# On parabolic partial differential equations with Hölder continuous diffusion
Coefficients

Majed Sofiani  Department of Mathematics, University of Kansas, Lawrence, KS
66045, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We investigate existence and regularity of weak solutions of a 1-dimensional
parabolic differential equation with a non-constant Hölder diffusion
coefficient and a rough forcing term. Such an equation appears in studying the
1-dimensional Ericksen-Leslie model for nematic liquid crystals where our
result applies. The result presented here uses the Hölder continuity of the
diffusion coefficient which comes from the physical background and the
analysis of the Ericksen-Leslie model. Moreover, the dependence of the Hölder
exponent of the solution is explicit on the Hölder exponent of the diffusion
coefficient.

##  1\. Introduction

Consider the following Cauchy problem

$\displaystyle w_{t}-k(x,t)w_{xx}+\gamma w$
$\displaystyle=f(x,t)+G(x,t)\quad\text{on}\quad\Omega_{T},$ (1.1)
$\displaystyle w(x,0)$ $\displaystyle=w_{0}(x)\quad\text{at}\quad t=0,$ (1.2)

where $T>0$ is arbitrary but fixed and $\Omega_{T}:=\mathbb{R}\times(0,T].$ We
assume

$\displaystyle\begin{split}k(x,t):&\Omega_{T}\to\mathbb{R}^{+}\in
C^{1/2}(\Omega_{T}),\\\ k_{x}(x,t)&\in
L^{\infty}([0,T],L^{2}(\mathbb{R})),\end{split}$ (1.3)

and for some $k_{L},k_{U}>0,$

$k_{L}\leq k(x,t)\leq k_{U};$
$\displaystyle\begin{split}f(x,t):\overline{\Omega}_{T}\to\mathbb{R}&\in
L^{\infty}([0,T],L^{2}(\mathbb{R}));\\\
G(x,t):\overline{\Omega}_{T}\to\mathbb{R}&\in
L^{\infty}([0,T],C(\mathbb{R}))\cap L^{\infty}(\Omega_{T});\\\ G_{x}(x,t)&\in
L^{\infty}([0,T],L^{1}(\mathbb{R}));\\\ \gamma&\in\mathbb{R};\end{split}$
(1.4)

and

$\displaystyle w(x,0)=w_{0}(x)\in H^{1}(\mathbb{R}).$ (1.5)

######  Remark 1.

The function $G$ has a specific form, namely

$G(x,t)=\int_{-\infty}^{x}f^{2}(z,t)\,dz,$

where $f$ is the function in (1.1).

Equation (1.1) with the specific data (1.3) and (1.4) appears in studying the
Poiseuille flows of nematic liquid crystals via the Ericksen-Leslie model [4,
9, 10, 11, 1, 13, 7]. This work is particularly inspired by the relatively
recent work in [1]. A brief background of equation (1.1) will be presented in
§1.1.

To state our main result, we need to introduce

######  Definition 1.

A function $w(x,t)$ is a weak solution to (1.1) and (1.2) if the following
holds,

  1. (1)

For any $\alpha\in(0,1/4),$

$w(x,t),w_{x}(x,t)\in L^{\infty}\cap C^{\alpha}(\Omega_{T});$

  2. (2)

for any $\phi\in C_{c}^{1}(\Omega_{T}),$

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}-\phi_{t}w+(k\phi)_{x}w_{x}+\gamma\phi
w\,dx\,dt=\int_{0}^{T}\int_{\mathbb{R}}(f+G)\phi\,dx\,dt;$ (1.6)

  3. (3)

The initial data is satisfied in the following sense,

$w(x,t)\to w_{0}(x)\quad\text{point-wise as}t\to 0^{+}.$

######  Theorem 1.

For any fixed $T>0,$ under the assumptions (1.3)-(1.5) the Cauchy problem
(1.1) and (1.2) has a weak solution in the sense of Definition 1 over the
domain $\Omega_{T}.$

###  1.1. Physical background and motivation

As mentioned above, this work is motivated by a study of Ericksen-Leslie model
for Poiseulle flow of nematic liquid crystals. The state variables for
nematics are the velocity vector $\bf u$ and the director field $\bf n.$ In
the case where the flow moves in the $z$-direction and is uniform in $x,y$ and
the molecules lie on the $x-z$ plane, we write the velocity field as
$\mathbf{u}(x,t)=(0,0,u(x,t))^{T}$ and the director field as
$\mathbf{n}(x,t)=(\sin\theta(x,t),0,\cos\theta(x,t))^{T},$ where $\theta$ is
the angle measured from the $z$ axis. The system reduces to

$\displaystyle\begin{split}\displaystyle
u_{t}&=\left(g(\theta)u_{x}+h(\theta)\theta_{t}\right)_{x},\\\
\theta_{tt}+\gamma_{1}\theta_{t}&=c(\theta)\big{(}c(\theta)\theta_{x}\big{)}_{x}-h(\theta)u_{x},\end{split}$
(1.7)

where the functions $c,$ $g,$ and $h$ are explicitly given in the literature

$\displaystyle\begin{split}g(\theta):=&\alpha_{1}\sin^{2}\theta\cos^{2}\theta+\frac{\alpha_{5}-\alpha_{2}}{2}\sin^{2}\theta+\frac{\alpha_{3}+\alpha_{6}}{2}\cos^{2}\theta+\frac{\alpha_{4}}{2},\\\
h(\theta):=&\alpha_{3}\cos^{2}\theta-\alpha_{2}\sin^{2}\theta=\frac{\gamma_{1}+\gamma_{2}\cos(2\theta)}{2},\\\
c^{2}(\theta):=&K_{1}\cos^{2}\theta+K_{3}\sin^{2}\theta.\end{split}$ (1.8)

In (1.8), $K_{1},K_{3}$ are Frank’s constants for splay and bending
deformation of the material and $\alpha_{j}^{\prime}s$ with
($\gamma_{1}=\alpha_{3}-\alpha_{2},\,\gamma_{2}=\alpha_{6}-\alpha_{5}$) are
the Leslie dynamics parameters. See [1] for a detailed discussion and
derivation of the Ericksen-Leslie system for Poiseulle flows.

