0}\geq 1+m,&\mbox{if }1+m>\frac{N}{2s},\\\
p_{0}>\frac{N}{2s},&\mbox{if }\frac{N}{2s}\geq 1+m.\end{cases}$

Then, for every $1\leq p\leq\infty$ it holds that

$\frac{c_{0}}{1+t^{\frac{\alpha}{m}}}\leq\|u(t)\|_{p}\leq\frac{c_{1}}{t^{\frac{\alpha}{m}}}\qquad\forall
t>0\,,$ (11)

where $c_{0},c_{1}>0$ depends on $N,s,\alpha,p,m,\Omega,\,\|u_{0}\|_{p_{0}}$.

The proof of (11) follows by the smoothing effects, abound from above the
$L^{\infty}$-norm, and a lower bound for a weighted $L^{1}_{\Phi_{1}}$-norm,
inspired by [50]. More detailed comments and a complete proof can be found in
Section 5.

###  1.2 Assumptions and functional setting of the operator $\mathcal{L}$

Let us present the main assumptions on the operator $\mathcal{L}$ and its
inverse operator $\mathcal{L}^{-1}$. These assumptions are the standard ones
in the literature, see [11, 12, 13, 15, 17, 18].

Assumptions on $\mathcal{L}^{-1}$. Our approach relies on the dual formulation
of (CPME), so that we assume the existence of an inverse operator
$\mathcal{L}^{-1}:L^{2}(\Omega)\rightarrow L^{2}(\Omega)$ for $\mathcal{L}$
represented through a kernel $\mathbb{G}_{\Omega}$:

$\mathcal{L}^{-1}u(x):=\int_{\Omega}\mathbb{G}_{\Omega}(x,y)u(y)\,{\rm d}y\,.$

The main assumption on the Green function $\mathbb{G}_{\Omega}$ is that there
exists $c_{1,\Omega}>0$ such that

$0\leq\mathbb{G}_{\Omega}(x,y)\leq\frac{c_{1,\Omega}}{|x-y|^{N-2s}}\,,$ (G1)

for some $s\in(0,1]$. Assumption (G1) guarantees the compactness of the
operator $\mathcal{L}^{-1}$ in $L^{2}(\Omega)$, see [11, Proposition 5.1]. As
a consequence, the operator $\mathcal{L}^{-1}$ has a discrete spectrum and a
$L^{2}(\Omega)$ orthonormal basis of eigenfunctions that we denote by
$\left(\lambda_{k}^{-1},\Phi_{k}\right)_{k\geq 1}$. Accordingly, we can
express

$\mathcal{L}^{-1}u(x)=\sum_{k=1}^{\infty}\lambda_{k}^{-1}\hat{u}_{k}\,\Phi_{k}(x)\,\qquad\mbox{with}\quad\hat{u}_{k}=\int_{\Omega}u(y)\Phi_{k}(y)\,{\rm
d}y\,.$

Furthermore, assumption (G1) also implies that $\mathcal{L}$ has the discrete
spectrum $\left(\lambda_{k},\Phi_{k}\right)_{k\geq 1}$ and then

$\mathcal{L}u(x)=\sum_{k=1}^{\infty}\lambda_{k}\hat{u}_{k}\,\Phi_{k}(x)\,.$
(12)

Hence, assumption (G1) is enough to construct the Hilbert space
$H^{*}(\Omega)$ – the functional space where fractional gradient flow theory
will be applied – as the topological dual space of

$H(\Omega)=\left\\{v\in
L^{2}(\Omega)\quad:\quad\|v\|_{H(\Omega)}^{2}=\sum_{k=1}^{\infty}\lambda_{k}\,\left|\hat{v}_{k}\right|^{2}<+\infty\right\\}\,.$

Note that the scalar product and the norm of $H^{*}(\Omega)$ now read as

$\langle
u,v\rangle_{H^{*}(\Omega)}=\sum_{k=1}^{\infty}\lambda_{k}^{-1}\hat{u}_{k}\,\hat{v}_{k}\qquad\mbox{and}\qquad\|u\|^{2}_{H^{*}(\Omega)}=\sum_{k=1}^{\infty}\lambda_{k}^{-1}|\hat{u}_{k}|^{2}\,.$

We refer to [15] for a proper identification of the functional spaces
$H(\Omega)$ and $H^{*}(\Omega)$ depending on the specific choice of the
operator $\mathcal{L}$. See also [11, 12, 18].

When dealing with boundary issues it is convenient to assume that the kernel
$\mathbb{G}_{\Omega}$ satisfies the following condition for some
$s,\gamma\in(0,1]$: there exist $c_{0,\Omega},c_{1,\Omega}>0$ such that for
a.e. $x,y\in\Omega$

$c_{0,\Omega}{\delta_{\gamma}}(x){\delta_{\gamma}}(y)\leq\mathbb{G}_{\Omega}(x,y)\leq\frac{c_{1,\Omega}}{\lvert
x-y\rvert^{N-2s}}\left(\frac{{\delta_{\gamma}}(x)}{\lvert
x-y\rvert^{\gamma}}\wedge 1\right)\left(\frac{{\delta_{\gamma}}(y)}{\lvert
x-y\rvert^{\gamma}}\wedge 1\right)\,,$ (G2)

where $\delta_{\gamma}(x)=\mbox{dist}(x,\partial\Omega)^{\gamma}$. It was
proven in [11] that (G2) implies that the first eigenfunction of
$\mathcal{L}$, namely, $\mathcal{L}\Phi_{1}=\lambda_{1}\Phi_{1}$ with
$\Phi_{1}\geq 0$ and some constant $\lambda_{1}>0$, satisfies
$\Phi_{1}(x)\asymp\mbox{dist}(x,\partial\Omega)^{\gamma}$. Thus, we can
rewrite assumption (G2) as follows

$\mathsf{c}_{0,\Omega}\Phi_{1}(x)\Phi_{1}(y)\leq\mathbb{G}_{\Omega}(x,y)\leq\frac{\mathsf{c}_{1,\Omega}}{\lvert
x-y\rvert^{N-2s}}\left(\frac{\Phi_{1}(x)}{\lvert x-y\rvert^{\gamma}}\wedge
1\right)\left(\frac{\Phi_{1}(y)}{\lvert x-y\rvert^{\gamma}}\wedge 1\right)\,.$
(G3)

Assumptions (G2) and (G3) will be used only in Theorem 1.5 and Theorem 1.6. In
the rest of the results we just assume (G1) on $\mathbb{G}_{\Omega}$.

Assumptions on $\mathcal{L}$. Let $\mathcal{L}:\mbox{\rm
dom}(\mathcal{L})\subset L^{2}(\Omega)\rightarrow L^{2}(\Omega)$ be a
positive, possibly unbounded and self-adjoint operator. From assumption (G1),
we know that the operator $\mathcal{L}$ can be expressed in its spectral form
as in (12), and hence the integration by parts formula follows:

$\int_{\Omega}u\mathcal{L}v\,{\rm
d}x=\int_{\Omega}\mathcal{L}^{\frac{1}{2}}u\mathcal{L}^{\frac{1}{2}}v\,{\rm
d}x=\int_{\Omega}v\mathcal{L}u\,{\rm d}x\qquad\forall u,v\in H(\Omega)\,.$

The main assumption that we need on the operator $\mathcal{L}$ is the
following weak form of Kato’s inequality: for any convex function $h\in
C^{1}(\mathbb{R})$ with $h(0)=0$ and any function $u\in L^{1}_{\rm
loc}(\Omega)$ with $h(u)\in H^{*}(\Omega)$, it holds that (recall
$\Phi_{1}\geq 0$ is the first eigenfunction of $\mathcal{L}$)

$\int_{\Omega}\mathcal{L}\left[h(u)\right]\Phi_{1}\,{\rm
d}x\leq\int_{\Omega}h^{\prime}(u)\mathcal{L}u\,\Phi_{1}\,{\rm d}x\,.$ (K)

######  Remark 1.

We are not aware of any positive, linear operator satisfying (G1) which does
not satisfy Kato’s inequality. Indeed, for the classical Dirichlet Laplacian,
$\mathcal{L}=(-\Delta_{\Omega})$, Kato’s inequality holds pointwise since
$h^{\prime\prime}(r)\geq 0$ when $h$ is a convex function. For nonlocal
operators of the form

$\mathcal{L}u(x)=\int_{\mathbb{R}^{N}}\left(u(x)-u(y)\right)\mathbb{K}(x,y)\,{\rm
d}y+B(x)u(x)\,,$ (13)

with $\mathbb{K}(x,y)\geq 0$ and $B(x)\geq 0$, Kato’s inequality also holds in
a stronger form than (K) as it can be seen in the Appendix of [13].

Notations.  We will write the $L^{p}$ norm in $\Omega$ as
$\|f\|_{p}:=(\int_{\Omega}|f|^{p}\,{\rm d}x)^{1/p}$, but for weighted norms
with the first eigenvalue $\Phi_{1}$ of the operator $\mathcal{L}$ we use
$\|f\|_{L^{p}_{\Phi_{1}}(\Omega)}:=\left(\int_{\Omega}|f|^{p}\Phi_{1}\,{\rm
d}x\right)^{1/p}$ in the space $L^{p}_{\Phi_{1}}(\Omega):=\\{f\in L^{1}_{\rm
loc}(\Omega):\|f\|_{L^{p}_{\Phi_{1}}(\Omega)}<+\infty\\}$. The Heaviside
function will be denoted by $\mbox{sign}_{+}(f):=\max\\{0,f\\}$, and positive
and negative part as $(f)_{+}:=\max\\{0,f\\}$ and $(f)_{-}=\max\\{0,-f\\}$,
respectively. We will use $f\asymp g$ when there exists $c_{1},c_{2}>0$ such
that $c_{1}g(x)\leq f(x)\leq c_{2}g(x)$ for a.e. $x$.

###  1.3 Examples of $\mathcal{L}$

The assumptions that we state above describe a wide class of operators of
local and nonlocal type. Let us enumerate some of the most important ones
here. For a more extended list of examples we refer to [13] and [18, Section
3].

The Dirichlet Laplacian. We can consider the classical Dirichlet Laplacian
$\mathcal{L}=(-\Delta_{\Omega})$ with $s=1$, where the Green function
$\mathbb{G}_{\Omega}$ is bounded by the classical Newtonian potential. For
this operator (G2) is satisfied with $\gamma=1$.

The main examples of nonlocal operators on bounded domains are the three
fractional Laplacians that we state below. Note that, on the whole space
$\mathbb{R}^{N}$, the three definitions are equivalent, but on bounded domains
they represent entirely different nonlocal operators, as highlighted by the
behaviour of the first eigenfunction for each of them. Indeed, from a
probabilistic standpoint, these operators are the infinitesimal generator of
distinct Lévy processes.

The Restricted Fractional Laplacian (RFL). We define this fractional Laplacian
operator acting on a bounded domain by using the integral representation on
the whole space in terms of a hypersingular kernel, namely,

$(-\Delta_{|\Omega})^{s}f(x)=c_{N,s}\;\mbox{P.V.
}\int_{\mathbb{R}^{N}}\frac{f(x)-f(y)}{|x-y|^{N+2s}}\,{\rm d}y\,,$ (14)

where $s\in(0,1)$ and we restrict the operator to functions that are zero
outside $\Omega$. For this operator (G2) is satisfied with $\gamma=s$.

Spectral Fractional Laplacian (SFL). Consider the so-called _spectral
definition_ of the fractional power $s\in(0,1)$ of the classical Dirichlet
Laplacian $\Delta_{\Omega}$ on $\Omega$ defined by a formula in terms of
semigroups, namely

$(-\Delta_{\Omega})^{s}f(x)=\frac{1}{\Gamma(-s)}\int_{0}^{\infty}(e^{t\Delta_{\Omega}}f(x)-f(x))\frac{\,{\rm
d}t}{t^{1+s}}=\sum_{j=1}^{\infty}\lambda_{j}^{s}\hat{f}_{j}\phi_{j},$ (15)

where $(\lambda_{k},\phi_{k})_{k\geq 1}$ are the eigenvalues and
eigenfunctions of the classical Dirichlet Laplacian on $\Omega$,
$\hat{f}=\int_{\Omega}f(x)\phi_{k}(x)\,{\rm d}x$, and $\|\phi_{k}\|_{2}=1$.
Since the eigenvalues are the same as in the Dirichlet Laplacian, (G2) is
fulfilled with $\gamma=1$.

Censored Fractional Laplacian (CFL).  This third kind of Fractional Laplacian
was introduced in [21], in connection with censored stable processes. The
operator takes the form:

$\mathcal{L}f(x)=\mathrm{P.V.}\int_{\Omega}\frac{f(x)-f(y)}{|x-y|^{N+2s}}\,{\rm
d}y\,,\qquad\mbox{with }\frac{1}{2}<s<1\,.$ (16)

The restriction on the parameter $s$ is due to the lack of trace inequality
for $s\in(0,1/2]$. This operator satisfies (G2) with $\gamma=2s-1$.

Other examples: As explained in [18, Section 3], our theory may also be
applied to: i) the operators above with uniformly elliptic, $C^{1}$ and
symmetric coefficients; ii) sums of two fractional operators; iii) the sum of
the classical Laplacian and a nonlocal operator kernel; iv) Schrödinger
equations for non-symmetric diffusions; v) gradient perturbation of restricted
fractional Laplacians.

