\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{cite}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 96, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/96\hfil H\"older continuity]
{H\"older continuity with exponent $(1+\alpha )/2$ in the time variable for
 solutions of parabolic equations}

\author[J. Aramaki \hfil EJDE-2015/96\hfilneg]
{Junichi Aramaki}

\address{Junichi Aramaki \newline
Division of  Science, Faculty of Science and Engineering,
Tokyo Denki University, \newline
Hatoyama-machi, Saitama 350-0394, Japan}
\email{aramaki@mail.dendai.ac.jp}

\thanks{Submitted February 16, 2015. Published April 13, 2015.}
\makeatletter
\@namedef{subjclassname@2010}{\textup{2010} Mathematics Subject Classification}
\makeatother
\subjclass[2010]{35A09, 35K10, 35D35}
\keywords{H\"older continuity;  parabolic equation}

\begin{abstract}
 We consider the regularity of solutions for some parabolic equations.
 We show  H\"older continuity with exponent $(1+\alpha )/2$, with respect
 to the time variable, when the gradient in the space variable of the
 solution has the H\"older continuity with exponent $\alpha $.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article we consider the  H\"older continuity of solutions for the 
equation.
\begin{equation}
Lu:= \sum _{i,j=1}^n a_{ij}(x,t) \frac{\partial^2u}{\partial x_i \partial x_j}
+ \sum _{i=1}^n b_i(x,t) \frac{\partial u}{\partial x_i}
-\frac{\partial u}{\partial t}=f \quad\text{in } Q \label{e1.1}
\end{equation}
where  $Q= \Omega \times (0,T]$, $\Omega \subset \mathbb{R}^n $ is a domain and 
$T>0$.
For the classical solution $u(x,t)$ of \eqref{e1.1}, we shall show 
the H\"older continuity with exponent $(1+\alpha )/2$ in the time variable $t$, 
when the gradient of $u$ with respect to  the space variable $x$ has 
H\"older continuity with exponent $\alpha $.

We assume that:
\begin{itemize}
\item[(H1)] $L$ is parabolic, i.e., for any $(x,t) \in Q$,
\[
\sum _{i,j~1}^n a_{ij}(x,t) \xi _i \xi _j >0 \quad 
\text{for all } 0\neq \xi= (\xi _1,\ldots ,\xi _n) \in \mathbb{R}^n.
\]
Note that $L$ is not necessary uniformly parabolic.

\item[(H2)] $a_{ij} ,b _i \in C(Q)$ for $i,j=1,\dots ,n$ where 
$C(Q)$ denotes the space of continuous functions in $Q$.

\item[(H3)] There exist constants $\mu _1,\mu _2>0$ such that
$$
\sum _{i=1}^n a_{ii}(x,t) \le \mu _1, \quad
 \sum _{i=1}^n |b_i(x,t) |\le \mu _2\quad \text{for all } (x,t) \in Q.
$$

\item[(H4)] $f=f(x,t)$ is a bounded continuous function in $Q$ satisfying
$$
|f(x,t)|\le \mu _3 \quad\text{for all } (x,t) \in Q.
$$
\end{itemize}
In the following, for non-negative integers $k,l$ and any set 
$A\subset \mathbb{R}^n$, we denote  the space of functions 
$u  \in C(A \times (0,T])$   such that $ u$ has continuous partial derivatives 
$ \partial_x^{\alpha }u$ for $|\alpha |\le k$ and 
$\partial_t^j  u$ for $j\le l $  in $A\times (0,T]$  by  
$C^{k,l}(A\times (0,T])$. 
Here 
\[
\partial^{\alpha }_xu= \frac{\partial^{|\alpha |}u}{\partial x_1^{\alpha _1}
 \cdots \partial x_n^{\alpha _n}}
\]
for any multi-index $\alpha = (\alpha _1, \ldots ,\alpha _n)$ 
and $ |\alpha |= \sum _{i=1}^n \alpha _i$. We also use the notation
 $u_t= \partial_t u$, $u_{x_i}= \partial_{x_i} u$, 
$u_{x_ix_j}= \partial_{x_i}\partial_{x_j}u$ etc.
Now we are in a position to state our main result.

\begin{theorem} \label{thm1}
Under the hypotheses {\rm (H1)--(H4)}, let $u \in C^{2,1}(Q)$ be a solution 
of \eqref{e1.1} in $Q$. Assume that there exist $\alpha \in (0,1]$ and  
constants $C_1 ,C_2\ge 0$ such that
\begin{equation}
|\nabla u(x,t) - \nabla u (y,t)|\le C_1|x-y |^{\alpha } \label{e1.2}
\end{equation}
for all $(x,t), (y,t) \in Q$, and
\begin{equation}
|\nabla u(x,t)|\le C_2 \label{e1.3}
\end{equation}
for all $(x,t) \in Q$. Here and hereafter $\nabla $ denotes the gradient
operator with respect to the space variable $x$.