In [1], a special case ($g=h=1$ and $\gamma_{1}=2$) was investigated and a
crucial quantity $J:=u_{x}+\theta_{t}$ was realized. It turns out that $J$ has
better regularity than $\theta_{t}$ and $u_{x}$ individually. That is, each of
$\theta_{t}$ and $u_{x}$ might blow up in finite time but $J$ will always be
bounded and in fact Hölder continuous. The system is written as

$\displaystyle\displaystyle u_{t}$
$\displaystyle=\left(u_{x}+\theta_{t}\right)_{x},$ (1.9)
$\displaystyle\theta_{tt}+2\theta_{t}$
$\displaystyle=c(\theta)\big{(}c(\theta)\theta_{x}\big{)}_{x}-u_{x}.$ (1.10)

In terms of $J,$

$\displaystyle u_{t}$ $\displaystyle=J_{x},$ (1.11)
$\displaystyle\theta_{tt}+\theta_{t}$
$\displaystyle=c(\theta)(c(\theta)\theta_{x})_{x}-J.$ (1.12)

To close the system, using the definition of $J$ and (1.11), it is easy to see
that

$J_{t}=J_{xx}+\theta_{tt}.$

Using the wave equation we obtain an equation for $J$

$\displaystyle
J_{t}=J_{xx}+c(\theta)(c(\theta)\theta_{x})_{x}-u_{x}-\theta_{t}.$ (1.13)

This parabolic PDE (1.13) has a corresponding parabolic PDE for the general
case (1.7), with non-constant $g$ and $h,$ satisfied by a quantity $A$
expressed as follows

$A(x,t):=\int_{-\infty}^{x}J(z,t)\,dz$

where $J$ in this case is slightly modified,

$J:=u_{x}+\frac{h}{g}\theta_{t}.$

For full context of the derivation of the quantity and its equation we refer
the readers to [2].

The parabolic PDE satisfied by $A$ has a similar form as equation (1.1)
considered in this paper. We will investigate this equation in a general setup
that is closely related to the situation for the system (1.7). The results
presented here are crucial for the proof of the global existence of weak
solution for the full Ericksen-Leslie model for Poiseulle flows of nematic
liquid crystals in [1]. That is mainly because the major difference is the
appearance of the function $g(\theta)$ as a diffusion coefficient in the
equations for the quantities $v$ and $A$ defined above.

###  1.2. A relevant work from [6] and Duhamel formula.

In this subsection we recall a result from [6] on the existence of classical
solutions to parabolic PDEs. The framework therein is relevant and is our
starting point for the analysis in this work. The results in [6] are shown for
the n-dimensional space and any Hölder exponent $0<\beta<1.$ We focus on the
relevant situation when $n=1$ and $\beta=1/2.$

Let $k(x,t)\in C^{\beta}$ with respect to both variables on $\Omega_{T}.$
Consider, the following homogeneous equation

$\displaystyle w_{t}-k(x,t)w_{xx}+\gamma w=0.$ (1.14)

Note that, for fixed $(\xi,\tau),$ the function

$\displaystyle
H^{\xi,\tau}(x-\xi,t-\tau)=\frac{1}{2\sqrt{k(\xi,\tau)}\sqrt{t-\tau}}e^{-\frac{(x-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}$
(1.15)

is the heat kernel with frozen coefficient $k(\xi,\tau),$ that is,

$H_{t}^{\xi,\tau}-k(\xi,\tau)H_{xx}^{\xi,\tau}=0.$

We can write the fundamental solution $\Gamma$ of (1.14) implicitly as follows

$\displaystyle\Gamma(x,t,\xi,\tau)$
$\displaystyle=H^{\xi,\tau}(x-\xi,t-\tau)+\int_{\tau}^{t}\int_{\mathbb{R}}H^{y,s}(x-y,t-s)\Phi(y,s;\xi,\tau)\,dy\,ds,$
(1.16)

where the function $\Phi$ is determined by the condition that

$\Gamma_{t}-k(x,t)\Gamma_{xx}+\gamma\Gamma=0.$

In [6], the author showed such function $\Phi$ exists and satisfies

$\displaystyle|\Phi(y,s;\xi,\tau)|\leq\frac{C}{(s-\tau)^{\frac{3-\beta}{2}}}e^{\frac{-d(y-\xi)^{2}}{4(s-\tau)}}=\frac{C}{(s-\tau)^{\frac{5}{4}}}e^{\frac{-d(y-\xi)^{2}}{4(s-\tau)}},$
(1.17)

where $C$ and $d$ are constants depending only on $k.$ We will use $C$ to
represent other constants that are not necessarily the same. In addition, the
author showed that if $q(x,t)$ is Hölder continuous on $\Omega_{T}$ and
$\psi_{0}(x)$ is continuous on $\mathbb{R}$, then the solution of the
following Cauchy problem

$-u_{t}+k(x,t)u_{xx}-\gamma u=q(x,t),$ $u(x,0)=\psi_{0}(x),$

is given by the Duhamel formula

$\displaystyle
u(x,t)=\int_{\mathbb{R}}\Gamma(x,t,\xi,0)\psi_{0}(\xi)\,d\xi+\int_{0}^{t}\int_{\mathbb{R}}\Gamma(x,t;\xi.\tau)q(\xi,\tau)\,d\xi\,d\tau,$
(1.18)

and it is a classical solution.

In addition, the following estimates were shown therein,

$\displaystyle|\Gamma(x,t;\xi,\tau)|\leq\frac{C}{\sqrt{t-\tau}}e^{-\frac{d(x-\xi)^{2}}{4(t-\tau)}}\approx
H(x-\xi,t-\tau),$ (1.19)
$\displaystyle|\Gamma_{x}(x,t;\xi,\tau)|\leq\frac{C}{{t-\tau}}e^{-\frac{d(x-\xi)^{2}}{4(t-\tau)}}\approx\frac{1}{\sqrt{t-\tau}}H(x-\xi,t-\tau),$
(1.20)

where $d$ is a constant depending on $k$ and $\approx$ means equality up to a
factor depending on $k(\xi,\tau).$

###  1.3. The main idea of this work

For the application in our work [2], we encounter a rougher, particularly not
Hölder continuous, data for the non-homogeneous term $q(x,t).$ For this
reason, the result in [6] can not be applied. Nevertheless, an extension of
the framework in [6] is needed in the sense of proving the existence of a weak
solution in the case when a rough source data imposed to the PDE. In addition,
Hölder estimates for the weak solution and its $x$-derivative are needed for
the work in [2]. We estimate the $L^{\infty},L^{2}$ and the $C^{\alpha}$ norms
of the solution and its derivative with respect to $x.$ We use the mollifying
technique to prove the existence of weak solution. Precisely, we mollify the
data $f$ and $G$ and then pass to the limit with respect to the $L^{\infty}$
norm. Briefly, if we mollify $f$ and $G$ we can write the solution in a
classical sense following [6]. Then, using the $L^{\infty}$ estimates of $u$
and $u_{x}$ we can take the limit of the weak formulation with respect to the
mollifier parameter. It is worth mentioning that Hölder continuity of
solutions to parabolic PDE’s has been extensively studied and many results
have been shown, for example [8, 15] and the references there. However, since
the equation here is motivated by a concrete model as discussed before, the
result here assumes Hölder continuity of the coefficient and gives more
explicit formula for the solution and also shows the dependence on the Hölder
exponent of the coefficient.