Functional spaces associated to $\mathcal{L}$ We recall that the three
fractional Laplacians on domains, (14), (15) and (16), can be expressed as in
(13), see Section 1.3 and [18] for more details and examples. Moreover, for
(14), (15) and (16), it is possible to identify $H(\Omega)$ and
$H^{*}(\Omega)$ with the standard fractional Sobolev spaces as follows: Let
$s\in(0,1]$

$H(\Omega)=\begin{cases}H_{0}^{s}(\Omega),&\mbox{if }s>1/2\\\
H^{1/2}_{00}(\Omega),&\mbox{if }s=1/2\\\ H^{s}(\Omega),&\mbox{if
}s<1/2,\end{cases}\qquad\mbox{and}\qquad
H^{*}(\Omega)=\begin{cases}H^{-s}(\Omega),&\mbox{if }s>1/2\\\
\left(H^{1/2}_{00}(\Omega)\right)^{*},&\mbox{if }s=1/2\\\
H^{-s}(\Omega),&\mbox{if }s<1/2.\end{cases}$

Recall that $H_{0}^{s}(\Omega)=H^{s}(\Omega)$ when $s\leq 1/2$, and hence the
dual coincide. If $s=1/2$, then $H^{s}_{00}(\Omega)\subsetneq
H^{s}_{0}(\Omega)$, which implies
$H^{-s}(\Omega)\subsetneq\left(H^{s}_{00}(\Omega)\right)^{*}$. See [15, Sec.
7.7] for more details. This identification holds for any operator
$\mathcal{L}$ with discrete spectrum such that
$0<\lambda_{k+1}/\lambda_{k}\leq\Lambda_{0}$ for some $\Lambda_{0}>0$.

##  2 Discretized problem

In this section we will provide some properties of the discretization of
(CPME) and then deduce the comparison principle. Several discretizations of
the Caputo derivative have been proposed through the years, see for instance
[27, 37, 38]. We choose the one in [37] based on a deconvolution scheme, since
it is enough to obtain the convergence results needed in this work.

######  Definition 2.1.

[37, Section 3.] Let $\tau=T/N>0$ be the time-step and $t_{k}=k\tau$ with
$0\leq k\leq N$ points of the interval $(0,T)$. Let us consider the Euler
implicit scheme for (CPME):

$\begin{cases}(D^{\alpha}_{\tau}U)_{k}=-\mathcal{L}U_{k}^{m}&\mbox{in}\quad\Omega,\quad\forall
k\geq 0\,,\\\ U_{0}=u_{0}&\mbox{in}\quad\Omega\,,\end{cases}$ (17)

where

$\left(D^{\alpha}_{\tau}U\right)_{k}:=\frac{1}{\tau^{\alpha}}\left(c_{k}^{k}(U_{k}-U_{0})+\sum_{i=1}^{k-1}c_{i}(U_{k}-U_{k-i})\right)\,,$
(18)

with $c_{k}^{k}:=\sum_{i=k}^{\infty}c_{i}>0$ and $c_{i}>0$ are defined
recursively by

$c_{i}:=\sum_{j=0}^{i-1}\left((i+1-j)^{\alpha}-(i-j)^{\alpha}\right)c_{j}\qquad\mbox{for
any }i\geq 1\,,$

with $c_{0}:=\Gamma(1+\alpha)=c_{k}^{k}+\sum_{i=1}^{k-1}c_{i}$. Equivalently,
we also will use

$(D^{\alpha}_{\tau}X)_{k}=\frac{1}{\tau^{\alpha}}\left(c_{0}(X_{k+1}-X_{0})-\sum_{i=1}^{k-1}c_{i}(X_{k-i}-X_{0})\right)\,.$
(19)

One of the drawbacks of the Caputo derivative is the lack of a chain rule.
However it is noted in [1, 51], that if the kernel $k$ belongs to
$H^{1}([0,T])$, then the “chain rule inequality” below holds for convex
functions $h\in C^{1}(\mathbb{R}^{N})$, see Lemma A.4. Namely, we have that

$\partial_{t}[k*(h(f)-h(f_{0}))](t)\leq
h^{\prime}(f(t))\partial_{t}[k*(f-f_{0})](t)\,.$

Another remarkable difference with the classical time derivative is the fact
that $D^{\alpha}_{t}f(t)\leq 0$ only implies $f(t)\leq f(0)$, but not the
monotonicity $f(t)\leq f(t_{0})$ for all $t\geq t_{0}$.

Also the discrete Caputo derivative (18) satisfies the monotonicity formula
and the chain rule inequality:

######  Proposition 2.2.

[37, Theorem 3.1.]

  1. i)

(Chain rule inequality). Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be a convex
function in $C^{1}(\mathbb{R})$. Then, for every sequence of functions
$\\{U_{k}\\}_{k\geq 0}\subset L^{1}_{loc}(\Omega)$ it holds that

$(D^{\alpha}_{\tau}\,h(U))_{k}\leq
h^{\prime}(U_{k})\,(D^{\alpha}_{\tau}U)_{k}\qquad\mbox{a.e. in }\Omega\,.$

  2. ii)

(Nonpositive discrete Caputo derivative). Assume that a sequence
$\\{X_{k}\\}_{k\geq 0}\subset\mathbb{R}$ satisfies
$(D^{\alpha}_{\tau}X)_{k}\leq 0$ for every $k\geq 1$, then

$X_{k}\leq X_{0}\qquad\forall k\geq 1\,.$

###### Proof.

  1. i)

Since $h\in C^{1}(\mathbb{R})$ is convex , we have that
$h^{\prime}(x)(x-y)\geq(h(x)-h(y))$ for every $x,y\in\mathbb{R}$. Then,

$\displaystyle h^{\prime}(U_{k})(D^{\alpha}_{\tau}u)_{k}$
$\displaystyle=\frac{1}{\tau^{\alpha}}\left(h^{\prime}(U_{k})(U_{k}-U_{0})c_{k}^{k}+\sum_{i=1}^{k-1}h^{\prime}(U_{k})(U_{k}-U_{k-i})c_{i}\right)$
$\displaystyle\geq\frac{1}{\tau^{\alpha}}\left(c_{k}^{k}\left(h(U_{k})-h(U_{0})\right)+\sum_{i=1}^{k-1}c_{i}(h(U_{k})-h(U_{k-i}))\right).$

  2. ii)

We prove the result by induction. For the case $k=1$ is trivial since
$\tau,c_{0}>0$. Now, assume that $X_{k}\leq X_{0}$ for every $k=1,\dots n$,
and let us prove it for $k=n+1$. From (19) and $(D^{\alpha}_{c}X)_{k+1}\leq 0$
we know that

$(D^{\alpha}_{\tau}X)_{k+1}=\frac{1}{\tau^{\alpha}}\left((c_{0}(X_{k+1}-X_{0})-\sum_{i=1}^{k}c_{i}(X_{k+1-i}-X_{0})\right)\leq
0\,,$

which by the induction hypothesis give us

$c_{0}(X_{k+1}-X_{0})\leq\sum_{i=1}^{k}c_{i}(X_{k+1-i}-X_{0})\leq 0\,,$

since $c_{i}>0$ for every $i\geq 0$.

∎

###  2.1 Existence and uniqueness for the discretized problem

The method used to prove existence of solutions for (17) is a time-fractional
version of the classical minimizing movements scheme [37, 38]. As in the case
of classical time derivative [13], the energy used to construct the gradient
flow solutions is

${\mathcal{E}}(u):=\begin{cases}\frac{1}{1+m}\int_{\Omega}|u(x)|^{1+m}\,{\rm
d}x,&\mbox{if }u\in L^{1+m}(\Omega),\\\ +\infty,&\mbox{otherwise}.\end{cases}$

Notice that this energy functional is convex and lower semicontinuous in the
Hilbert space $H^{*}(\Omega)$, which is the (topological) dual space of
$H(\Omega)$, the domain of the quadratic form associated to $\mathcal{L}$,

$H(\Omega)=\left\\{u\in
L^{2}(\Omega)\;:\;\|u\|^{2}_{H(\Omega)}=\int_{\Omega}u\mathcal{L}u\,{\rm
d}x<+\infty\right\\}\,.$

Let us recall that the space $H^{*}(\Omega)$ is a Hilbert space endowed with
the scalar product and the norm

$\langle u,v\rangle_{H^{*}(\Omega)}=\int_{\Omega}u\mathcal{L}^{-1}v\,{\rm
d}x\qquad\mbox{and}\qquad\|u\|^{2}_{H^{*}(\Omega)}=\int_{\Omega}u\mathcal{L}^{-1}u\,{\rm
d}x\,.$

The classical gradient flow theory in Hilbert spaces – in particular, in
$H^{*}(\Omega)$ – provides solutions of

$-\partial_{t}u(t)\in\partial{\mathcal{E}}(u(t))\qquad\mbox{for a.e. }t>0\,,$
(20)

where the subdifferential of ${\mathcal{E}}$ is defined by

$\xi\in\partial{\mathcal{E}}(u)\quad\Leftrightarrow\quad{\mathcal{E}}(v)-{\mathcal{E}}(u)\geq\langle\xi,v-u\rangle_{H^{*}(\Omega)}\qquad\forall
v\in H^{*}(\Omega)\,.$

In our case, note that (20) is related to the classical PME on domains; note
in fact that for any $u$ such that

$u\in D(\partial{\mathcal{E}})=\\{v\in L^{1+m}(\Omega)\cap
H^{*}(\Omega)\;:\;|v|^{m-1}v\in H(\Omega)\\}\,,$

we have, in the $H^{*}(\Omega)$ sense,

$\xi\in\partial{\mathcal{E}}(u)\qquad\Leftrightarrow\qquad\mathcal{L}^{-1}\xi=|u|^{m-1}u\,.$

In the fractional gradient flow theory, the classical time derivative
$\partial_{t}u$ in (20) is substituted by a fractional time derivative, such
as the Caputo derivative, $D^{\alpha}_{t}u$. The well-posedness of fractional
gradient flow solutions in Hilbert spaces has been proven in [1, 37, 38]. The
first step in this approach is to provide the existence and uniqueness of the
time discretized equation (17), that we stated below.

######  Theorem 2.3.

[37, 38] Let $m>0$ and assume (G1). For any $U_{0}\in L^{1+m}(\Omega)\cap
H^{*}(\Omega)$ fixed there exists a unique sequence of functions
$\\{U_{k}\\}_{k\geq 0}\subset D(\partial{\mathcal{E}})$ satisfying

$-(D^{\alpha}_{\tau}U)_{k}\in\partial{\mathcal{E}}(U_{k})\qquad\forall k\geq
0\,,$ (21)

which is (17) in the $H^{*}$-sense, namely, for every $k\geq 0$

$\int_{\Omega}(D^{\alpha}_{\tau}U)_{k}\mathcal{L}^{-1}\varphi\,{\rm
d}x=-\int_{\Omega}U_{k}^{m}\varphi\,{\rm d}x\qquad\forall\varphi\in
H^{*}(\Omega)\,.$ (22)

Moreover,

${\mathcal{E}}(U_{k})\leq{\mathcal{E}}(U_{0})\qquad\forall k\geq 1\,.$ (23)

The idea of the proof is the same of the nonlinear theory of Brezis for
gradient flows [20], see also Komura [36] and the excellent lecture notes [4].
Due to the convexity of ${\mathcal{E}}$ we can construct recursively the next
function $U_{k+1}$ with $k\geq 0$ by the minimizing movements scheme, namely,
for any $k\geq 0$

$U_{k+1}:=\underset{u\in
H^{*}(\Omega)}{\mbox{argmin}}\left[\frac{1}{2\tau^{\alpha}}\left(c_{k+1}^{k+1}\|u-U_{0}\|^{2}_{H^{*}(\Omega)}+\sum_{i=1}^{k}c_{i}\|u-U_{n-i}\|_{H^{*}(\Omega)}^{2}\right)+{\mathcal{E}}(u)\right]\,.$
(24)

By the lower semicontinuity and convexity of the energy functional the
sequence $U_{k}$ is unique and satisfies (21). The decay of the energy (23)
follows from Proposition 2.2, namely,

$\displaystyle(D^{\alpha}_{\tau}{\mathcal{E}}(U))_{k+1}$
$\displaystyle\leq\int_{\Omega}U^{m}_{k+1}(D^{\alpha}_{\tau}U)_{k+1}\,{\rm
d}x=-\int_{\Omega}U_{k+1}^{m}\,\mathcal{L}U^{m}_{k+1}\,{\rm
d}x=-\|U_{k+1}^{m}\|^{2}_{H^{*}(\Omega)}\leq 0\,.$

###  2.2 Discrete Comparison Principle

The main result of this section is the proof of the comparison principle for
the discrete solutions obtained in Theorem 2.3. For this purpose, we first
prove a T-contractivity estimate in the Banach space
$L^{1}_{\Phi_{1}}(\Omega)$. Recall that assumption (G2) implies that the
operator $\mathcal{L}$ has a discrete spectrum $(\lambda_{k},\Phi_{k})_{k\geq
1}$ with $\Phi_{k}\in L^{\infty}(\Omega)$. Indeed, eigenfunctions are
classical in the interior, i.e. $C^{2s+\gamma}$, whenever the operator allows
it, and have sharp boundary regularity $C^{\gamma}(\overline{\Omega})$.
Moreover,

$\underline{\kappa}\;\mbox{dist}(x,\partial\Omega)^{\gamma}\leq\Phi_{1}(x)\leq\overline{\kappa}\;\mbox{dist}(x,\partial\Omega)^{\gamma}\,,$

see [11, Theorem 7.1. and 7.2.] for more details.