{\rm (i)} Let $\Omega ' \subset \Omega $ be a subdomain such that  
$\operatorname{dist} (\Omega ',\partial\Omega )\ge d>0$, 
and define $Q'=\Omega ' \times (0,T]$. Then there exist $\delta >0$ 
depending only on $\mu _1,\mu _2, \mu _3$ and   $\alpha $, $K>0$ depending 
only on $\mu _1,\mu _2,\mu _3,d, \alpha ,C_1$ and $C_2$ such that
\begin{equation}
|u(x,t) - u(x,t_0)|\le K|t-t_0 |^{(1+\alpha )/2} \label{e1.4}
\end{equation}
for all $(x,t), (x,t_0) \in Q'$ with $|t-t_0 |<\delta $.

{\rm (ii)} Furthermore, if we assume that $\partial\Omega \neq \emptyset $ 
and $u \in C^{1,0}(\overline{\Omega }\times (0,T])$ satisfies that there 
exist $\beta  \in (0,1]$ and a constant $D\ge 0$ such that
$$
|\nabla u (x,t) -\nabla u(x,t_0)|\le D|t-t_0|^{(1+\beta )/2}
$$
for all $x \in \partial\Omega $ and $t,t_0 \in (0,T]$, then for any 
$\sigma >0$ there exists $K>0$ depending only on $\mu _1,\mu _2, \mu _3, C_1,C_2,D$ 
and $\sigma $ such that
$$
|u(x,t) -u(x,t_0)|\le K |t-t_0|^{(1+\gamma )/2}, \quad 
\gamma = \min \{ \alpha ,\beta \}
$$
for any $(x,t),(x,t_0) \in Q$ with $|t -t_0 |<\sigma $.
\end{theorem}

\begin{remark} \label{rmk2} \rm
Gilding \cite{Gil} assumed that $|u(x,t) -u(y,t) |\le C_1|x-y|^{\alpha }$ 
instead of \eqref{e1.2} and \eqref{e1.3}, and obtained 
\[
|u(x,t) -u(x,t_0)|\le K|t-t_0|^{\alpha }
\]
instead of \eqref{e1.4}. Note that the papers of Brandt \cite{Br} 
and Knerr \cite{Kne} can be viewed as precursors to the present study.
 See also the discussion of Ladyzhenskaja et al \cite{LSU} in \cite{Kne}. 
Then the author of \cite{Gil} applied the result to the Cauchy problem 
for the porous media equation in one dimension. See also Aronson \cite{Aron} 
and B\'enilan \cite{Ben}. On the other hand,  our result can be  
applied to the regularity for a quasilinear parabolic type system 
associated with the Maxwell equation. For such application, see Aramaki \cite{Ar}.
\end{remark}

\section{Proof of Theorem \ref{thm1}}
 We shall use a modification of the arguments in  \cite{Gil}.