Following the discussion in §1.2, we can represent the solution to (1.1) and
(1.2) as follows,

$\displaystyle
w(x,t)=\int_{\mathbb{R}}\Gamma(x,t,\xi,0)w_{0}(\xi)\,d\xi+\int_{0}^{t}\int_{\mathbb{R}}\Gamma(x,t;\xi.\tau)[G(\xi,\tau)+f(\xi,\tau)]\,d\xi\,d\tau.$
(1.21)

We will show that this representation of the solution indeed satisfies the
weak formulation in Definition 1. We will consider the three terms of the
expression (1.21) separately and denote each term as follows,

$\displaystyle W(x,t):=\int_{\mathbb{R}}\Gamma(x,t,\xi,0)w_{0}(\xi)\,d\xi,$
$\displaystyle
W_{G}(x,t):=\int_{0}^{t}\int_{\mathbb{R}}\Gamma(x,t;\xi.\tau)G(\xi,\tau)\,d\xi\,d\tau,$

and

$\displaystyle
W_{f}(x,t):=\int_{0}^{t}\int_{\mathbb{R}}\Gamma(x,t;\xi.\tau)f(\xi,\tau)\,d\xi\,d\tau.$

Analogously, denote the $x$ derivative of the expressions as

$\displaystyle
W_{x}(x,t):=\int_{\mathbb{R}}\Gamma_{x}(x,t,\xi,0)w_{0}(\xi)\,d\xi.$
$\displaystyle
W_{G,x}(x,t):=\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi.\tau)G(\xi,\tau)\,d\xi\,d\tau,$

and

$\displaystyle
W_{f,x}(x,t):=\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi.\tau)f(\xi,\tau)\,d\xi\,d\tau.$

The key estimates on the solution $w$ are presented in the next propositions.

######  Proposition 1.1.

Fix $T>0.$ Under the assumptions (1.3)- (1.5), the functions
$W_{G},W_{f},W_{G,x},$ and $W_{f,x}$ belong to the following space,

$L^{\infty}\cap C^{\alpha}(\bar{\Omega}_{T}),$

for any $0<\alpha<1/4.$ Moreover, for any compact set $F\subset\Omega_{T},$
one has

$\max\big{\\{}\|W\|_{(L^{\infty}\cap
C^{\alpha})(F)},\|W_{x}\|_{(L^{\infty}\cap C^{\alpha})(F)}\big{\\}}<\infty,$

and

$W(x,t),W_{x}(x,t)\to w_{0}(x),w^{\prime}_{0}(x)\mbox{ point-wise as }t\to
0^{+}.$

As a consequence of the first assertion in Proposition 1.1, we have that for
any constant $K>0,$ there exists $\delta>0,$ depending on $G,f$ and $k,$
sufficiently small such that

$\|W_{\\{\square\\}}(x,t)\|_{(L^{\infty}\cap
C^{\alpha})(\overline{\Omega}_{\delta})}\leq K,$

for $\\{\square\\}:=\\{G\\},\\{f\\},\\{G,x\\}$ and $\\{f,x\\}.$

Due to the lack of decay of the source term $G,$ we can only obtain $L^{2}$
decay of the terms $W_{f},W_{f,x}$ and $W_{G,x}.$ More precisely,

######  Proposition 1.2.

For any constant $K>0,$ there exists $\delta>0,$ depending on $G,f$ and $k,$
sufficiently small such that

$\|W_{\\{\circ\\}}(x,t)\|_{L^{2}(\overline{\Omega}_{\delta})}\leq K,$

for $\\{\circ\\}:=\\{f\\},\\{f,x\\},$ and $\\{G,x\\}.$

The rest of the paper is organised as follows. In Section 2, we present a
proof of Proposition 1.1. In Section 3, we prove the global existence of the
weak solution (Theorem 1). Finally, we prove Proposition 1.2 in Section 4.

##  2\. Proof of Proposition 1.1

To avoid repetitions, we will show the estimates for $W_{G,x}$ and $W_{f,x}$.
The other terms have no derivatives on $\Gamma$ so they have even weaker
integrable singularities and can be estimated in a similar fashion.

###  2.1. The $L^{\infty}$ estimates

In this subsection, we consider the $L^{\infty}$ estimates for $W_{G,x}$ and
$W_{f,x}.$

For the term

$\displaystyle
W_{G,x}:=\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi.\tau)G(\xi,\tau)\,d\xi\,d\tau,$
(2.1)

we use (1.20) to get,

$\displaystyle|W_{G,x}|\lesssim\|G\|_{L^{\infty}(\Omega_{T})}\int_{0}^{t}\int_{\mathbb{R}}\frac{1}{{t-\tau}}e^{-\frac{d(x-\xi)^{2}}{4(t-\tau)}}\,d\xi\,d\tau\lesssim
T^{1/2}\|G\|_{L^{\infty}(\Omega_{T})},$

which yields,

$\displaystyle\|W_{G,x}\|_{L^{\infty}(\Omega_{T})}\lesssim
T^{1/2}\|G\|_{L^{\infty}(\Omega_{T})}.$ (2.2)

Similarly, for the term

$\displaystyle
W_{f,x}:=\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi.\tau)f(\xi,\tau)\,d\xi\,d\tau,$
(2.3)

we use (1.20) and Hölder inequality,

$\displaystyle|W_{f,x}|\lesssim$
$\displaystyle\bigg{[}\int_{0}^{t}\int_{\mathbb{R}}\frac{1}{{|t-\tau|}^{2-2r}}e^{-\frac{d(x-\xi)^{2}}{2(t-\tau)}}\,d\xi\,d\tau\bigg{]}^{1/2}\bigg{[}\int_{0}^{t}\int_{\mathbb{R}}\frac{1}{{|t-\tau|}^{2r}}f^{2}\,d\xi\,d\tau\bigg{]}^{1/2}$
$\displaystyle\lesssim\bigg{[}\int_{0}^{t}\frac{1}{{|t-\tau|}^{\frac{3}{2}-2r}}\,d\tau\bigg{]}^{1/2}\bigg{[}\int_{0}^{t}\frac{1}{{|t-\tau|}^{2r}}\,d\tau\bigg{]}^{1/2}\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}.$

For $r=\frac{3}{8}$ we obtain:

$|W_{f,x}|\lesssim T^{1/4}\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))},$

which yields,

$\displaystyle\|W_{f,x}\|_{L^{\infty}(\Omega_{T})}\lesssim
T^{1/4}\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}.$ (2.4)

###  2.2. The $C^{\alpha}$ estimates

Recall,

$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi,\tau)\square(\xi,\tau)\,d\xi\,d\tau$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}H_{x}^{\xi,\tau}(x-\xi,t-\tau)\square(\xi,\tau)\,d\xi\,d\tau$
(2.5)
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}}\bigg{[}\int_{\tau}^{t}\int_{\mathbb{R}}H_{x}^{y,s}(x-y,t-s)\Phi(y,s;\xi,\tau)\,dy\,ds\bigg{]}\square(\xi,\tau)\,d\xi\,d\tau.$
(2.6)

We call (2.5) the ’principal term’ and (2.6) the ’lower order term.’ We need
to estimate the $C^{\alpha}$ norm in $x$ for both terms.