Let us remark that for nonnegative functions $u$ the belong to $H^{*}(\Omega)$
also belongs to $L^{1}_{\Phi_{1}}(\Omega)$:

$\|u\|_{L^{1}_{\Phi_{1}}(\Omega)}=\int_{\Omega}u\,\Phi_{1}\,{\rm
d}x=\int_{\Omega}\mathcal{L}^{-1/2}u\,\mathcal{L}^{1/2}\Phi_{1}\leq\lambda_{1}^{1/2}\|u\|_{H^{*}(\Omega)},$
(25)

since $u\geq 0$ and $\|\Phi_{1}\|_{L^{2}(\Omega)}=1$.

The theorem below follows the formal proof of the classical T-contractivity
estimate in $L^{1}_{\Phi_{1}}$ [13, Theorem 2.4.] and the rigorous techniques
used to show the $L^{1}$-contractivity for the linear equation with Caputo
derivative [51, Theorem 7.1.]. Both techniques require extra regularity on the
time derivative, more precisely $\partial_{t}u\in
L^{1}((0,T):L^{1}_{\Phi_{1}}(\Omega))$ or $D^{\alpha}_{t}u\in
L^{\infty}((0,T):L^{1}(\Omega))$, respectively. Since we only expect,
a-priori, that $D^{\alpha}_{t}u\in L^{2}((0,T):H^{*}(\Omega))$, we first prove
T-contractivity at the discrete level, and this allows to show comparison
principle as $\tau\to 0$.

######  Theorem 2.4.

(Discrete T-contractivity in $L^{1}_{\Phi_{1}}(\Omega)$). Let $m>0$ and assume
(G1) and (K). Consider $\\{U_{k}\\}_{k\geq 0},\\{V_{k}\\}_{k\geq 0}\subset
L^{1+m}(\Omega)\cap H^{*}(\Omega)$ two sequence of solutions for (17). Then,

$\|\left(U_{k}-V_{k}\right)_{+}\|_{L^{1}_{\Phi_{1}}(\Omega)}\leq\|(U_{0}-V_{0})_{+}\|_{L^{1}_{\Phi_{1}}(\Omega)}\qquad\forall
k\geq 1\,.$

###### Proof.

Notice that $U_{k},V_{k}\in L^{1+m}(\Omega)\subset L^{1}_{\Phi_{1}}(\Omega)$
for every $k\geq 0$, since $\Phi_{1}\in L^{\infty}(\Omega)$. Therefore, from
the definition of discrete Caputo derivative, we obtain

$(D^{\alpha}_{\tau}U)_{k},\;(D^{\alpha}_{\tau}V)_{k}\in
L^{1}_{\Phi_{1}}(\Omega)\qquad\forall k\geq 1\,.$

We will show that
$\left(D^{\alpha}_{\tau}\|(U-V)_{+}\|_{L^{1}_{\Phi_{1}}(\Omega)}\right)_{k}\leq
0$ for every $k\geq 1$ and then apply Proposition 2.2. For this purpose, let
us consider $h_{\varepsilon}\in C^{\infty}(\mathbb{R})$ with $\varepsilon>0$
definde by

$h_{\varepsilon}(x)=\sqrt{(x_{+})^{2}+\varepsilon^{2}}-\varepsilon,\qquad
x\in\mathbb{R}\,,$

which is a convex and smooth approximation of $(x)_{+}$. Moreover,
$|h_{\varepsilon}(x)|\leq|x|$ and $|h^{\prime}_{\varepsilon}(x)|\leq 1$ since

$h^{\prime}_{\varepsilon}(x)=\frac{(x)_{+}}{\sqrt{x^{2}+\varepsilon^{2}}}.$

Next, we test equation (22) for $U_{k}$ and $V_{k}$ with
$\mathcal{L}(h^{\prime}_{\varepsilon}(U_{k}^{m}-V_{k}^{m})\Phi_{1})\in
H^{*}(\Omega)$; after subtracting the resulting equations, we get

$\int_{\Omega}\left(D^{\alpha}_{\tau}(U-V)\right)_{k}h^{\prime}_{\varepsilon}(U^{m}_{k}-V^{m}_{k})\Phi_{1}\,{\rm
d}x=-\int_{\Omega}\mathcal{L}(U^{m}_{k}-V^{m}_{k})h^{\prime}_{\varepsilon}(U_{k}^{m}-V_{k}^{m})\Phi_{1}\,{\rm
d}x\,.$ (26)

Using the convexity, the nonnegativity of $h_{\varepsilon}$, and Kato’s
inequality (K), we bound the right hand side as

$\displaystyle-\int_{\Omega}\mathcal{L}(U^{m}_{k}-V^{m}_{k})\,h^{\prime}_{\varepsilon}(U_{k}^{m}-V_{k}^{m})\Phi_{1}\,{\rm
d}x$
$\displaystyle\leq-\int_{\Omega}\mathcal{L}\left[h_{\varepsilon}(U_{k}^{m}-V_{k}^{m})\right]\Phi_{1}\,{\rm
d}x$
$\displaystyle=-\lambda_{1}\int_{\Omega}\left(h_{\varepsilon}(U^{m}_{k}-V^{m}_{k})\right)\Phi_{1}\,{\rm
d}x\leq 0\,,$

since $\Phi_{1}$ is the first eigenfunction and $\Phi_{1}\geq 0$. Thus, we
deduce from (26) and Proposition 2.2 that

$\displaystyle\left(D^{\alpha}_{\tau}\int_{\Omega}h_{\varepsilon}(U_{k}-V_{k})\Phi_{1}\,{\rm
d}x\right)_{k}$
$\displaystyle\leq\int_{\Omega}\left(D^{\alpha}_{\tau}(U-V)\right)_{k}h^{\prime}_{\varepsilon}(U_{k}-V_{k})\Phi_{1}\,{\rm
d}x$
$\displaystyle=\int_{\Omega}\left(D^{\alpha}_{\tau}(U-V)\right)_{k}h^{\prime}_{\varepsilon}(U^{m}_{k}-V^{m}_{k})\Phi_{1}\,{\rm
d}x$
$\displaystyle\quad+\int_{\Omega}(D^{\alpha}_{\tau}(U-V))_{k}\left(h^{\prime}_{\varepsilon}(U_{k}-V_{k})-h^{\prime}_{\varepsilon}(U^{m}_{k}-V^{m}_{k})\right)\Phi_{1}\,{\rm
d}x$ $\displaystyle\leq I_{\varepsilon}\,,$

where

$I_{\varepsilon}:=\int_{\Omega}\left|(D^{\alpha}_{\tau}(U-V))_{k}\right|\left|h^{\prime}_{\varepsilon}(U_{k}-V_{k})-h^{\prime}_{\varepsilon}(U^{m}_{k}-V^{m}_{k})\right|\Phi_{1}\,{\rm
d}x\,.$

To prove that $I_{\varepsilon}\rightarrow 0$ when $\varepsilon\rightarrow 0$,
we use dominated convergence theorem: note that
$(D^{\alpha}_{\tau}(U-V))_{k}\in L^{1}_{\Phi_{1}}(\Omega)$ for every $k\geq
1$, $|h_{\varepsilon}^{\prime}(x)-h_{\varepsilon}^{\prime}(y)|\leq 2$ for
every $x,y\in\mathbb{R}$, and $h^{\prime}_{\varepsilon}\rightarrow h^{\prime}$
pointwise. On the other hand $|h_{\varepsilon}(x)|\leq|x|$, and therefore

$\left|\left(D^{\alpha}_{\tau}h_{\varepsilon}(U-V)\right)_{k}\right|\leq\sum_{j=0}^{k}\lambda_{j}\left(|U_{j}|+|V_{j}|\right)\in
L^{1}_{\Phi_{1}}(\Omega)\,$

for some $\lambda_{j}>0$. Hence, the limit $\varepsilon\rightarrow 0$ yields

$\left(D^{\alpha}_{\tau}\int_{\Omega}h_{\varepsilon}(U_{k}-V_{k})\Phi_{1}\,{\rm
d}x\right)_{k}\rightarrow\left(D^{\alpha}_{\tau}\int_{\Omega}(U_{k}-V_{k})_{+}\Phi_{1}\,{\rm
d}x\right)_{k}\qquad\forall k\geq 1\,,$

since $h_{\varepsilon}(x)\rightarrow(x)_{+}$. Therefore,

$\left(D^{\alpha}_{\tau}\int_{\Omega}(U_{k}-V_{k})_{+}\Phi_{1}\,{\rm
d}x\right)_{k}\leq 0\qquad\forall k\geq 1\,,$

and the result follows by Proposition 2.2. ∎

Using the theorem above we prove the following discrete comparison principle:

######  Corollary 2.5.

(Discrete Comparison Principle). Let $m>0$ and assume (G1) and (K). Consider
two sequence of solutions of (17), namely, $\\{U_{k}\\}_{k\geq
0},\\{V_{k}\\}_{k\geq 0}\subset L^{1+m}(\Omega)\cap H^{*}(\Omega)$. Then, if
$U_{0}(x)\leq V_{0}(x)$ for a.e. $x\in\Omega$, it holds that for every $k\geq
1$

$U_{k}(x)\leq V_{k}(x)\qquad\mbox{for a.e. }x\in\Omega\,.$

Moreover, if $V_{0}\geq 0$ a.e. in $\Omega$ then $V_{k}\geq 0$ a.e. in
$\Omega$.

###### Proof.

By Theorem 2.4 we know that for every $k\geq 1$

$\|(U_{k}-V_{k})_{+}\|_{L^{1}_{\Phi
1}(\Omega)}\leq\|(U_{0}-V_{0})_{+}\|_{L^{1}_{\Phi 1}(\Omega)}\,.$

Since $U_{0}\leq V_{0}$ a.e. in $\Omega$, we have that
$\|(U_{0}-V_{0})_{+}\|_{L^{1}_{\Phi 1}(\Omega)}=0$. Hence,

$\|(U_{k}-V_{k})_{+}\|_{L^{1}_{\Phi_{1}}(\Omega)}=0\,\qquad\mbox{ for all
}k\geq 1\,.$

The comparison principle follows since
$\Phi_{1}(x)\geq\underline{\kappa}\;\mbox{dist}(x,\partial\Omega)$. For the
nonnegativity of $V_{k}$ we just take $U_{k}=0$ for every $k\geq 0$, which is
the trivial solution. ∎

##  3 Continuous problem

In this section we prove Theorem 1.2. For this purpose, we recall the result
in [37] about the convergence of the discrete solutions of (21) to
$H^{*}$-solutions of (CPME). In addition, we recollect some other results
about existence, uniqueness and estimates for solutions exhibited in [1]. At
the end of this section, we provide a proof of the comparison principle for
the $H^{*}$-solutions by using the discrete comparison principle of Theorem
2.5 and the convergence of the discrete scheme. The statements in [1] and [37]
are in the general framework of subdifferentials of $\lambda$-convex energy
functionals in Hilbert spaces; here we adjust these statements for the Porous
Medium equation.

###  3.1 Existence and uniqueness

In [37], the authors show that solutions of the implicit Euler scheme (21)
converge with certain rates to the unique $H^{*}$-solution, following the
steps of the classical theory of gradient flow of Brezis [20] and Komura [36].
The theorem below will be crucial in the proof of the comparison principle.

######  Theorem 3.1.

[37, Theorem 5.10] Let $m>0$ and assume (G1). Then, for every initial datum
$u_{0}\in L^{1+m}(\Omega)\cap H^{-1}(\Omega)$ there exits a unique $H^{*}$
solution $u=u(t,x)$ of (CPME). Moreover, it satisfies that

$\displaystyle\|u(t+\tau)-u(t)\|_{H^{*}(\Omega)}\leq
C|\tau|^{\alpha/2}\qquad\forall\tau>0\,,$ (27)

and the following estimates for the error with respect to the discrete
solution (24) holds,

$\sup\limits_{k}\|U_{k}-u(t_{k})\|_{H^{*}(\Omega)}\leq
C(u_{0},T)\;\tau^{\alpha/4}\qquad\forall\tau>0\,.$ (28)

Convergence of the discrete scheme in a more general setting has been proven
very recently by Li and Salgado in. [38, Theorem 4.5.].

A different proof of existence and uniqueness of $H^{*}$-solution of (CPME) is
provided by Akagi in [1]. Akagi approximates the problem by adding a classical
time derivative $\varepsilon\,\partial_{t}u_{\varepsilon}$. After proving the
existence for the equation
$\varepsilon\partial_{t}u_{\varepsilon}+D^{\alpha}_{t}u_{\varepsilon}=-\mathcal{L}u_{\varepsilon}^{m}$
using the theory of maximal monotone operators, he passes to the limit
$\varepsilon\rightarrow 0$ thanks to uniform estimates. We will use some of
these estimates in the proof of smoothing effects. Below we recall the most
important ones:

######  Theorem 3.2.

[1, Theorem 2.3. and 6.2] Let $m>0$ and assume (G1). Let $u_{0}\in
L^{1+m}(\Omega)\cap H^{-1}(\Omega)$. Then, there exits a unique $H^{*}$
solution $u=u(t,x)$ of (CPME). This solution $u$ satisfies:

$\begin{split}u\in L^{\infty}((0,T):H^{*}(\Omega)),\quad|u|^{m-1}u\in
L^{2}((0,T):H(\Omega))\,,\\\
\ell*\left\|D^{\alpha}_{t}u\right\|_{H^{*}(\Omega)}^{2}\in
L^{\infty}(0,T)\qquad\qquad\,,\end{split}$ (29)

and

$\displaystyle{\mathcal{E}}(u(t))\leq{\mathcal{E}}(u_{0})\qquad t>0\,.$ (30)

Moreover, contractivity in $H^{*}(\Omega)$ holds, that is, if $u$ and $v$ are
$H^{*}$ solutions with initial datum $u_{0},v_{0}\in L^{1+m}(\Omega)\cap
H^{-1}(\Omega)$, respectively, then

$\|u(t)-v(t)\|_{H^{*}(\Omega)}\leq\|u_{0}-v_{0}\|_{H^{*}(\Omega)}\qquad\;\mbox{for
a.e. }\;t\in(0,T)\,.$ (31)

Since there is uniqueness of $H^{*}$-solution, Theorem 3.1 and Theorem 3.2
describe the same solution for a fixed $u_{0}\in L^{1+m}(\Omega)\cap
H^{*}(\Omega)$. Despite the difference in the approximation procedures, the
limit solution coincide. Hence, we can use the estimates from both theorems
simultaneously.