(i) Let $\Omega '\subset \Omega $ be a subdomain with 
$\operatorname{dist} (\Omega ',\partial\Omega )\ge d>0$ and define 
$Q'=\Omega ' \times (0,T]$.  Fix arbitrary points $(x_0,t_0), (x_0,t_1)\in Q'$ 
with $0<t_0<t_1\le T$ and choose $0< \rho <d$, and define $\mu $ and $C$ so that
$$
\mu = \max \{ \mu _1,\mu _2, \mu _2 C_2+  \mu _3 \}\quad\text{and}\quad
 C=\frac{C_1}{1+\alpha }.
$$
Moreover, we define a set and  functions
\begin{gather*}
N=\{x\in \mathbb{R}^n ; |x-x_0|< \rho \} \times (t_0,t_1]\subset Q, \\
\begin{aligned}
v^{\pm } (x,t)
&= \mu \{ 1+ 2s \rho ^{-2}(1+\rho )\}(t-t_0) + s\rho ^{-2}|x-x_0|^2 
 + C\rho ^{1+\alpha }\\
&\quad \pm \{ u(x,t)-u(x_0,t_0)-\nabla u (x_0,t_0)\cdot (x-x_0)\}
\end{aligned}
\end{gather*}
where ``$\cdot $'' denotes the inner product in $\mathbb{R}^n$. Let
\[
s = \sup _{t_0\le t\le t_1, x \in \Omega '}|u(x,t)-u(x,t_0)|.
\]
Since
\begin{gather*}
v_t^{\pm } = \mu \{ 1+2s \rho ^{-2} (1+\rho )\}\pm u_t (x,t),\\
v_{x_i}^{\pm } = 2s \rho ^{-2} (x_i-x_{0,i})\pm \{ u_{x_i}(x,t)-u_{x_i}(x_0,t_0)\},\\
v_{x_ix_j}^{\pm } = 2s \rho ^{-2} \delta _{ij}\pm u_{x_ix_j} (x,t)
\end{gather*}
where $\delta _{ij}$ denotes the Kronecker delta, we have
\begin{equation}
\begin{aligned}
Lv^{\pm }
&= -\mu -2s \rho ^{-2} \mu (1+\rho )+ 2s \rho ^{-2}
\bigl\{ \sum _{i=1}^n a_{ii}(x,t) + \sum _{i=1}^n b_i(x,t) (x_i-x_{0,i})\bigr\} \\
& \quad \pm Lu(x,t) \mp \sum _{i=1}^n b_i(x,t) u_{x_i}(x_0,t_0) \\
& \le  -\mu -2s \rho ^{-2} ( \mu + \mu \rho )+2s \rho ^{-2} (\mu _1+ \mu _2 \rho )
 + |f(x,t)|\\
& \quad  + \sum _{i=1}^n |b_i(x,t)||u_{x_i}(x_0,t_0)|\\
&\le -\mu -2s\rho ^{-2} (\mu + \mu \rho )+ 2s \rho ^{-2} (\mu _1+ \mu _2 \rho )
 + \mu _3+ C_2 \mu _2
\le  0. \label{e2.1}
\end{aligned}
\end{equation}
Here we used the definition of $\mu $.