Starting with the principal term, we write $H_{x}^{\xi,\tau}$ explicitly:

$\displaystyle
H_{x}^{\xi,\tau}(x-\xi,t-\tau)=-\frac{(x-\xi)}{4\sqrt{\pi}k(\xi,\tau)^{3/2}(t-\tau)^{3/2}}e^{-\frac{(x-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}.$

Without loss of generality, fix $x_{1}<x_{2}$ and consider the intervals

$B_{1}:=|x_{1}-\xi|<|x_{2}-\xi|\mbox{ and }B_{2}:=|x_{2}-\xi|<|x_{1}-\xi|.$

On $B_{1}$ we use

$\displaystyle\big{|}H_{x}(x_{2}-\xi,t-\tau)-H_{x}(x_{1}-\xi,t-\tau)\big{|}$
$\displaystyle\leq\frac{1}{k^{3/2}_{L}}\bigg{|}\frac{x_{1}-x_{2}}{(t-\tau)^{3/2}}e^{-\frac{(x_{2}-\xi)^{2}}{4k_{U}(t-\tau)}}\bigg{|}+\frac{1}{k^{3/2}_{L}}\bigg{|}\frac{x_{1}-\xi}{(t-\tau)^{3/2}}\bigg{[}e^{-\frac{(x_{1}-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}-e^{-\frac{(x_{2}-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}\bigg{]}\bigg{|},$

while on $B_{2}$ we use

$\displaystyle\big{|}H_{x}(x_{2}-\xi,t-\tau)-H_{x}(x_{1}-\xi,t-\tau)\big{|}$
$\displaystyle\leq\frac{1}{k^{3/2}_{L}}\bigg{|}\frac{x_{1}-x_{2}}{(t-\tau)^{3/2}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}\bigg{|}+\frac{1}{k^{3/2}_{L}}\bigg{|}\frac{x_{2}-\xi}{(t-\tau)^{3/2}}\bigg{[}e^{-\frac{(x_{2}-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}-e^{-\frac{(x_{1}-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}\bigg{]}\bigg{|}.$

We will consider the case on $B_{1}.$ The estimates on $B_{2}$ follow
similarly.

Dividing the inequality by $|x_{2}-x_{1}|^{\alpha}$ we get

$\displaystyle\frac{\big{|}H_{x}(x_{2}-\xi,t-\tau)-H_{x}(x_{1}-\xi,t-\tau)\big{|}}{|x_{2}-x_{1}|^{\alpha}}$
$\displaystyle\lesssim\frac{|x_{2}-x_{1}|^{1-\alpha}}{|t-\tau|^{3/2}}e^{-\frac{(x_{2}-\xi)^{2}}{4k_{U}(t-\tau)}}+\frac{|x_{1}-\xi|}{|t-\tau|^{3/2}|x_{2}-x_{1}|^{\alpha}}\bigg{|}e^{-\frac{(x_{1}-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}-e^{-\frac{(x_{2}-\xi)^{2}}{4k(\xi,\tau)(t-\tau)}}\bigg{|}.$
(2.7)

Now for the principal part (2.5) we have two terms produced by (2.7). We start
with the first term in (2.7):

$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{|x_{2}-x_{1}|^{1-\alpha}}{|t-\tau|^{3/2}}e^{-\frac{(x_{2}-\xi)^{2}}{4k_{U}(t-\tau)}}|\square(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle\leq\underbrace{2\int_{0}^{t}\int_{B_{1}}\frac{|x_{2}-\xi|^{1-\alpha}}{|t-\tau|^{3/2}}e^{-\frac{(x_{2}-\xi)^{2}}{4k_{U}(t-\tau)}}|\square(\xi,\tau)|\,d\xi\,d\tau}_{I_{\square}}.$

For the second term in (2.7) we have,

$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{3/2}|x_{2}-x_{1}|^{\alpha}}\bigg{|}e^{-\frac{(x_{1}-\xi)^{2}}{4k(t-\tau)}}-e^{-\frac{(x_{2}-\xi)^{2}}{4k(t-\tau)}}\bigg{|}|\square(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{3/2}|x_{2}-x_{1}|^{\alpha}}e^{-\frac{(x_{1}-\xi)^{2}}{4k(t-\tau)}}\bigg{|}1-e^{\frac{(x_{1}-\xi)^{2}}{4k(t-\tau)}-\frac{(x_{2}-\xi)^{2}}{4k(t-\tau)}}\bigg{|}|\square(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{3/2}|x_{2}-x_{1}|^{\alpha}}e^{-\frac{(x_{1}-\xi)^{2}}{4k(t-\tau)}}\bigg{|}1-e^{-\frac{(x_{2}-x_{1})(x_{2}+x_{1}-2\xi)}{4k(t-\tau)}}\bigg{|}|\square(\xi,\tau)|\,d\xi\,d\tau.$
(2.8)

Note that for $\xi\in B_{1}$ we have

$\frac{(x_{2}-x_{1})(x_{2}+x_{1}-2\xi)}{4k(\xi,\tau)(t-\tau)}>0.$

The following lemma will be needed,

######  Lemma 2.1.

For all $a\geq 0$ and $0<\lambda\leq 1,$ one has

$1-e^{-a}\leq\frac{1}{\lambda}a^{\lambda}.$

###### Proof.

The case $\lambda=1$ can be shown straightforwardly by the mean value theorem.
For $0<\lambda<1,$ define $F(a)=\frac{1}{\lambda}a^{\lambda}+e^{-a}-1.$ This
is a continuous function on $[0,\infty)$ for any $\lambda\in(0,1).$ Since
$F(0)=0$ and $\lim_{a\to\infty}F(a)=\infty.$ It suffices to show that the
values of $F$ at local extrema are all non-negative.

If $F$ has no extrema then it is monotone and we are done. Otherwise, suppose
$\tilde{a}>0$ is a local extremum of $F,$ that is $F^{\prime}(\tilde{a})=0,$
which gives

${\tilde{a}}^{\lambda-1}=e^{-{\tilde{a}}}.$

Now we have,

$F(\tilde{a})=\frac{1}{\lambda}\tilde{a}^{\lambda}+{\tilde{a}}^{\lambda-1}-1:=\tilde{F}.$

We claim that $\tilde{F}>0$ for all $\tilde{a}>0$ and hence if $\tilde{a}$ is
a local minimum then $F(\tilde{a})$ is still positive. To show that we notice,

$\tilde{F}\to\infty,$

as $\tilde{a}\to 0^{+},$ and

$\tilde{F}\to\infty,$

as $\tilde{a}\to\infty.$ Moreover, there is only one extremum point $a^{*}$
that is

${a^{*}}^{\lambda-1}+(\lambda-1){a^{*}}^{\lambda-2}=0,$

which gives

$a^{*}=1-\lambda.$

It must be a global minimum and finally

$\tilde{F}(1-\lambda)=\frac{1}{\lambda}(1-\lambda)^{\lambda}+(1-\lambda)^{\lambda-1}-1>0$

if and only if

$\frac{1-\lambda}{\lambda}+1-(1-\lambda)^{1-\lambda}>0,$

if and only if

$1-\lambda(1-\lambda)^{1-\lambda}>0,$

which is clearly true since $0<\lambda<1.$ ∎

Now using Lemma 2.1 for (2.8) with $\lambda=\frac{1}{2}-\epsilon,$ for some
$\epsilon$ to be determined later, and