###  3.2 Comparison Principle

Now, we are in a position to prove the comparison principle for the
$H^{*}$-solutions of (CPME). Let us recall the statement which holds for every
$m>0$.

######  Theorem 3.3.

Let $m>0$ and assume (G1) and (K). Let $u$ and $v$ be two $H^{*}$-solutions of
(CPME) with initial data $u_{0},v_{0}\in L^{1+m}(\Omega)\cap H^{*}(\Omega)$,
respectively. If $u_{0}\leq v_{0}$ a.e. in $\Omega$, then $\forall t>0$

$u(t)\leq v(t)\qquad\mbox{ a.e. in }\Omega\,.$

Proof. The idea is to inherit the comparison principle from the discrete
comparison principle shown in Corollary 2.5. Let us consider two solutions
$u,v$ of (CPME) with initial data $u_{0},v_{0}\in L^{1+m}(\Omega)\cap
H^{*}(\Omega)$ respectively, satisfying

$u_{0}(x)\leq v_{0}(x)\qquad\mbox{ for a.e. }x\in\Omega\,.$ (32)

We proceed by contradiction. Assume that there exist $t_{0}>0$ and
$\widetilde{\Omega}\subset\Omega$ with $|\widetilde{\Omega}|>0$ such that

$u(t_{0},x)-v(t_{0},x)>0\qquad\mbox{ for a.e. }x\in\widetilde{\Omega}\,.$ (33)

Then, we can define the sequence of time indexes $\\{n_{\tau}\\}_{\tau>0}$
such that for all $\tau>0$ we have $t_{n_{\tau}}=\tau\,n_{\tau}$ and

$t_{0}\in[t_{n_{\tau}},t_{n_{\tau}+1})\,.$

Let us consider the respective discrete approximations of $u$ and $v$ as in
(24), namely, the sequences $\\{U_{k}\\}_{k\geq 0},\,\\{V_{k}\\}_{k\geq
0}\subset L^{1+m}(\Omega)\cap H^{*}(\Omega)$ with $U_{0}=u_{0}$ and
$V_{0}=v_{0}$. Notice that from (32) and the discrete comparison principle in
Theorem 2.5, it holds that for every $k\geq 0$

$U_{k}(x)-V_{k}(x)=-\left(U_{k}(x)-V_{k}(x)\right)_{-}\qquad\mbox{ for a.e.
}x\in\Omega\,.$

Hence, we deduce from (28) that

$\displaystyle\|(u(t_{n_{\tau}})-v(t_{n_{\tau}}))+\left(U_{n_{\tau}}-V_{n_{\tau}}\right)_{-}\|_{H^{*}(\Omega)}\rightarrow
0\qquad\mbox{as }\tau\rightarrow 0\,.$ (34)

Note that by (27), we also know that

$\|(u(t_{n_{\tau}})-v(t_{n_{\tau}}))-(u(t_{0})-v(t_{0}))\|_{H^{*}(\Omega)}\rightarrow
0\qquad\mbox{as }\tau\rightarrow 0\,,$

therefore each of the terms $u(t_{n_{\tau}})-v(t_{n_{\tau}})$ and
$U_{n_{\tau}}-V_{n_{\tau}}$ in (34) converge (strongly in $H^{*}(\Omega)$) to
$u(t_{0})-v(t_{0})\in L^{1+m}(\Omega)$.

In addition, the control of the energy in (23) and (30) implies the following
uniform bound of the $L^{1+m}$-norm

$\|(u(t_{n_{\tau}})-v(t_{n_{\tau}}))+\left(U_{n_{\tau}}-V_{n_{\tau}}\right)_{-}\|_{L^{1+m}}\leq
2\left(\|u_{0}\|_{L^{1+m}}+\|v_{0}\|_{L^{1+m}}\right)\qquad\forall\tau>0\,.$

Thus, the sequence
$\left\\{(u(t_{n_{\tau}})-v(t_{n_{\tau}}))+\left(U_{n_{\tau}}-V_{n_{\tau}}\right)_{-}\right\\}_{\tau>0}$
is weakly compact in $L^{1+m}(\Omega)$ and by (34) and uniqueness of the weak-
strong limit, we obtain that up to a subsequence

$(u(t_{n_{\tau}})-v(t_{n_{\tau}}))+\left(U_{n_{\tau}}-V_{n_{\tau}}\right)_{-}\rightharpoonup
0\qquad\mbox{ in }L^{1+m}(\Omega)\,,$ (35)

or equivalently,

$\int_{\Omega}\left[(u(t_{n_{\tau}})-v(t_{n_{\tau}}))+\left(U_{n_{\tau}}-V_{n_{\tau}}\right)_{-}\right]\psi\,{\rm
d}x\rightarrow 0\qquad\forall\psi\in L^{\frac{1+m}{m}}(\Omega)\,.$ (36)

We proceed similarly for the sequence
$\left\\{(u(t_{n_{\tau}})-v(t_{n_{\tau}}))-(u(t_{0})-v(t_{0}))\right\\}_{\tau>0}$
and thanks to

$\|(u(t_{n_{\tau}})-v(t_{n_{\tau}}))-(u(t_{0})-v(t_{0}))\|_{1+m}\leq
2\left(\|u_{0}\|_{1+m}+\|v_{0}\|_{1+m}\right)\qquad\forall\tau>0\,,$

we obtain

$\int_{\Omega}\left[(u(t_{n_{\tau}})-v(t_{n_{\tau}}))-\left(u(t_{0})-v(t_{0})\right)\right]\psi\,{\rm
d}x\rightarrow 0\qquad\forall\psi\in L^{\frac{1+m}{m}}(\Omega)\,.$ (37)

Let us choose $\widetilde{\psi}(x):=\chi_{\widetilde{\Omega}}(x)\in
L^{\infty}(\Omega)$ to be the characteristic function of $\widetilde{\Omega}$,
in order to have

$\displaystyle A_{\tau}:=$
$\displaystyle\int_{\Omega}\left[(u(t_{n_{\tau}})-v(t_{n_{\tau}}))+(U_{n_{\tau}}-V_{n_{\tau}})_{-}\right]\widetilde{\psi}\,{\rm
d}x$ $\displaystyle\geq$
$\displaystyle\int_{\Omega}\left(u(t_{n_{\tau}})-v(t_{n_{\tau}}))-(u(t_{0})-v(t_{0})\right)\widetilde{\psi}\,{\rm
d}x+\int_{\Omega}\left(u(t_{0})-v(t_{0})\right)\widetilde{\psi}\,{\rm d}x$
$\displaystyle=$
$\displaystyle\underbrace{\int_{\Omega}\left(u(t_{n_{\tau}})-v(t_{n_{\tau}}))-(u(t_{0})-v(t_{0})\right)\widetilde{\psi}\,{\rm
d}x}\limits_{B_{\tau}\rightarrow
0}+\underbrace{\int_{\widetilde{\Omega}}\left(u(t_{0})-v(t_{0})\right)\,{\rm
d}x}\limits_{C>0}\,.$

Finally, we obtain a contradiction, since (36) and (37) implies
$A_{\tau}\rightarrow 0$ and $B_{\tau}\rightarrow 0$ when $\tau\rightarrow 0$,
but from the contradiction hypothesis (33) we know that $C>0$ and it does not
depend on $\tau$.

$\square$

##  4 Smoothing effects

The aim of this section is to prove the $L^{p}-L^{\infty}$ smoothing effects
in Theorem 1.4. We divide the proof in several steps. First, in Proposition
4.2 we use the dual formulation of the problem and the Fundamental Theorem of
Calculus for Caputo derivative to obtain a fundamental pointwise formula.
Then, in Theorem 4.3 we obtain a time monotonicity formulae for the function
$t^{\frac{\alpha}{m-1}}u(t)$. The time monotonicity formulae we obtain are
consistent with the case $\alpha=1$, since they depend only on the time
scaling and the comparison principle. Next, we apply the time monotonicity
formulae in the fundamental pointwise formula in order to obtain the upper
bounds of $u^{m}(t,x)$, as described in Proposition 4.4. We conclude the proof
of Theorem 1.4 and Theorem 4.6 using assumptions (G1) and (G2), respectively,
and Hölder’s inequality.

Before starting with the first proposition of this section, let us present a
useful lemma about the differentiability of the time convolution.

######  Lemma 4.1.

Let $f\in L^{1}((0,T))$ and $g\in W^{1,2}((0,T))$ with $g(0)=0$, then $f*g\in
W^{1,2}((0,T))$ and

$\frac{\rm d}{\,{\rm d}t}(f*g)=\left(f*\frac{{\rm d}g}{\,{\rm d}t}\right)\,.$

###### Proof.

We obtain that $f*g\in L^{2}((0,T))$ by Youngs inequality. For any $\varphi\in
C_{c}^{\infty}([0,T])$ let us consider

$\displaystyle\int_{0}^{T}(f*g)(t)\varphi^{\prime}(t)\,{\rm d}t$
$\displaystyle=\int_{0}^{T}\int_{0}^{t}f(t-\tau)g(\tau)\varphi^{\prime}(\tau)\,{\rm
d}\tau\,{\rm
d}t=\int_{0}^{T}\left(\int_{\tau}^{T}f(t-\tau)\varphi^{\prime}(\tau)\,{\rm
d}t\right)g(\tau)\,{\rm d}\tau.$ (38)

Now, we show that for a.e. $\tau\in(0,T)$

$\int_{\tau}^{T}f(t-\tau)\varphi^{\prime}(\tau)\,{\rm
d}t=\int_{0}^{T-\tau}f(t)\varphi^{\prime}(\tau+t)\,{\rm
d}t=\left(\int_{0}^{T-\tau}f(t)\varphi(\tau+t)\,{\rm
d}t\right)^{\prime}=\left(\int_{\tau}^{T}f(t-\tau)\varphi(\tau)\,{\rm
d}t\right)^{\prime}.$

For this purpose, let us study the incremental quotient

$\displaystyle\frac{1}{h}\left[\int_{0}^{T-\tau-h}f(t)\varphi(t+\tau+h)\,{\rm
d}t-\int_{0}^{T-\tau}f(t)\varphi(t+\tau)\,{\rm d}t\right]$ $\displaystyle=$
$\displaystyle\int_{0}^{T-\tau}f(t)\chi_{[0,T-\tau-h]}(t)\frac{\varphi(t+\tau+h)-\varphi(t+\tau)}{h}\,{\rm
d}t-\frac{1}{h}\int_{T-\tau-h}^{T-\tau}f(t)\varphi(t+\tau)\,{\rm d}t.$

The first term converge by dominated convergence theorem when $h$ goes to 0.
For the second term we use Lebesgue differentiation theorem and the fact that
$\varphi(T)=0$ to prove that it converge to 0 as $h$ goes to 0. Hence, we
conclude from (38) and integration by parts that

$\displaystyle\int_{0}^{T}(f*g)(t)\varphi^{\prime}(t)\,{\rm d}t$
$\displaystyle=\int_{0}^{T}\left(\int_{\tau}^{T}f(t-\tau)\varphi(t)\right)^{\prime}g(\tau)\,{\rm
d}\tau=-\int_{0}^{T}\int_{\tau}^{T}f(t-\tau)\varphi(t)g^{\prime}(\tau)\,{\rm
d}\tau$ $\displaystyle=\int_{0}^{T}(f*g^{\prime})(t)\varphi(t)\,{\rm d}t,$

since $g(0)=0$. ∎

Taking in to account estimate (29) from the Existence Theorem 3.2, we make the
dual formulation of the problem rigourous and we prove the Fundamental Theorem
of Calculus for Caputo in the following proposition.

######  Proposition 4.2 (Fundamental pointwise formula).

Let $u$ be a $H^{*}$-solution of (CPME) with

$\ell*\|D^{\alpha}_{t}u\|^{2}_{H^{*}(\Omega)}\in L^{\infty}(\Omega)\,.$ (39)

Then, for a.e. $t\in(0,T)$ and a.e. $x_{0}\in\Omega$ it holds that

$\int_{\Omega}(u_{0}(x)-u(t,x))\mathbb{G}_{\Omega}(x_{0},x)\,{\rm
d}x=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\frac{u^{m}(\tau,x_{0})}{(t-\tau)^{1-\alpha}}\,{\rm
d}\tau.$ (40)

###### Proof.

The intuition behind the proof is to formally use the following test function
in (3),

$\psi(\tau,x):=\frac{\chi_{(0,t)}(\tau)}{(t-\tau)^{1-\alpha}}\;\delta_{x_{0}}(x),$

and integrate on $[0,T]$. Formally, on the left hand side we would have

$\displaystyle\int_{0}^{T}\int_{\Omega}D^{\alpha}_{t}u(\tau,x)\mathcal{L}^{-1}\psi(\tau,x)\,{\rm
d}x\,{\rm d}\tau$
$\displaystyle=\ell*\left(\int_{\Omega}D^{\alpha}_{t}u\mathcal{L}^{-1}\delta_{x_{0}}\,{\rm
d}x\right)=\ell*k*\left(\frac{\rm d}{\,{\rm
d}t}\int_{\Omega}(u(t,x)\mathbb{G}_{\Omega}(x_{0},x)\,{\rm d}x\right)$
$\displaystyle=\int_{\Omega}(u(t,x)-u_{0}(x))\mathbb{G}_{\Omega}(x_{0},x)\,{\rm
d}x\,,$

and on the right hand side

$-\int_{0}^{T}\int_{\Omega}u^{m}(\tau,x)\psi(\tau,x)\,{\rm d}x\,{\rm
d}\tau=-\int_{0}^{t}\frac{u^{m}(\tau,x_{0})}{(t-\tau)^{1-\alpha}}\,{\rm
d}\tau\,.$

In order to give a rigorous proof, we split the proof in two steps.