$a=\frac{(x_{2}-x_{1})(x_{2}+x_{1}-2\xi)}{4k(\xi,\tau)(t-\tau)},$

we have

$\displaystyle\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{3/2}|x_{2}-x_{1}|^{\alpha}}e^{-\frac{(x_{1}-\xi)^{2}}{4k(t-\tau)}}\bigg{|}1-e^{-\frac{(x_{2}-x_{1})(x_{2}+x_{1}-2\xi)}{4k(t-\tau)}}\bigg{|}|\square(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle\lesssim\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{3/2}|x_{2}-x_{1}|^{\alpha}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}\bigg{|}\frac{(x_{2}-x_{1})(x_{2}+x_{1}-2\xi)}{4(t-\tau)}\bigg{|}^{\frac{1}{2}-\epsilon}|\square(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle\leq\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{3/2}|x_{2}-x_{1}|^{\alpha}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}\frac{(x_{2}-x_{1})^{1-2\epsilon}+[2(x_{2}-x_{1})|x_{1}-\xi|]^{\frac{1}{2}-\epsilon}}{[4(t-\tau)]^{\frac{1}{2}-\epsilon}}|\square(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle\lesssim\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}{(x_{2}-x_{1})^{1-2\epsilon-\alpha}}|\square(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle+\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|^{\frac{3}{2}-\epsilon}}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}{(x_{2}-x_{1})^{\frac{1}{2}-\epsilon-\alpha}}|\square(\xi,\tau)|\,d\xi\,d\tau\lesssim
II_{\square}+III_{\square}.$ (2.9)

where $II_{\square}$ and $III_{\square}$ are given by

$\displaystyle
II_{\square}:=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}$
$\displaystyle|\square(\xi,\tau)|\,d\xi\,d\tau,$ $\displaystyle
III_{\square}:=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|^{\frac{3}{2}-\epsilon}}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}|\square(\xi,\tau)|\,d\xi\,d\tau.$

The last inequality in (2.2) follows from the requirements
$1-2\epsilon-\alpha>0$ and $\frac{1}{2}-\epsilon-\alpha>0.$ Note that the
second requirement implies the first. Now, we choose

$\displaystyle\epsilon<\frac{1}{2}-\alpha.$ (2.10)

Finally, we have the ”lower order term” (2.6).

$\displaystyle IV_{\square}:=$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\int_{\tau}^{t}\int_{B_{1}}\frac{\big{|}H_{x}(x_{2}-y,t-s)-H_{x}(x_{1}-y,t-s)\big{|}}{|x_{2}-x_{1}|^{\alpha}}|\Phi(y,s;\xi,\tau)|\,dy\,ds\,|\square(\xi,\tau)|\,d\xi\,d\tau$
(2.11)
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\int_{\tau}^{t}\int_{B_{1}}\|H_{x}(x-y,t-s)\|_{C^{\alpha}_{x}}|\Phi(y,s;\xi,\tau)|\,dy\,ds\,|\square(\xi,\tau)|\,d\xi\,d\tau,$
(2.12)

where, for short, we write

$\|H_{x}(x-y,t-s)\|_{C^{\alpha}_{x}}:=\frac{\big{|}H_{x}(x_{2}-y,t-s)-H_{x}(x_{1}-y,t-s)\big{|}}{|x_{2}-x_{1}|^{\alpha}}.$

Therefore, to estimate the $C^{\alpha}_{x}$ norm of $M_{\square}(x,t)$ it
suffices to bound the integrals

$I_{\square},II_{\square},III_{\square}\,\,\,\text{and}\,\,\,IV_{\square},$

for $\square=\\{G,x\\}$ and $\\{f,x\\}.$

######  Remark 2.

Notice that if $\alpha>0$ is the Hölder exponent of $H_{x}(x,t)$ with respect
to $x$ then $\alpha/2$ is the Hölder exponent of $H_{x}(x,t)$ with respect to
$t.$ This is due to the scaling difference of the $x$ and $t$ variables of the
heat kernel $H,$ that is

$(H_{x})_{t}=(H_{x})_{xx}.$

To this end, recall

$\displaystyle I_{\square}$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{2}-\xi|^{1-\alpha}}{|t-\tau|^{3/2}}e^{-\frac{(x_{2}-\xi)^{2}}{4k_{U}(t-\tau)}}|\square(\xi,\tau)|\,d\xi\,d\tau,$
$\displaystyle II_{\square}$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}|\square(\xi,\tau)|\,d\xi\,d\tau,$
$\displaystyle III_{\square}$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|^{\frac{3}{2}-\epsilon}}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}|\square(\xi,\tau)|\,d\xi\,d\tau,$

and

$\displaystyle
IV_{\square}:=\int_{0}^{t}\int_{\mathbb{R}}\int_{\tau}^{t}\int_{B_{1}}\|H_{x}(x-y,t-s)\|_{C^{\alpha}_{x}}|\Phi(y,s;\xi,\tau)|\,dy\,ds\,|\square(\xi,\tau)|\,d\xi\,d\tau.$

####  2.2.1. The $C^{\alpha}$ estimate for $W_{G,x}$

We consider the term

$\displaystyle
W_{G,x}:=\int_{0}^{t}\int_{B_{1}}\Gamma_{x}(x,t;\xi.\tau)G(\xi,\tau)\,d\xi\,d\tau.$
(2.13)

Using the u-substitution $u:=\frac{x_{2}-\xi}{\sqrt{k_{U}}\sqrt{t-\tau}},$ we
write

$\displaystyle I_{G,x}$
$\displaystyle\lesssim\|G\|_{L^{\infty}(\Omega_{T})}\int_{0}^{t}\int_{B_{1}}\frac{|u|^{1-\alpha}|t-\tau|^{1-\frac{\alpha}{2}}}{{|t-\tau|^{3/2}}}\,e^{-|u|^{2}}\,du\,d\tau$
$\displaystyle\leq
C\,\|G\|_{L^{\infty}(\Omega_{T})}\int_{0}^{t}\frac{1}{|t-\tau|^{1/2+\alpha/2}}\,d\tau=C\,\|G\|_{L^{\infty}(\Omega_{T})}\,t^{1/2-\alpha/2}.$
(2.14)

Using the u-substitution $u:=\frac{x_{1}-\xi}{\sqrt{k_{U}}\sqrt{t-\tau}},$ we
write

$\displaystyle II_{G,x}$
$\displaystyle\lesssim\|G\|_{L^{\infty}(\Omega_{T})}\int_{0}^{t}\int_{B_{1}}\frac{|u|}{{|t-\tau|^{1-\epsilon}}}\,e^{-|u|^{2}}\,du\,d\tau$
$\displaystyle\leq C\,\|G\|_{L^{\infty}(\Omega_{T})}t^{\epsilon}.$ (2.15)