$\bullet~{}$Step 1. First, we show that for any $\varphi\in H^{*}(\Omega)$ we
have

$\int_{\Omega}(u(t,x)-u_{0}(x))\mathcal{L}^{-1}\varphi(x)\,{\rm
d}x=-\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\int_{\Omega}\frac{u^{m}(\tau,x)}{(t-\tau)^{1-\alpha}}\varphi(x)\,{\rm
d}x\,{\rm d}\tau\,,$ (41)

by testing the equation with

$\tilde{\psi}(\tau,x)=\frac{\chi_{(0,t)}(\tau)}{(t-\tau)^{1-\alpha}}\,\varphi(x)\in
L^{1}(0,T;H^{*}(\Omega))\,.$

Note that condition (39) implies that $\tilde{\psi}$ is an admissible test
function, and $\ell*\left(D^{\alpha}_{t}u\mathcal{L}^{-1}\varphi\,{\rm
d}x\right)\in L^{\infty}(0,T)$, since

$\displaystyle\left|\int_{0}^{t}\int_{\Omega}\frac{D^{\alpha}_{t}u(\tau,x)\mathcal{L}^{-1}\varphi(x)}{(t-\tau)^{1-\alpha}}\,{\rm
d}x\,{\rm d}\tau\right|$
$\displaystyle\leq\|\varphi\|_{H^{*}(\Omega)}\int_{0}^{t}\frac{\|D^{\alpha}_{t}u(\tau)\|_{H^{*}(\Omega)}}{(t-\tau)^{1-\alpha}}\,{\rm
d}\tau$
$\displaystyle\leq\|\varphi\|_{H^{*}(\Omega)}\left(\int_{0}^{t}\frac{\|D^{\alpha}_{t}u(\tau)\|^{2}_{H^{*}(\Omega)}}{(t-\tau)^{1-\alpha}}\,{\rm
d}\tau\right)\left(\int_{0}^{t}\frac{\,{\rm
d}\tau}{(t-\tau)^{1-\alpha}}\right)^{1/2}$ $\displaystyle\leq
t^{\alpha/2}\|\varphi\|_{H^{*}(\Omega)}\;\left(\ell*\|D^{\alpha}_{t}u\|^{2}_{H^{*}(\Omega)}(t)\right)^{1/2}<+\infty\,.$

Hence, applying Fubini’s Theorem and Lemma 4.1 we obtain

$\displaystyle\ell*\left(\int_{\Omega}D^{\alpha}_{t}u\mathcal{L}^{-1}\varphi\,{\rm
d}x\right)\\!(t)\\!$
$\displaystyle=\\!\\!\int_{\Omega}\\!\\!\ell*\\!\left(\frac{\rm d}{\,{\rm
d}t}[k*(u-u_{0})]\right)\\!\\!(t)\mathcal{L}^{-1}\varphi\,{\rm
d}x=\\!\int_{\Omega}\\!\\!\frac{\rm d}{\,{\rm
d}t}\left(\ell*[k*(u-u_{0})](t)\right)\mathcal{L}^{-1}\varphi\,{\rm d}x$
$\displaystyle=\int_{\Omega}\frac{\rm d}{\,{\rm
d}t}(1*(u-u_{0})(t))\mathcal{L}^{-1}\varphi\,{\rm
d}x=\int_{\Omega}(u(t)-u_{0})\mathcal{L}^{-1}\varphi\,{\rm d}x\,.$

$\bullet~{}$Step 2. Now, we consider the sequence
$\varphi_{r}(x)=\frac{\chi_{B_{r}(x_{0})}(x)}{|B_{r}|}\in H^{*}(\Omega)$ with
$x_{0}\in\Omega$ being a Lebesgue point and we pass to the limit in (41) when
$r\rightarrow 0$. ∎

Next, we want to bound from below the integral on the right hand side of (40),
by some pointwise value of the solution of $u(t,x_{0})$. Therefore, we prove
the two time monotonicity formulae of Theorem 1.3, depending on whether
$0<m<1$ or $m>1$. This type of time monotonicity estimates are the analogous
of the Benilan-Crandall estimates for the classical problem. In the context of
Caputo derivatives, time monotonicity formulae are a surprising consequence of
the comparison principle and the time scaling of (CPME).

######  Theorem 4.3 (Time monotonicity formulae).

Let $u$ be a nonnegative $H^{*}$-solution for (CPME) under assumptions of
Theorem 1.2.

  1. 1.

If $0<m<1$, then the function

$t\mapsto t^{\frac{-\alpha}{1-m}}u(t,x)$

is non increasing for any $t>0$ and a.e $x\in\Omega$.

  2. 2.

If $m>1$, then the function

$t\mapsto t^{\frac{\alpha}{m-1}}u(t,x)$

is non decreasing for any $t>0$ and a.e $x\in\Omega$

###### Proof.

The proof follows by time scaling and the comparison principle of Theorem 1.2.
For any $\lambda>0$, we define the rescaled solution

$u_{\lambda}(t,x):=\lambda^{-\frac{\alpha}{1-m}}u(\lambda t,x)$

which is a solution to (CPME) since

$D^{\alpha}_{t}\left(\lambda^{-\frac{\alpha}{1-m}}u(\lambda
t)\right)=\lambda^{\alpha-\frac{\alpha}{1-m}}\left(D^{\alpha}_{t}u\right)(\lambda
t)=-\lambda^{-\frac{\alpha m}{1-m}}\mathcal{L}u^{m}(\lambda
t)=-\mathcal{L}\left(\lambda^{-\frac{\alpha}{1-m}}u(\lambda t)\right)^{m}\,.$

Case $0<m<1$: Let us choose $0<t_{0}<t_{1}$ and fix
$\lambda=\frac{t_{1}}{t_{0}}\geq 1$, then it holds that

$\displaystyle
u_{\lambda}(0,x)=\left(\frac{t_{1}}{t_{0}}\right)^{-\frac{\alpha}{1-m}}u_{0}(x)\leq
u_{0}(x)\qquad\mbox{a.e.}\quad x\in\Omega\,.$

By comparison principle of Theorem 1.2, we have that $u_{\lambda}(t,x)\leq
u(t,x)$ for any $t>0$ and a.e. $x\in\Omega$. Hence, we can take $t=t_{0}$ in
order to conclude that

$\left(\frac{t_{1}}{t_{0}}\right)^{-\frac{\alpha}{1-m}}u(t_{1},x)\leq
u(t_{0},x)\qquad\mbox{for a.e.}\quad x\in\Omega\,.$

Case $m>1$: Let us consider any $0<t_{0}<t_{1}$ and fix
$\lambda=\frac{t_{0}}{t_{1}}\leq 1$, then

$u_{\lambda}(0,x)=\left(\frac{t_{0}}{t_{1}}\right)^{\frac{\alpha}{m-1}}u_{0}(x)\leq
u_{0}(x)\qquad\mbox{a.e.}\quad x\in\Omega\,.$

As in the previuos case, we use comparison principle of Theorem 1.2 in order
to obtain $u_{\lambda}(t,x)\leq u(t,x)$ for any $t>0$ and a.e. $x\in\Omega$.
Therefore, we choose $t=t_{1}$ to get

$\left(\frac{t_{0}}{t_{1}}\right)^{\frac{\alpha}{m-1}}u(t_{0},x)\leq
u(t_{1},x)\qquad\mbox{for a.e.}\quad x\in\Omega\,.$

∎

Afterwards, we implement the time monotonicity formulae of Theorem 4.3 in the
equality (40) in order to obtain the following fundamental upper bounds.

######  Proposition 4.4 (Fundamental upper bounds).

Assume (G1) and (K) and let $u$ be a bounded and nonnegative $H^{*}$-solution
for (CPME). Then for a.e. $x_{0}\in\Omega$ and a.e. $t\in(0,T)$ it holds that

  1. 1.

If $0<m<1$, then

$u^{m}(t,x_{0})\leq\frac{\mathsf{c}}{t^{\alpha}}\int_{\Omega}(u_{0}(x)-u(t,x))\mathbb{G}_{\Omega}(x,x_{0})\,{\rm
d}x,$ (42)

with $\mathsf{c}=\mathsf{c}(m,\alpha)>0$.

  2. 2.

If $m>1$, then

$u^{m}(t,x_{0})\leq\frac{\mathsf{c}}{t^{\alpha}}\int_{\Omega}(u_{0}(x)-u(2t,x))\mathbb{G}_{\Omega}(x,x_{0})\,{\rm
d}x,$ (43)

with $\mathsf{c}=\mathsf{c}(m,\alpha)>0$.

###### Proof.

The idea of the proof is to use Benilan-Crandall estimates (4.3) on the right
hand side of the fundamental integral formula (40). Namely, it sufices to
prove the following inequality

$\mathsf{c}\;t^{\alpha}\,u^{m}(t,x_{0})\leq\int_{0}^{t_{1}}\frac{u^{m}(\tau,x_{0})}{(t_{1}-\tau)^{\alpha}}\,{\rm
d}\tau,$ (44)

with $\mathsf{c}=\mathsf{c}(m,\alpha)>0$ and $t_{1}=t$ if $0<m<1$ and
$t_{1}=2t$ for $m>1$.
$\bullet~{}$Case $0<m<1$: By Theorem 4.3, we have that

$u^{m}(\tau,x_{0})\geq\left(\frac{\tau}{t}\right)^{\frac{\alpha
m}{1-m}}u^{m}(t,x_{0})\qquad\mbox{for}\quad 0<\tau<t\,.$

Then,

$\begin{split}\int_{0}^{t}\frac{u^{m}(\tau,x_{0})}{(t-\tau)^{1-\alpha}}\,{\rm
d}\tau&\geq\frac{u^{m}(t,x_{0})}{t^{\frac{\alpha
m}{1-m}}}\int_{0}^{t}\frac{\tau^{\frac{\alpha
m}{1-m}}}{(t-\tau)^{1-\alpha}}\,{\rm d}\tau\\\
&=t^{\alpha}u^{m}(t,x_{0})\int_{0}^{1}\frac{r^{\frac{\alpha
m}{1-m}}}{(1-r)^{1-\alpha}}{\rm d}r\\\ &=B\left(\frac{\alpha
m}{1-m}+1,\alpha\right)t^{\alpha}u^{m}(t,x_{0})\,,\end{split}$

where we used the change of variable $tr=\tau$ and the definition of the beta
function $B(z_{1},z_{0})=\int_{0}^{1}\tau^{z_{1}-1}(1-\tau)^{z_{2}-1}{\rm
d}\tau$, which is well finite for $z_{1},z_{2}>0$.

$\bullet~{}$Case $m>1$: By Theorem 4.3, we have that

$u^{m}(\tau)\geq\left(\frac{t}{\tau}\right)^{\frac{\alpha
m}{m-1}}u^{m}(t,x_{0})\qquad\mbox{for}\quad 0<t<\tau<2t\,.$

Then,

$\displaystyle\int_{0}^{2t}\frac{u^{m}(\tau,x_{0})}{(2t-\tau)^{1-\alpha}}\,{\rm
d}\tau$
$\displaystyle\geq\int_{t}^{2t}\frac{u^{m}(\tau,x_{0})}{(2t-\tau)^{1-\alpha}}\,{\rm
d}\tau$ $\displaystyle\geq
t^{\frac{\alpha(m)}{m-1}}u^{m}(t,x_{0})\int_{t}^{2t}\frac{\,{\rm
d}\tau}{(2t-\tau)^{1-\alpha}\;\tau^{\frac{\alpha m}{m-1}}}$
$\displaystyle\geq\frac{1}{2^{\frac{\alpha
m}{m-1}}}\;u^{m}(t,x_{0})\int_{t}^{2t}\frac{\,{\rm
d}\tau}{(2t-\tau)^{1-\alpha}}$ $\displaystyle=\frac{1}{\alpha\,2^{\frac{\alpha
m}{m-1}}}\;t^{\alpha}\,u^{m}(t,x_{0})\,.$

The proof follows by combining (40) and (44). ∎

Next, we recall a result from [11] involving $L^{q}$ estimates of the Green
function.

######  Lemma 4.5 (Green function estimates, [11]).

Let $\mathbb{G}_{\Omega}$ be the kernel of $\mathcal{L}^{-1}$, and assume that
(G1) holds. Then, for all $0<q<{N}/(N-2s)$, there exist a constant
$c_{2,\Omega}(q)>0$ such that

$\sup_{x_{0}\in\Omega}\int_{\Omega}\mathbb{G}_{\Omega}^{q}(x,x_{0})\,{\rm
d}x\leq c_{2,\Omega}(q)\,.$ (45)

The constant $c_{2,\Omega}$ depends only on $s,N,q,\Omega$, and have an
explicit expression, cf. [11].