Similarly,

$\displaystyle III_{G,x}$
$\displaystyle\lesssim\|G\|_{L^{\infty}(\Omega_{T})}\int_{0}^{t}\int_{B_{1}}\frac{|u|^{\frac{3}{2}-\epsilon}}{{|t-\tau|^{\frac{3}{4}-\frac{\epsilon}{2}}}}\,e^{-|u|^{2}}\,du\,d\tau$
$\displaystyle\leq
C\,\|G\|_{L^{\infty}(\Omega_{T})}t^{\frac{1}{4}+\frac{\epsilon}{2}},$ (2.16)

and

$\displaystyle IV_{G,x}$
$\displaystyle\lesssim\,\|G\|_{L^{\infty}(\Omega_{T})}\int_{0}^{t}\int_{B_{1}}\bigg{[}\int_{0}^{s}\frac{d\tau}{(s-\tau)^{3/4}}\bigg{]}\|H_{x}(x-y,t-s)\|_{C^{\alpha}_{x}}\,dy\,ds$
$\displaystyle\leq\|G\|_{L^{\infty}(\Omega_{T})}\,t^{1/4}\int_{0}^{t}\int_{B_{1}}\frac{\big{|}H_{x}(x_{2}-y,t-s)-H_{x}(x_{1}-y,t-s)\big{|}}{|x_{2}-x_{1}|^{\alpha}}\,dy\,ds$
$\displaystyle\leq
C\,\|G\|_{L^{\infty}(\Omega_{T})}\,t^{1/4}\,\max\\{t^{\frac{1}{2}-\alpha/2},t^{\epsilon},t^{\frac{1}{4}+\frac{\epsilon}{2}}\\}.$
(2.17)

####  2.2.2. The $C^{\alpha}$ estimate for $W_{f,x}$

We consider the term

$\displaystyle
W_{f,x}:=\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi.\tau)f(\xi,\tau)\,d\xi\,d\tau.$
(2.18)

Recall,

$\displaystyle I_{f,x}$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{2}-\xi|^{1-\alpha}}{|t-\tau|^{3/2}}e^{-\frac{(x_{2}-\xi)^{2}}{4k_{U}(t-\tau)}}|f(\xi,\tau)|\,d\xi\,d\tau,$
$\displaystyle II_{f,x}$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}|f(\xi,\tau)|\,d\xi\,d\tau,$
$\displaystyle III_{f,x}$
$\displaystyle=\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|^{\frac{3}{2}-\epsilon}}{|t-\tau|^{2-\epsilon}}e^{-\frac{(x_{1}-\xi)^{2}}{4k_{U}(t-\tau)}}|f(\xi,\tau)|\,d\xi\,d\tau,$

and

$\displaystyle
IV_{f,x}=\int_{0}^{t}\int_{B_{1}}\bigg{[}\int_{0}^{s}\int_{\mathbb{R}}\frac{const}{(s-\tau)^{5/4}}e^{\frac{-d(y-\xi)^{2}}{4(s-\tau)}}\,|f(\xi,\tau)|\,d\xi\,d\tau\bigg{]}\|H_{x}(x-y,t-s)\|_{C^{\alpha}_{x}}\,dy\,ds.$

We start with $I_{f,x},$

$\displaystyle I_{f,x}$
$\displaystyle\leq\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|x_{2}-\xi|^{2-2\alpha}}{|t-\tau|^{3-2r}}e^{-\frac{(x_{2}-\xi)^{2}}{2k_{U}(t-\tau)}}\,d\xi\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}|f(\xi,\tau)|^{2}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|u|^{2-2\alpha}|t-\tau|^{3/2-\alpha}}{|t-\tau|^{3-2r}}e^{-u^{2}}\,du\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}|f(\xi,\tau)|^{2}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}\bigg{(}\int_{0}^{t}\frac{1}{|t-\tau|^{\frac{3}{2}-2r+\alpha}}\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}t^{-\frac{1}{4}+r-\frac{\alpha}{2}}\,t^{\frac{1}{2}-r}=t^{\frac{1}{4}-\frac{\alpha}{2}}\,\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))},$
(2.19)

for $\alpha+\frac{1}{4}<r<\frac{1}{2}.$ Similarly,

$\displaystyle II_{f,x}$
$\displaystyle\leq\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|^{2}}{|t-\tau|^{4-2\epsilon-2r}}e^{-\frac{(x_{1}-\xi)^{2}}{2k_{U}(t-\tau)}}\,d\xi\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}|f(\xi,\tau)|^{2}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|u|^{2}}{|t-\tau|^{\frac{5}{2}-2\epsilon-2r}}e^{-u^{2}}\,du\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}|f(\xi,\tau)|^{2}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}\bigg{(}\int_{0}^{t}\frac{1}{|t-\tau|^{\frac{5}{2}-2\epsilon-2r}}\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}t^{-\frac{3}{4}+\epsilon+r}\,t^{\frac{1}{2}-r}=t^{-\frac{1}{4}+\epsilon}\,\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))},$
(2.20)

for $\frac{3}{4}-\epsilon<r<\frac{1}{2}.$ Note here we need $\epsilon>1/4.$
This with the previous condition (2.10) on $\epsilon$ we get

$\frac{1}{4}<\epsilon<\frac{1}{2}-\alpha.$

By similar estimates, we have

$\displaystyle III_{f,x}$
$\displaystyle\leq\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|x_{1}-\xi|^{3-2\epsilon}}{|t-\tau|^{4-2\epsilon-2r}}e^{-\frac{(x_{1}-\xi)^{2}}{2k_{U}(t-\tau)}}\,d\xi\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}|f(\xi,\tau)|^{2}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{|u|^{3-2\epsilon}}{|t-\tau|^{2-\epsilon-2r}}e^{-u^{2}}\,du\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}|f(\xi,\tau)|^{2}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}\bigg{(}\int_{0}^{t}\frac{1}{|t-\tau|^{2-\epsilon-2r}}\,d\tau\bigg{)}^{1/2}\bigg{(}\int_{0}^{t}\int_{B_{1}}\frac{1}{|t-\tau|^{2r}}\,d\xi\,d\tau\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}t^{-\frac{1}{2}+\frac{\epsilon}{2}+r}\,t^{\frac{1}{2}-r}=t^{\epsilon/2}\,\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))},$
(2.21)

for $\frac{1}{2}-\frac{\epsilon}{2}<r<\frac{1}{2}.$ For the lower order term

$\displaystyle IV_{f,x}=$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\int_{\tau}^{t}\int_{B_{1}}\|H_{x}(x-y,t-s)\|_{C^{\alpha}_{x}}|\Phi(y,s;\xi,\tau)|\,dy\,ds\,|f(\xi,\tau)|\,d\xi\,d\tau$
$\displaystyle\leq\int_{0}^{t}\int_{\mathbb{R}}\int_{\tau}^{t}\int_{B_{1}}\|H_{x}(x-y,t-s)\|_{C^{\alpha}_{x}}\frac{const}{(s-\tau)^{5/4}}e^{\frac{-d(y-\xi)^{2}}{4(s-\tau)}}\,dy\,ds\,|f(\xi,\tau)|\,d\xi\,d\tau.$
(2.22)

Making the change of variables: $y=u+\xi$ and $s=v+\tau$ we obtain

$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\int_{0}^{t-\tau}\int_{\tilde{B}_{1}}\|H_{x}(x-u-\xi,t-v-\tau)\|_{C^{\alpha}_{x}}\frac{const}{(v)^{5/4}}e^{\frac{-d(u)^{2}}{4(v)}}|f(\xi,\tau)|\,dV$
(2.23)

where $dV=du\,dv\,\,d\xi\,d\tau.$ Similarly, we denote

$\|H_{x}(x-u-\xi,t-s)\|_{C^{\alpha}_{x}}:=\frac{\big{|}H_{x}(x_{2}-u-\xi,t-v-\tau)-H_{x}(x_{1}-u-\xi,t-v-\tau)\big{|}}{|x_{2}-x_{1}|^{\alpha}}.$

Interchanging the integrals we get

$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\bigg{[}\int_{0}^{t-v}\int_{\mathbb{R}}\|H_{x}(x-u-\xi,t-s)\|_{C^{\alpha}_{x}}\,|f(\xi,\tau)|\,d\xi\,d\tau\bigg{]}\frac{const}{(v)^{5/4}}e^{-d\frac{u^{2}}{4v}}\,du\,dv.$
$\displaystyle\leq\max\\{t^{\frac{1}{4}-\frac{\alpha}{2}},t^{-\frac{1}{4}+\epsilon},t^{\epsilon/2}\\}\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}\int_{0}^{t}\int_{\mathbb{R}}\frac{const}{(v)^{5/4}}e^{-d\frac{u^{2}}{4v}}\,du\,dv$
$\displaystyle\lesssim
t^{3/4}\,\max\\{t^{\frac{1}{4}-\frac{\alpha}{2}},t^{-\frac{1}{4}+\epsilon},t^{\epsilon/2}\\}\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))}.$
(2.24)

Altogether, we obtain that $W_{G,x},W_{f,x}\in
C^{\alpha}_{x}(\overline{\Omega}_{T})$ for $\alpha\in(0,1/2).$ Moreover, due
to the specific scaling of space and time in the heat kernel, that is

$(H)_{t}=(H)_{xx},$

we have that

$W_{G,x},W_{f,x}\in C^{\alpha}_{t}(\overline{\Omega}_{T}),$

for $\alpha\in(0,1/4).$

###  2.3. The initial data

We will show

First, note that $W(x,t)$ satisfies, in a classical sense, the Cauchy problem

$\displaystyle\begin{split}w_{t}-k(x,t)w_{xx}+\gamma w=0,\\\
w(x,0)=w_{0}(x)\in H^{1}(\mathbb{R}).\end{split}$ (2.25)

The calculation to show $W(x,t)\to w_{0}(x)$ is similar to the calculation in
[6] (Chapter 1, Theorem 11).

To show that $W_{x}(x,t)\to w^{\prime}_{0}(x)$ we use standard second order
parabolic equations theory. Denote $w_{x}:=u,$ using (2.25) we write the
equation satisfied by $u$

$\displaystyle\begin{split}u_{t}-\big{(}k(&x,t)u_{x}\big{)}_{x}+\gamma u=0\\\
u(x,0)&=u_{0}(x)\in L^{2}(\mathbb{R}).\end{split}$ (2.26)

It is known that the solution $u$ to (2.26) belongs to
$C([0,T],L^{2}(\mathbb{R}))$ and $u(x,0)=u_{0}(x)$ [5]. Hence, by definition
of $u$ we obtain $W_{x}(x,0)=w^{\prime}_{0}(x).$

Finally, we emphasize that the regularity of the initial data $w_{0}(x)$
assumed gives that $W(x,t)$ and $W_{x}(x,t)$ are also Hölder continuous with
the same exponent $\alpha$ away from $t=0.$ The calculations are quite
standard and similar to the proof above so we omit this part.

In the next section, we show the existence of weak solution as a limit with
respect to a mollifier’s parameter. We will make use of the $L^{\infty}$
estimates of $w$ and $w_{x}.$

##  3\. Global existence of weak solution

In this section we use standard results regarding mollifying functions with
specific properties. Detailed proofs can be found in the appendix of [5].
Recall (1.4) and Remark 1,

$f(x,t)\in L^{2}(\Omega_{T})\quad\text{and}\quad
G(x,t)=\int_{-\infty}^{x}f^{2}(z,t)\,dz\in L^{\infty}(\Omega_{T}).$

Denote the mollification of $f$ by $f^{\varepsilon}.$ Therefore, the
mollification of $G$ is

$G^{\varepsilon}=\int_{-\infty}^{x}{f^{\varepsilon}}^{2}\,dz.$

Since both $f$ and $G$ are $L^{2}_{loc},$ We know that

$f^{\varepsilon}\to f\quad\text{and}\quad G^{\varepsilon}\to G\mbox{ in
}L^{2}_{loc}\mbox{ as }\varepsilon\to 0.$

Following the discussion in §1.2, the Cauchy problem

$\displaystyle w^{\varepsilon}_{t}-k(x,t)w^{\varepsilon}_{xx}+\gamma
w^{\varepsilon}$ $\displaystyle=G^{\varepsilon}(x,t)+f^{\varepsilon}(x,t),$
(3.1) $\displaystyle w(x,0)$ $\displaystyle=w_{0},$ (3.2)

has a classical solution given by

$\displaystyle
w^{\varepsilon}(x,t)=\int_{\mathbb{R}}\Gamma(x,t,\xi,0)w_{0}(\xi)\,d\xi+\int_{0}^{t}\int_{\mathbb{R}}\Gamma(x,t;\xi.\tau)[G^{\varepsilon}(\xi,\tau)+f^{\varepsilon}(\xi,\tau)]\,d\xi\,d\tau.$
(3.3)

Therefore, the following identity holds

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}-\phi_{t}w^{\varepsilon}+(k\phi)_{x}w^{\varepsilon}_{x}+\gamma\phi
w^{\varepsilon}\,dx\,dt=\int_{0}^{T}\int_{\mathbb{R}}(f^{\varepsilon}+G^{\varepsilon})\phi\,dx\,dt,$
(3.4)

for any $\phi\in C_{c}^{1}(\Omega_{T}).$ Using the $L^{\infty}$ estimates of
$w,w_{x}$ in Section 2, we take $\varepsilon\to 0$ to obtain