Finally, we are in the position to prove the $L^{p}-L^{\infty}$ smoothing
effects of Theorem 1.4, namely

$\|u(t)\|_{\infty}\leq\overline{\kappa}\frac{\|u_{0}\|^{\frac{1}{m}}_{p}}{t^{\frac{\alpha}{m}}}\qquad\forall
t>0\,.$

Proof of Theorem 1.4. First, we prove the estimate for nonnegative solutions
and then generalize the result for sign-changing solutions.

$\bullet~{}$Step 1. (Smoothing effect for nonnegative solutions). If we
combine Proposition 4.4 and Lemma 4.5, the boundedness of solutions $u(t)$
follows for any $t>0$ given that $0\leq u_{0}\in L^{q^{\prime}}(\Omega)$ with
$q^{\prime}>\frac{N}{2s}$:

$\displaystyle\|u(t)\|_{\infty}^{m}$
$\displaystyle\leq\sup\limits_{x_{0}\in\Omega}\frac{\mathsf{c}}{t^{\alpha}}\int_{\Omega}(u_{0}(x)-u(t_{1},x))\mathbb{G}_{\Omega}(x,x_{0})\,{\rm
d}x\leq\sup\limits_{x_{0}\in\Omega}\frac{\mathsf{c}}{t^{\alpha}}\int_{\Omega}u_{0}(x)\mathbb{G}_{\Omega}(x,x_{0})\,{\rm
d}x$
$\displaystyle\leq\frac{\mathsf{c}}{t^{\alpha}}\|u_{0}\|_{q^{\prime}}\sup\limits_{x_{0}\in\Omega}\|\mathbb{G}_{\Omega}(\cdot,x_{0})\|_{q}\leq
C\frac{\|u_{0}\|_{q^{\prime}}}{t^{\alpha}},$

where we have used the nonnegativity of $u(t_{1},x)$ with $t_{1}=t$ if $0<m<1$
and $t_{1}=2t$ if $m>1$, and Young’s inequality. The constant $C$ depends on
$\alpha,m,s,N$ and $\Omega$.

$\bullet~{}$Step 2. (Smoothing effect for sign-changing solutions). We reduce
the problem to nonnegative initial data using the comparison principle. We
split the signed initial datum,

$u_{0}=(u_{0})_{+}-(u_{0})_{-}\,,$

and consider the nonnegative $H^{*}$-solutions $\tilde{u}^{+}$ and
$\tilde{u}^{-}$ with nonnegative initial data $(u_{0})_{+},(u_{0})_{-}$,
respectively. First, we have to check that $(u_{0})_{+},(u_{0})_{-}\in
L^{p}(\Omega)\cap H^{*}(\Omega)$ with $p$ as in (5). Since $u_{0}\in
L^{p}(\Omega)\cap H^{*}(\Omega)$ with $p$ satisfying (5), then
$(u_{0})_{+},(u_{0})_{-}\in L^{p}(\Omega)$ with the same $p$. It remains to
prove that they also belong to $H^{*}(\Omega)$. For this purpose, we have to
use the Hardy-Littlewood-Sobolev inequality associated to the quadratic form
of the operator: There exists $\mathcal{H}_{\mathcal{L}}>0$ such that

$\|f\|_{H^{*}(\Omega)}\leq\mathcal{H}_{\mathcal{L}}\|f\|_{\frac{2N}{N+2s}}\,,$
(46)

which holds under assumption (G1), see [15, Theorem 7.5]. Since $N>2s$ then
$\frac{2N}{N+2s}<\frac{N}{2s}$, and hence

$\|(u_{0})_{\pm}\|_{H^{*}(\Omega)}\leq\mathcal{H}_{\mathcal{L}}\|(u_{0})_{\pm}\|_{\frac{2N}{N+2s}}\leq\mathcal{H}_{\mathcal{L}}\left|\Omega\right|^{\frac{N-2s}{2N}}\|(u_{0})_{\pm}\|_{\frac{N}{2s}}\,.$

Now, we use the smoothing effects for nonnegative initial data of Step 1 to
obtain

$\|\tilde{u}^{\pm}(t)\|_{\infty}\leq
C\,\frac{\|(u_{0})_{\pm}\|_{p}^{\frac{1}{m}}}{t^{\frac{\alpha}{m}}}\qquad\forall
t>0\,.$

By the comparison principle of Theorem 1.2, $u_{0}\leq(u_{0})_{+}$ a.e. in
$\Omega$, therefore we have for all $t>0$

$u(t,x)\leq\tilde{u}^{+}(t,x)\leq
C\,\frac{\|(u_{0})_{+}\|_{p}^{\frac{1}{m}}}{t^{\frac{\alpha}{m}}}\qquad\mbox{
for a.e. }x\in\Omega\,.$

For the lower bound, note that $-u_{0}\leq(u_{0})_{-}$ a.e. in $\Omega$.
Hence, by comparison principle it holds that for every $t>0$

$-u(t,x)\leq\tilde{u}^{-}(t,x)\leq
C\,\frac{\|(u_{0})_{-}\|_{p}^{\frac{1}{m}}}{t^{\frac{\alpha}{m}}}\qquad\mbox{
for a.e. }x\in\Omega\,,$

and the result follows.

$\square$

On the other hand, if instead of using assumption (G1) we use (G2), we are
able to prove the following $L^{\infty}$ estimate involving the distance to
the boundary of $\Omega$.

######  Theorem 4.6 (Upper boundary estimates).

Let $m>0$ with $m\neq 1$ and $N>2s>\gamma$. Assume (G2) and (K). Then, for any
$u$ being a $H^{*}$-solution with nonnegative initial datum $u_{0}\in
L^{p}(\Omega)\cap H^{*}(\Omega)$ with $p>\frac{N}{2s-\gamma}$ and $p\geq 1+m$
it holds that

$\left\|\frac{u(t)}{\mbox{\rm
dist}(\cdot,\partial\Omega)^{\gamma/m}}\right\|_{\infty}\leq\overline{\kappa}\frac{\|u_{0}\|_{p}^{1/m}}{t^{\frac{\alpha}{m}}}\qquad\forall
t>0\,.$ (47)

###### Proof.

Let us combine Proposition 4.4 with assumption (G2) in order to obtain

$\displaystyle u^{m}(t,x_{0})$
$\displaystyle\leq\frac{\mathsf{c}}{t^{\alpha}}\int_{\Omega}\left(u_{0}(x)-u(t_{1},x)\right)\mathbb{G}_{\Omega}(x,x_{0})\,{\rm
d}x\leq\frac{\mathsf{c}}{t^{\alpha}}\int_{\Omega}\frac{u_{0}(x)\,\mbox{\rm
dist}(x_{0},\partial\Omega)^{\gamma}}{|x-x_{0}|^{N-2s+\gamma}}\,{\rm d}x$
$\displaystyle\leq c_{\Omega}\frac{\|u_{0}\|_{p}}{t^{\alpha}}\,\mbox{\rm
dist}(x_{0},\partial\Omega)^{\gamma}\,,$

where we have used the nonnegativity of $u(t_{1},x)$ with $t_{1}=t$ if $0<m<1$
and $t_{1}=2t$ if $m>1$ and Hölder’s inequality with $p>\frac{N}{2s-\gamma}$.
The result follows by dividing both sides of the inequality by $\mbox{\rm
dist}(\cdot,\partial\Omega)^{\gamma}$ and taking the essential supremum of
$x_{0}$ in $\Omega$. ∎

As it is stated in Theorem 1.5, the upper boundary estimate above and the
comparison principle allow us to provide a Global Harnack Principle for a
class of $H^{*}$-solutions.

######  Theorem 4.7 (Global Harnack Principle for “big” initial data).

Let $m>\frac{N-2s}{N+2s}$ with $m\neq 1$ and $N>2s>\gamma$. Assume (G2) and
(K). If the initial datum $u_{0}\in L^{p}(\Omega)\cap H^{*}(\Omega)$ with
$p>\frac{N}{2s-\gamma}$ and $p\geq 1+m$ satisfies

$u_{0}(x)\geq c\,\mbox{\rm dist}(x,\partial\Omega)^{\gamma/m}\qquad\mbox{ for
a.e. }x\in\Omega\,,$

for certain $c>0$, then it holds that for every $t>0$

$\frac{c_{0}}{1+t^{\alpha}}\leq\frac{u^{m}(t,x)}{\mbox{\rm
dist}(x,\partial\Omega)^{\gamma}}\leq\frac{c_{1}}{t^{\alpha}}\qquad\mbox{ for
a.e. }x\in\Omega\,.$ (48)

###### Proof.

The upper bound follows directly by Theorem 4.6. For the lower bound we use
the comparison principle of Theorem 1.2 with respect to the separate-variables
solution, which exists if $m>\frac{N-2s}{N+2s}$. This solution has the form

$v(t,x)=F(t)S(x)\,,$

with $S$ solving the elliptic problem $\mathcal{L}S^{m}=S$ on $\Omega$ and $F$
solving the fractional ODE of the form $D^{\alpha}_{t}F=-F^{m}$ for $t>0$ with
$F(0)=F_{0}>0$. We give a precise description of these solutions in Section 5.
As it was proved in [11], under assumption (G2) the function $S(x)$ behaves
like

$S(x)\asymp\mbox{\rm dist}(x,\partial\Omega)^{\frac{\gamma}{m}}\qquad\mbox{for
a.e. }x\in\Omega\,,$

and hence, from the assumption on the initial data we have

$u_{0}(x)\geq c\,\mbox{dist}(x,\partial\Omega)^{\frac{\gamma}{m}}\geq
C_{0}S(x)\qquad\mbox{ a.e. in }\Omega\,.$

Now, choosing $F_{0}=C_{0}>0$ we obtain

$u_{0}(x)\geq F_{0}S(x)=v(0,x)\qquad\mbox{ a.e. in }\Omega\,.$

Finally, by comparison principle in Theorem 1.2, for a.e. $x\in\Omega$ and
every $t>0$ it holds that

$u(t,x)\geq v(t,x)=F(t)S(x)\geq
c\,F(t)\,\mbox{dist}(x,\partial\Omega)^{\gamma/m}\geq
c_{0}\,\frac{\mbox{dist}(x,\partial\Omega)^{\gamma/m}}{1+t^{\alpha}}\,,$

since we have used the lower estimates for $F(t)$ proved in [49], see Theorem
A.3. ∎

##  5 Sharp time decay of $L^{p}$-norms

This section contains the proof Theorem 1.7; in particular, we show that every
$H^{*}$-solution does not extinguish in finite time.

We first show, heuristically, why we should not expect extinction in finite
time as in the case with classical time derivative. Let us consider solutions
of for (CPME) of the form

$v(t,x)=F(t)S(x).$

Then, $F$ and $S$ satisfy the relation

$\frac{\mathcal{L}S^{m}(x)}{S(x)}=-\frac{D^{\alpha}_{t}F(t)}{F^{m}(t)}=\lambda\,.$

The equation satisfied by $S(x)$ is the same as in the classical time
derivative case. Defining $V(x):=S^{m}(x)$ and $p:=\tfrac{1}{m}$ we get the
well-known semilinear equation of Lane-Emden-Fowler type:

$\begin{cases}\mathcal{L}V(x)&=\lambda
V^{p}(x)\qquad\mbox{in}\quad\Omega\,,\\\
V(x)&=0\qquad\qquad\;\;\mbox{on}\quad\partial\Omega\,.\end{cases}$ (49)

Due to the Pohozaev identity, in the local and the nonlocal setting,
nontrivial solutions to (49) fail to exists for $p<\frac{N+2s}{N-2s}$,
equivalently, for $m>m_{s}:=\frac{N-2s}{N+2s}$. As shown in [11], under
assumption (G2), solutions to (49) behave like

$V(x)\asymp\mbox{\rm dist}(x,\partial\Omega)^{\gamma}\asymp\Phi_{1}(x)\,,$

with the parameter $\gamma>0$ depending on the operator $\mathcal{L}$.

The time-dependent function $F(t)$, solution to

$\begin{cases}D^{\alpha}_{t}F(t)=-\lambda F^{m}(t)\,,\\\
F(0)=F_{0}>0\,,\end{cases}$ (50)

has a completely different behaviour than in the classical case: by Theorem
A.3, proved in [49], we know that $F(t)$ is bounded above and below by a
function of the form $\frac{1}{1+t^{\frac{\alpha}{m}}}$, more precisely

$\frac{c_{1}}{1+t^{\frac{\alpha}{m}}}\leq
F(t)\leq\frac{c_{2}}{1+t^{\frac{\alpha}{m}}}\qquad\forall t>0\,,$ (51)

for some constants $c_{1},c_{2}>0$ that only depend on $F_{0},\lambda,\alpha$
and $m$. The lower bound in (51) implies that $v(x,t)$ does not vanish in
finite time. Note also that the decay rate of any $L^{p}$-norm of $v(x,t)$
coincides with the regularization rate shown in Theorem 1.4.

An important question is whether this non-extinction phenomenon in finite time
occurs for all nonnegative solution of (CPME). As we will prove in this
section, the answer is affirmative and the time decay for the $L^{p}$-norms
for all $1\leq p\leq\infty$ is sharp.

The proof starts with a lower bound for the $L^{1}_{\Phi_{1}}$-norm of the
solution with a rate of $t^{-\alpha/m}$. Then, we conclude with an
interpolation argument between the $L^{1}_{\Phi_{1}}$-norm and the
$L^{\infty}$-norm.