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}-\phi_{t}w+(k\phi)_{x}w_{x}+\gamma\phi
w\,dx\,dt=\int_{0}^{T}\int_{\mathbb{R}}(f+G)\phi\,dx\,dt,$ (3.5)

for any $\phi\in C_{c}^{1}(\Omega_{T}).$

##  4\. Proof of Proposition 1.2

In this section, we show the $L^{2}$ estimates for $W_{G,x},W_{f,x}$ and
$W_{f}.$

###  4.1. The $L^{2}$ estimate for $W_{G,x}$

We treat the term

$\displaystyle
W_{G,x}:=\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi.\tau)G(\xi,\tau)\,d\xi\,d\tau.$

Recall,

$\displaystyle
W_{G}:=\int_{0}^{t}\int_{\mathbb{R}}\Gamma(x,t,\xi,\tau)G(\xi,\tau)\,d\xi\,d\tau$

Moreover, by the discussion in Section 1 we define

$W^{\varepsilon}_{G}:=\int_{0}^{t}\int_{\mathbb{R}}\Gamma(x,t,\xi,\tau)G^{\varepsilon}(\xi,\tau)\,d\xi\,d\tau$

where $G^{\varepsilon}$ is the mollification of $G.$ We have the following
weak formulation satisfied

$\displaystyle\int\int
W^{\varepsilon}_{G}\phi_{t}-(\phi\,k)_{x}\,W^{\varepsilon}_{G,x}-\gamma_{1}W^{\varepsilon}_{G}\,\phi\,dx\,dt=-\int\int
G^{\varepsilon}\phi\,dx\,dt.$ (4.1)

for any $\phi\in C^{1}_{0}(\Omega_{T}).$ In fact, $W^{\varepsilon}_{G}$ is a
classical solution to the equation corresponding to the weak formulation.
Choose $\phi=\eta_{x},$

$\displaystyle\int\int-W^{\varepsilon}_{G,x}\eta_{t}+\eta_{x}\,k\,W^{\varepsilon}_{G,xx}+\gamma_{1}W^{\varepsilon}_{G,x}\,\eta\,dx\,dt=\int\int
G^{\varepsilon}_{x}\eta\,dx\,dt.$ (4.2)

It is easy to see that $W^{\varepsilon}_{G,x}$ satisfies

$\displaystyle(W^{\varepsilon}_{G,x})_{t}-\big{(}k(x,t)(W^{\varepsilon}_{G,x})_{x}\big{)}_{x}+\gamma_{1}W^{\varepsilon}_{G,x}=G^{\varepsilon}_{x}$
(4.3)

in a classical sense. Now, for short we denote $W^{\varepsilon}_{G,x}$ by
$B^{\varepsilon}_{x}.$ Multiplying (4.3) by $B^{\varepsilon}_{x}$ and
integrating,

$\displaystyle\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}}(B^{\varepsilon}_{x})^{2}\,dx+\int_{\mathbb{R}}k(x,t)(B^{\varepsilon}_{xx})^{2}+\gamma_{1}(B^{\varepsilon}_{x})^{2}\,dx=\int_{\mathbb{R}}G^{\varepsilon}_{x}B^{\varepsilon}_{x}\,dx.$
(4.4)

Integrating in $t,$

$\displaystyle\frac{1}{2}\int_{\mathbb{R}}(B^{\varepsilon}_{x})^{2}\,dx+\int_{0}^{t}\int_{\mathbb{R}}k(x,s)(B^{\varepsilon}_{xx})^{2}+\gamma_{1}(B^{\varepsilon}_{x})^{2}\,dx\,ds=\int_{0}^{t}\int_{\mathbb{R}}G^{\varepsilon}_{x}B^{\varepsilon}_{x}\,dx\,ds.$
(4.5)

Now the given form of $G,$ that is $G_{x}=f^{2},$ implies that
$G^{\varepsilon}_{x}\to G_{x}$ in $L^{1}.$ and we also have, from the previous
sections, $B^{\varepsilon}_{x}\to B_{x}$ in $L^{\infty}.$ Together, we obtain
for small $\varepsilon,$

$\int_{0}^{t}\int_{\mathbb{R}}G^{\varepsilon}_{x}B^{\varepsilon}_{x}\,dx\leq\frac{1}{2}\int_{0}^{t}\int_{\mathbb{R}}G_{x}B_{x}\,dx\leq\frac{1}{2}\,T\,\|B_{x}\|_{L^{\infty}([0,T],\mathbb{R})}\|G_{x}\|_{L^{\infty}([0,T],L^{1}(\mathbb{R}))}.$

Hence, by (4.5) and (2.2) we have

$\displaystyle\|B^{\varepsilon}_{x}\|_{L^{2}([0,T],\mathbb{R})}\leq\frac{1}{2}T^{3/2}\|G\|_{L^{\infty}([0,T],\mathbb{R})}\|G_{x}\|_{L^{\infty}([0,T],L^{1}(\mathbb{R}))}.$
(4.6)

Since $B^{\varepsilon}_{x}\to B_{x}$ in $L^{\infty},$ there exists a
subsequence, still denoted by $B^{\varepsilon}_{x},$ converges weakly to
$B_{x}$ in $L^{2}.$ Hence, $W_{G,x}\in L^{2}$ and

$\|W_{G,x}\|_{L^{2}([0,t],\mathbb{R})}\leq\frac{1}{2}T^{3/2}\|G\|_{L^{\infty}([0,T],\mathbb{R})}\|G_{x}\|_{L^{\infty}([0,T],L^{1}(\mathbb{R}))}.$

###  4.2. The $L^{2}$ estimate for $W_{f,x}$ and $W_{f}$

We consider the term

$\displaystyle
W_{f,x}:=\int_{0}^{t}\int_{\mathbb{R}}\Gamma_{x}(x,t;\xi.\tau)f(\xi,\tau)\,d\xi\,d\tau,$
(4.7)

and use Young’s inequality with the help of (1.20) we get,

$\displaystyle\|W_{f,x}\|_{L^{2}}$
$\displaystyle\leq\bigg{\|}\int_{0}^{t}\int_{\mathbb{R}}\frac{cont}{{t-\tau}}e^{-\frac{d(x-\xi)^{2}}{4(t-\tau)}}|f(\xi,\tau)|\,d\xi\,d\tau\bigg{\|}_{L^{2}}$
(4.8) $\displaystyle\approx\|\tilde{H}\ast
f\|_{L^{2}}\leq\|\tilde{H}\|_{L^{1}}\|f\|_{L^{2}}\leq
T^{1/2}\|f\|_{L^{\infty}((0,T),L^{2}(\mathbb{R}))},$ (4.9)

where $\tilde{H}=\frac{1}{\sqrt{t}}H.$ Similarly, one can show the $L^{2}$ of
$W_{f}.$

Acknowledgement: I would like to thank my advisors Prof. Geng Chen and Prof.
Weishi Liu for their valuable comments and guidance. This paper is motivated
by a project that we have conducted recently. I would also like to thank Prof.
Lihe Wang for referring me to some of the relevant previous results in the
literature.

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