But first let us show the following estimate for the $L^{1}_{\Phi_{1}}$-norm
of the Caputo derivative.

######  Lemma 5.1.

Let $m>0$ and $u$ be a $H^{*}$-solution of (CPME) with $u_{0}\in
L^{1+m}(\Omega)\cap H^{*}(\Omega)$. Then, estimate (29), which we recall here,

$\ell*\|D^{\alpha}_{t}u(t)\|^{2}_{H^{*}(\Omega)}\in L^{\infty}((0,T))\,,$

implies that

$\ell*\|D^{\alpha}_{t}u(t)\|_{L^{1}_{\Phi_{1}}(\Omega)}\in
L^{\infty}((0,T))\,.$ (52)

###### Proof.

Since the kernel $\ell$ is nonnegative and $\Phi_{1}\in H(\Omega)$, we have
that

$\displaystyle\left|\ell*\left(\int_{\Omega}D^{\alpha}_{t}u(t,x)\Phi_{1}(x)\,{\rm
d}x\right)\right|$
$\displaystyle=\left|\int_{0}^{t}\int_{\Omega}\ell(t-s)D^{\alpha}_{t}u(s,x)\Phi_{1}(x)\,{\rm
d}x\,{\rm d}t\right|$
$\displaystyle\leq\|\Phi_{1}\|_{H(\Omega)}\int_{0}^{t}\ell(t-s)\|D^{\alpha}_{t}u(s)\|_{H^{*}(\Omega)}\,{\rm
d}s$
$\displaystyle\leq\frac{\|\Phi_{1}\|_{H(\Omega)}}{2}\int_{0}^{t}\ell(t-s)\left(1+\|D^{\alpha}_{t}u(s)\|_{H^{*}(\Omega)}^{2}\right)\,{\rm
d}s$
$\displaystyle\leq\frac{\|\Phi_{1}\|_{H(\Omega)}}{2}\left(\|\ell\|_{L^{1}(0,T)}+\ell*\|D^{\alpha}_{t}u(t)\|_{H^{*}(\Omega)}^{2}\right)\,,$

where in the second inequality we have used Young’s inequality. We conclude by
taking the essential supremum in $t\in(0,T)$ and using estimate (29). ∎

The last lemma is used in the proof of a lower and upper bounds for the
$L^{1}_{\Phi_{1}}$-norm of the $H^{*}$-solutions.

######  Proposition 5.2.

Let $0<m\leq 1$ and u be a $H^{*}$-solution of (CPME) with $0\leq u_{0}\in
L^{1}_{\Phi_{1}}(\Omega)$. Then,

$\frac{c}{1+t^{\frac{\alpha}{m}}}\leq\|u(t)\|_{L^{1}_{\Phi_{1}}}\leq\|u_{0}\|_{L^{1}_{\Phi_{1}}(\Omega)}\qquad\quad\forall
t>0\,,$ (53)

with $c$ depending on $\alpha,m,\lambda_{1},\|\Phi_{1}\|_{L^{1}(\Omega)}$ and
$\|u_{0}\|_{L^{1}_{\Phi_{1}}}$.

Moreover, if $0\leq u_{0}\in L^{p}(\Omega)\cap H^{*}(\Omega)$ with
$p>\frac{N}{2s}\geq 1+m$ or $p\geq 1+m>\frac{N}{2s}$, with $m>1$, then

$\frac{c_{0}}{1+t^{\frac{\alpha}{m}}}\leq\|u(t)\|_{L^{1}_{\Phi_{1}}}\leq\|u_{0}\|_{L^{1}_{\Phi_{1}}(\Omega)}\qquad\quad\forall
t>0\,.$ (54)

###### Proof.

Inspired by [50], we look for a fractional differential inequality for the
$L^{1}_{\Phi_{1}}$-norm of the solution and then use a comparison argument.

$\bullet~{}$Step 1. Upper bound. From the definition of the nonnegative
$H^{*}$-solution $u$ in (3), we deduce

$\displaystyle\int_{\Omega}\partial_{t}[k*(u-u_{0})](t)\Phi_{1}\,{\rm
d}x=-\lambda_{1}\int_{\Omega}u^{m}(t)\Phi_{1}\,{\rm d}x\leq 0\qquad\mbox{ for
a.e. }t>0\,,$

using as a test function $0\leq\lambda_{1}\Phi_{1}\in H(\Omega)$. We now
convolve both sides of the equation with the nonnegative kernel $\ell$ and
obtain

$\displaystyle\ell*\left(\int_{\Omega}\partial_{t}[k*(u-u_{0})](t)\Phi_{1}\,{\rm
d}x\right)=\int_{\Omega}\left(\ell*\partial_{t}[k*(u-u_{0})]\right)(t)\Phi_{1}\,{\rm
d}x\leq 0\,.$

Since $\ell*k=1$ and
$\partial_{t}\left[1*(u(t,x)-u_{0}(x))\right]=u(t,x)-u_{0}(x)$ for a.e.
$x\in\Omega$, we conclude that

$\int_{\Omega}\left(u(t)-u_{0})\right)\Phi_{1}\,{\rm d}x\leq 0\qquad\forall
t>0\,.$

$\bullet~{}$Step 2. Lower regularized differential inequality. We first
regularize the kernel $k$ to apply a comparison principle for fractional ODEs.
We know from (29) that $Y(t):=\|u(t)\|_{L^{1}_{\Phi_{1}}}\in
L^{\infty}((0,T))$ and $k*(u-u_{0})\in H^{1}([0,T])$. Moreover, there exists a
nonnegative kernel $h_{n}\in H^{1}([0,T])$ (see for instance [49, 51])
satisfying

$\displaystyle h_{n}*g\rightarrow g\qquad\mbox{in }L^{p}(0,T)\quad\mbox{when
}n\rightarrow\infty\,,$

for any function $g\in L^{p}((0,T))$. The aim of this step is to prove the
following inequality for the function $Y\in L^{\infty}((0,T))$:

$\partial_{t}[k_{n}*(Y-Y_{0})](t)\geq-h_{n}*f(t,Y(t))\qquad\mbox{for a.e.
}t\in(0,T)\,,$ (55)

with $k_{n}:=h_{n}*k$ and $f(t,x)$ nondecreasing in $x$ for a.e. $t>0$ and
$f(t,Y(t))\in L^{1}((0,T))$.

We start with the definition of the $H^{*}$-solution $u$ in (3) with the
admissible test function $\lambda_{1}\Phi_{1}\in H(\Omega)$,

$\int_{\Omega}\partial_{t}[k*(u-u_{0}](t)\Phi_{1}\,{\rm
d}x=-\lambda_{1}\int_{\Omega}u^{m}(t)\Phi_{1}\,{\rm d}x\qquad\mbox{ for a.e.
}t>0.$ (56)

  1. i)

Case $m=1$: We convolve both sides of (56) with the nonnegative kernel
$h_{n}\in H^{1}([0,T])$ and obtain for a.e. $t>0$

$\int_{0}^{t}\int_{\Omega}h_{n}(t-\tau)\partial_{t}[k*(u-u_{0})](\tau)\Phi_{1}\,{\rm
d}x\,{\rm
d}\tau=-\lambda_{1}\int_{0}^{t}\int_{\Omega}h_{n}(t-s)u(\tau)\Phi_{1}\,{\rm
d}x\,{\rm d}\tau\,.$

Then, choosing $f(t,x)=\lambda_{1}x$ we deduce (55) from Fubini-Tonelli’s
Theorem and the regularity of $h_{n}\in H^{1}([0,T])$.

  2. ii)

Case $0<m<1$: We use Hölder’s inequality in (56) and get

$\int_{\Omega}\partial_{t}[k*(u-u_{0}](t)\Phi_{1}\,{\rm
d}x\geq-\lambda_{1}\|\Phi_{1}\|_{1}^{1-m}\left(\int_{\Omega}u(t)\Phi_{1}\,{\rm
d}x\right)^{m}\qquad\mbox{ for a.e. }t>0.$

Then, we convolve the inequality above with the nonnegative kernel $h_{n}\in
H^{1}([0,T])$,

$\int_{0}^{t}\int_{\Omega}h_{n}(t-\tau)\partial_{t}[k*(u-u_{0})](\tau)\Phi_{1}\,{\rm
d}x\,{\rm
d}\tau\geq-\lambda_{1}\|\Phi_{1}\|_{1}^{1-m}\\!\int_{0}^{t}h_{n}(t-s)\left(\int_{\Omega}u(\tau)\Phi_{1}\,{\rm
d}x\right)^{m}\,{\rm d}\tau,$

for a.e. $t>0$. We deduce (55) with
$f(t,x)=\lambda_{1}\|\Phi_{1}\|_{1}^{1-m}|x|^{m-1}x$ using Fubini-Tonelli’s
Theorem and the regularity of the kernel $h_{n}\in H^{1}([0,T])$.

  3. iii)

Case $m>1$: We use smoothing effects of Theorem 1.4 in (56)

$\displaystyle\int_{\Omega}\partial_{t}[k*(u-u_{0}](t)\Phi_{1}\,{\rm d}x$
$\displaystyle=-\lambda_{1}\int_{\Omega}u^{m}(t)\Phi_{1}\,{\rm
d}x\geq-\lambda_{1}\|u(t)\|_{\infty}^{m-1}\int_{\Omega}u(t)\Phi_{1}\,{\rm d}x$
$\displaystyle\geq-\lambda_{1}\overline{\kappa}\frac{\|u_{0}\|_{p}^{\frac{m-1}{m}}}{t^{\frac{\alpha(m-1)}{m}}}\int_{\Omega}u(t)\Phi_{1}\,{\rm
d}x\,.$

Hence, convolving with the nonnegative kernel $h_{n}\in H^{1}([0,T])$ we
obtain (55) as before, but with
$f(t,x)=\lambda_{1}\overline{\kappa}\|u_{0}\|_{p}^{\frac{m-1}{m}}\,t^{-\frac{\alpha(m-1)}{m}}x$.

$\bullet~{}$Step 3. Lower regularized integral inequality. Now, we combine the
inequality for solution $Y$ of (55) with the solution $Z$ of the equation

$\partial_{t}[k_{n}*(Z-Z_{0})](t)=-h_{n}*f(t,Z(t))\qquad\mbox{for a.e.
}t\in(0,T)\,,$ (57)

with $Z_{0}=Y_{0}$ and we prove that

$\frac{1}{2}[k_{n}*(Z-Y)_{+}^{2}](T)+\int_{0}^{T}(Z(t)-Y(t))_{+}h_{n}*(f(t,Z(t))-f(t,Y(t)))\,{\rm
d}t\leq 0\,.$ (58)

For this purpose, we subtract the equation of $Y$ in (55)from (57) and we
multiply the resulting equation by $(Z-Y)_{+}\in L^{\infty}(0,T)$. This yields

$(Z-Y)_{+}(t)\partial_{t}\left[k_{n}*(Z-Y)\right](t)+(Z-Y)_{+}(t)\,h_{n}*(f(t,Z(t))-f(t,Y(t)))\leq
0\,.$

By the fundamental identity of Lemma A.4 with $h(r)=\frac{1}{2}(r_{+})^{2}$,
we obtain

$\displaystyle\frac{1}{2}\partial_{t}\left[k_{n}*(Z-Y)_{+}^{2}\right](t)+(Z-Y)_{+}(t)\,h_{n}*(f(t,Z(t))-f(t,Y(t)))$
$\displaystyle\leq$
$\displaystyle(Z-Y)_{+}(t)\partial_{t}\left[k_{n}*(Z-Y)\right](t)+(Z-Y)_{+}(t)\,h_{n}*(f(t,Z(t))-f(t,Y(t)))\leq
0.$

Integrating in $[0,T]$ and recalling that $Y_{0}=Z_{0}$ a.e. in $\Omega$, we
obtain (58) .

$\bullet~{}$Step 4. Comparison with $Z$. We want to take the limit in (58) and
obtain

$k*(Z-Y)_{+}^{2}(T)\leq 0\,,$ (59)

which implies $Z(t)\leq Y(t)$ for a.e. $t>0$ since $k(t)>0$ for all $t>0$.

Recalling that $(Z-Y)_{+}^{2}\in L^{\infty}((0,T))$, we deduce

$\displaystyle\left|k_{n}*(Z-Y)_{+}^{2}(T)-k*(Z-Y)_{+}^{2}(T)\right|\leq\|(Z-Y)_{+}^{2}\|_{\infty}\int_{0}^{T}|(h_{n}*k)(t)-k(t)|\,{\rm
d}t\rightarrow 0\,,$

when $n\rightarrow\infty$. On the other hand, since $f(t,Y(t)),f(t,Z(t))\in
L^{1}((0,T))$ we obtain

$\displaystyle\left|\int_{0}^{T}\\!\\!(Z(t)-Y(t))_{+}h_{n}*(f(t,Z(t))-f(t,Y(t)))\,{\rm
d}t-\int_{0}^{T}(Z(t)-Y(t))_{+}(f(t,Z(t))-f(t,Y(t)))\,{\rm d}t\right|$
$\displaystyle\leq$
$\displaystyle\|(Z-Y)_{+}\|_{\infty}\int_{0}^{T}\\!\\!|[h_{n}*(f(t,Z(t))-f(t,Y(t)))](t)-(f(t,Z(t))-f(t,Y(t)))(t)|\,{\rm
d}t\rightarrow 0\,,$

when $n\rightarrow\infty$. Thus, taking the limit in (58) we obtain

$\frac{1}{2}[k*(Z-Y)_{+}^{2}](T)+\int_{0}^{T}(Z(t)-Y(t))_{+}(f(t,Z(t))-f(t,Y(t)))\,{\rm
d}t\leq 0\,.$

Notice that $(Z(t)-Y(t))_{+}(f(t,Z(t))-f(t,Y(t)))\geq 0$ for a.e. $t>0$ by the
nondecreasing assumption on $f$, and hence (59) holds. The result follows from
(59) and the fact that $k(t)>0$ for every $t>0$.

$\bullet~{}$Step 5. Lower bound. Finally, we show that if $Z_{0}=Y_{0}$, then

$\frac{c}{1+t^{\frac{\alpha}{m}}}\leq Z(t)\leq Y(t)\qquad\forall t>0\,,$

which give us the lower bound for $Y(t)=\|u(t)\|_{L^{1}_{\Phi_{1}}(\Omega)}$.

Let us recall the form of $Z$ for the three particular case of the fractional
ODEs in (57):

  1. i)

If $m=1$ and $f(t,x)=cx$, $Z$ is the Mittag-Leffler function
$Z(t)=Y_{0}E_{\alpha}[-ct^{\alpha}]$, see (61).

  2. ii)

If $0<m<1$ and $f(t,x)=c|x|^{m-1}x$, we do not know the explicit solution of
$Z$, but we know its behaviour by the result of Vergara and Zacher in [49],
see Theorem A.3.

  3. iii)

If $m>1$ and $f(t,x)=ct^{-\frac{\alpha(m-1)}{m}}x$, $Z$ is the so-called
Kilbas-Saigo function of the form
$Z(t)=Y_{0}E_{\alpha,\frac{1}{m},\frac{1-m}{m}}[-ct^{\frac{\alpha}{m}}]$, see
Lemma A.1.

Furthermore, by (62), Theorem A.3 and Proposition A.2, the three functions $Z$
defined above have the same decay

$Z(t)\asymp\frac{1}{1+t^{\frac{\alpha}{m}}}\,,$

which implies the desired lower bound ∎

Now that we have a lower bound for the decay of the $L^{1}_{\Phi_{1}}$-norm of
solutions of (CPME), we are in a position to prove the sharp time decay of the
$L^{p}$-norm stated in Theorem 1.7.

Proof of Theorem 1.7. The proof follows by an interpolation argument, using
the lower bound of the $L^{1}_{\Phi_{1}}$-norm of solutions and the
$L^{p}-L^{\infty}$ smoothing effects. By Proposition 5.2 and Theorem 1.4 we
obtain that for every $t>0$

$\displaystyle\frac{c}{1+t^{\frac{\alpha}{m}}}$
$\displaystyle\leq\|u(t)\|_{L^{1}_{\Phi_{1}}}\leq\|u(t)\|_{p}\;\|\Phi_{1}\|_{p^{\prime}}\leq|\Omega|\|u(t)\|_{\infty}\,\|\Phi_{1}\|_{\infty}\leq|\Omega|\,\|\Phi_{1}\|_{\infty}\,\overline{\kappa}\,\frac{\|u_{0}\|^{\frac{1}{m}}_{p_{0}}}{t^{\frac{\alpha}{m}}}\,,$
(60)

for any $1\leq p\leq\infty$ and $p^{\prime}$ being its conjugate exponent.
Hence, for the case $1\leq p<\infty$, we divide (60) by
$\|\Phi_{1}\|_{p^{\prime}}$ and for the case $p=\infty$ we divide inequality
(60) by $|\Omega|\,\|\Phi_{1}\|_{\infty}$ and conclude the proof.

$\square$

##  6 Open questions

In the study of the problem for the Caputo Porous Medium equation many
questions have arisen. We share the most important ones with the community.

  1. i)

T-contractivity: From the fractional gradient flow theory developed in [1, 37,
38] we know that the contractivity in the Hilbert space $H^{*}(\Omega)$ holds,
that is, if $u$ and $v$ are $H^{*}$-solutions with initial data $u_{0},v_{0}$,
respectively, then

$\|u(t)-v(t)\|_{H^{*}(\Omega)}\leq\|u_{0}-v_{0}\|_{H^{*}(\Omega)}\qquad\forall
t>0\,.$

In order to obtain the comparison principle of Theorem 1.2, we try to prove a
stronger version of property above called T-contractivity in $H^{*}(\Omega)$,
namely,

$\|(u(t)-v(t))_{+}\|_{H^{*}(\Omega)}\leq\|(u_{0}-v_{0})_{+}\|_{H^{*}(\Omega)}\qquad\forall
t>0\,.$

Since nonnegative functions of $H^{*}(\Omega)$ are in
$L^{1}_{\Phi_{1}}(\Omega)$ and
$\Phi_{1}(x)\asymp\mbox{dist}(x,\partial\Omega)^{\gamma}$, this
T-contractivity would imply the comparison principle of Theorem 1.2 directly.
But, due to the lack of regularity estimates for $D^{\alpha}_{t}u$, we have to
prove the T-contractivity in $L^{1}_{\Phi_{1}}(\Omega)$ at the discrete level
in Theorem 2.4 and inherit the comparison principle from the discrete
comparison principle.

We think that it would be enough to have $|D^{\alpha}_{t}u|\in
L^{2}((0,T):H^{*}(\Omega))$ or $|D^{\alpha}_{t}u|\in
L^{2}((0,T):L^{1}_{\Phi_{1}}(\Omega))$ in order to prove T-contractivity in
$H^{*}(\Omega)$ or $L^{1}_{\Phi_{1}}(\Omega)$, respectively. Let us remark
that T-contractivity in $L^{1}_{\Phi_{1}}(\Omega)$ is also an open problem in
the case of classical time derivative, when considering weak dual solutions of
the Porous Medium and Fast Diffusion equation. In [13, 18] it was proved only
for minimal weak dual solutions.

  2. ii)

Higher regularity: The classical way to prove Hölder continuity is via some
kind of Harnack inequalities for the solution. The problem with PDEs involving
Caputo time derivative is that obtaining lower bounds for solutions is
significantly more challenging. In this manuscript we prove a Global Harnack
Principle in Theorem 1.6 only for a class of solutions with “big” initial
data, but we did not succeed to prove it for general initial data. To our
knowledge, the only result in this direction appeared in [3] and [2], where
the authors apply the De Giorgi’s method to show Hölder regularity.

  3. iii)

Asymptotic behaviour: Theorem 1.7 shows that solutions of (CPME) does not
vanish in finite time for any $m>0$. One may wonder about the behaviour of the
solutions when $t\rightarrow\infty$. From the smoothing effects of Theorem
1.4, we deduce that solution vanish at infinity, but we would like to know if
there is any asymptotic profile $S(x)$ such that

$\lim\limits_{t\rightarrow\infty}\|t^{\frac{\alpha}{m}}u(t)-S\|_{X}=0\,,$

for certain norm $X$. From the classical theory, we expect that $S$ is the
solution of the elliptic problem described in Section 5 and that the solution
behaves at infinity like the separate-variable solution. In the classical
approach, the results relies on a change of variables which with Caputo time
derivative is more involved, see the fundamental identity for regular kernels
in Lemma A.4.

##  Appendix A Appendix. Fractional ODEs

In this appendix we collect some of the known results about the Caputo
derivative and the associated fractional ODEs that are useful for our study.
Let us start with the linear differential equation with $\alpha\in(0,1)$,
$v_{0}>0$ and $\lambda\in\mathbb{R}^{N}$

$\begin{cases}D^{\alpha}_{t}v(t)=\lambda v(t)\quad\mbox{in }(0,+\infty)\\\
v(0)=v_{0},\end{cases}$ (61)

which solution is the so-called Mittag-Leffler function of the form
$v(t)=v_{0}\,E_{\alpha}(\lambda t^{\alpha})$, where

$E_{\alpha}(x)=\sum_{k=1}^{\infty}\frac{x^{k}}{\Gamma(k\alpha+1)}\,.$

There is a large amount of literature about this function, cf [32, 33]. But,
let us remark that the behaviour at the infinity of $E_{\alpha}(-x)$ is given
by

$E_{\alpha}(-x)\asymp\frac{1}{1+x}\,,$ (62)

as it is shown in [39]. Moreover, we can see the Mittag-Leffler function
$E_{\alpha}(-x^{2})$ with $\alpha\in(0,1)$ as an interpolation between the
Gaussian and the polynomial with quadratic decay

$E_{1}(-x^{2})=e^{-x^{2}}\leq E_{\alpha_{1}}(-x^{2})\leq
E_{\alpha_{2}}(-x^{2})\leq\frac{1}{1+x^{2}}\,,$

with $0<\alpha_{1}\leq\alpha_{1}<1$. Now, let us consider another linear
fractional ODE which has an explicit solution.

######  Lemma A.1.

[35, Example 4.11] Let $0<\alpha<1$ and $\beta>-\alpha$. Then, the solution of

$\begin{cases}D^{\alpha}_{t}v(t)&=-\lambda t^{\beta}v(t)\\\
v(0)&=v_{0},\end{cases}$ (63)

is given by

$v(t)=v_{0}\,E_{\alpha,1+\frac{\beta}{\alpha},\frac{\beta}{\alpha}}[-\lambda
t^{\alpha+\beta}]\,.$

This function is called the Kilbas-Saigo function and its explicit form can be
found in [19]. However, for our study we only need the asymptotic behaviour
that we state below.

######  Proposition A.2.

[19, Proposition 6] Let $0<\alpha<1$, $r>0$ and $l>r-\frac{1}{\alpha}$, then

$E_{\alpha,r,l}(-x)\asymp\frac{1}{1+c\,x}\qquad\quad\mbox{as}\quad
x\rightarrow\infty\,.$

In the nonlinear setting, fractional ODEs becomes very delicate. Let us
present the following result due to Vergara and Zacher addressing the
behaviour of a basic nonlinear fractional ODE.

######  Theorem A.3.

[49, Theorem 7.1] Let $\alpha\in(0,1)$, $\lambda,m>0$ and $v_{0}>0$. Let $v\in
H^{1}_{\rm loc}(\mathbb{R}_{+})$ be the solution of

$\begin{cases}D^{\alpha}_{t}v(t)=-\lambda v^{m}(t)&\mbox{in }(0,+\infty)\\\
v(0)=v_{0}.\end{cases}$

Then, there exist $c_{1},c_{2}>0$ depending on $v_{0},m,\alpha$ and $\lambda$
such that

$\frac{c_{1}}{1+t^{\frac{\alpha}{m}}}\leq
v(t)\leq\frac{c_{2}}{1+t^{\frac{\alpha}{m}}}\qquad\forall t>0\,.$ (64)

The explicit form of the solution to this basic nonlinear fractional ODE is
not known, but in [49] they construct subsolution and supersolution to
estimate the lower and upper bounds. Note that the behaviour of $v$ described
in (64) is completely different from the solution of the analogous classical
ODE. In the classical case, the exponent $m$ plays a crucial role. Namely, for
the classical ODE of the form $\partial_{t}v=-\lambda v^{m}$ with $v(0)=v_{0}$
we have that:

  * •

If $0<m<1$, the solution vanish in finite time since

$v(t)=(v_{0}^{1-m}-\lambda(1-m)\,t)_{+}^{\frac{1}{1-m}}\,.$

  * •

If $m=1$, the solution decays exponentially, $v(t)=v_{0}\exp(-\lambda t)$.

  * •

If $m>1$, the solution decays polynomially

$v(t)=\frac{1}{\left(v_{0}^{-(m-1)}+\lambda(m-1)t\right)^{\frac{1}{m-1}}}\leq\frac{C}{t^{\frac{1}{m-1}}}\,.$

This shows at the level of ODEs how different are the solutions when we
consider Caputo time derivative. In particular, there is no continuity of
results when $\alpha\rightarrow 1$.

To conclude this appendix, we present a fundamental identity for fractional
derivatives with regular kernels. This formula plays the role of the chain
rule $\partial_{t}h(u(t))=h^{\prime}(u(t))\partial_{t}u(t)$ in the classical
sense.

######  Lemma A.4.

[49, Lemma 2.2] Let $T>0$ and $I$ an open set of $\mathbb{R}$. Further, let
$\tilde{k}\in W^{1,1}([0,T])$, $h\in C^{1}(I)$ and $u\in L^{1}((0,T))$ with
$u(t)\in I$ for a.e. $t\in(0,T)$. Suppose that the functions
$h(u),h^{\prime}(u)u$ and $h^{\prime}(u)(\tilde{k}^{\prime}*u)$ belong to
$L^{1}((0,T))$ (which is the case if $u\in L^{\infty}((0,T))$). Then, we have
for a.e. $t\in(0,T)$

$\begin{split}h^{\prime}(u(t))\frac{\rm d}{\,{\rm
d}t}(\tilde{k}*u)(t)=&\frac{\rm d}{\,{\rm
d}t}(\tilde{k}*h(u))(t)+\bigg{(}-h(u(t))+h^{\prime}(u(t))u(t)\bigg{)}\tilde{k}(t)\\\
&+\int_{0}^{t}\bigg{(}h(u(t-\tau))-h(u(t))-h^{\prime}(u(t))\left[u(t-\tau)-u(t)\right]\bigg{)}(-\tilde{k}^{\prime}(\tau))\,{\rm
d}\tau\,.\end{split}$

Let us remark that when the function $h$ is convex and the kernel $\tilde{k}$
is nonnegative and nonincreasing, we obtain the inequality

$\frac{\rm d}{\,{\rm d}t}(\tilde{k}*h(u))\leq h^{\prime}(u(t))\frac{\rm
d}{\,{\rm d}t}(\tilde{k}*u)(t)\,.$

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