When $t=t_0$ and $|x -x_0|\le \rho $, from the definition of $C$, we  see that
\begin{equation}
\begin{aligned}
v ^{\pm }(x,t_0)
&= s\rho ^{-2} |x-x_0|^2 + C\rho ^{1+\alpha }\\
& \quad \pm \{ u(x,t_0)-u(x_0,t_0)- \nabla u(x_0,t_0)\cdot (x-x_0)\} \\
&= s\rho ^{-2} |x-x_0|^2 + C\rho ^{1+\alpha } \\
& \quad \pm \int _0^1 (\nabla u(\theta x_0 + (1-\theta )x)
- \nabla u(x_0,t_0)) \cdot (x-x_0) d\theta \\
& \ge  s \rho ^{-2} |x-x_0|^2 +C\rho ^{1+\alpha }\\
& \quad  -C_1 \int _0^1 |\theta x_0+(1-\theta )x-x_0|^{\alpha } d\theta |x-x_0|\\
&\ge  s\rho ^{-2} |x-x_0|^2 + C\rho ^{1+\alpha } -\frac{C_1}{1+\alpha }
\rho ^{1+\alpha }
 \ge  0. \label{e2.2}
\end{aligned}
\end{equation}
When $|x-x_0|=\rho $ and $t_0<t\le t_1$, using the definition of $s$,
we can see that
\begin{equation}
\begin{aligned}
v^{\pm }(x,t)
&= \mu \{1+ 2s \rho ^{-2} (1+\rho )\}(t-t_0) + s+ C\rho ^{1+\alpha } \\
& \quad \pm \{ u(x,t) -u(x_0,t_0) -\nabla u(x_0,t_0)\cdot (x-x_0)\} \\
&= \mu \{1+ 2s \rho ^{-2} (1+\rho )\}(t-t_0) + s+ C\rho ^{1+\alpha } \\
& \quad \pm \{ u(x,t_0) -u(x_0,t_0) -\nabla u(x_0,t_0)\cdot (x-x_0)\} \\
& \quad \pm \{ u(x,t)-u(x,t_0)\}  \\
&\ge   \mu \{1+ 2s \rho ^{-2} (1+\rho )\}(t-t_0) + s+ C\rho ^{1+\alpha }
 -\frac{C_1}{1+\alpha } \rho ^{1+\alpha } -s \\
&\ge  0. \label{e2.3}
\end{aligned}
\end{equation}
Thus from \eqref{e2.1}, \eqref{e2.2} and \eqref{e2.3}, we see that
\begin{equation}
\begin{gathered}
Lv^{\pm }\le 0 \quad \text{in } N,\\
v^{\pm }\ge 0 \quad \text{on the parabolic boundary of }N.
\end{gathered}\label{e2.4}
\end{equation}
By the maximum principle (cf. Friedman \cite[p. 34]{Fr}
or Lieberman \cite[Chapter 2, Lemma 2.3]{Li}), it follows
that $v ^{\pm } \ge 0$ in $N$.
Hence we have
\begin{align*}
&\mp \{ u(x,t)- u(x_0,t_0)-\nabla u(x_0,t_0)\cdot (x-x_0)\} \\
&\le C\rho ^{1+\alpha } + \mu \{ 1+2s\rho ^{-2} (1+\rho )\} (t-t_0)
 + s\rho ^{-2} |x -x_0|^2.
\end{align*}
If we put $x=x_0$, then we see that
\[
|u(x_0,t)- u(x_0,t_0)|\le C\rho ^{1+\alpha }
+ \mu \{ 1+2s\rho ^{-2} (1+\rho )\} (t-t_0) .
\]
Since $x_0\in \Omega '$ and $t\in (t_0,t_1]$ are arbitrary, it follows that
\begin{equation}
\begin{aligned}
s&\le  C\rho ^{1+\alpha } + \mu \{ 1+2s \rho ^{-2} (1+\rho )\} (t_1-t_0) \\
&= C\rho ^{1+\alpha } + \mu (t-t_0) + \frac12 s  \{ 4\mu \rho ^{-2}
(1+\rho ) (t_1-t_0)\}.
\end{aligned} \label{e2.5}
\end{equation}
Let $\rho ^* $ be the positive root of the quadratic equation
$y^2= 4\mu (1+y) (t_1-t_0)$, i.e.,
\begin{equation}
\rho ^* = 2\mu (t_1-t_0) + 2\{ \mu (t_1-t_0)
+ \mu ^2 (t_1-t_0)^2\}^{1/2}.\label{e2.6}
\end{equation}
If we define $\delta = d^2/(4\mu (1+d))$, for  $t_1<t_0+ \delta $,
it is easily seen that $\rho ^*<d$. Thus we can replace $\rho $
in \eqref{e2.5} with $\rho ^*$. Therefore when $t_0<t_1<t_0+\delta $,
 we  see that
\begin{align*}
s& \le  C\bigl(2\mu (t_1-t_0)+ 2\{ \mu (t_1-t_0)
 + \mu ^2 (t_1-t_0)^2 \}^{1/2} \bigr)^{1+ \alpha }\\
&\quad  + \mu (t_1-t_0)+ \frac12 s \\
& = C\bigl(2\mu (t_1-t_0)^{1/2} + 2\{ \mu + \mu ^2 (t_1-t_0) \}^{1/2}
 \bigr)^{1+ \alpha }(t_1-t_0) ^{(1+\alpha )/2} \\
&\quad  + \mu (t_1-t_0)^{(1-\alpha )/2} (t_1-t_0)^{(1+\alpha )/2} + \frac12 s .
\end{align*}
Since $t_1-t_0<\delta $, we have
\[
s\le 2\bigl[C \bigl( 2\mu \delta ^{1/2} + 2\{ \mu
+ \mu ^2 \delta \}^{1/2} \bigr)^{1+ \alpha }+ \mu \delta ^{(1-\alpha )/2} \bigr]
(t_1-t_0) ^{(1+\alpha )/2}.
\]
Thus we have
\[
|u(x_0,t_1)-u(x_0,t_0)|\le K (t_1-t_0)^{(1+\alpha )/2}
\]
where
$$
K= 2\bigl[C \bigl( 2\mu \delta ^{1/2} + 2\{ \mu
+ \mu ^2 \delta \}^{1/2} \bigr)^{1+ \alpha }+ \mu \delta ^{(1-\alpha )/2} \bigr]
$$
for any $t_1<t_0+\delta $. Since $(x_0,t_0)$ and $ (x_0,t_1)$ with
$t_0< t_1\le T$ are arbitrary points in $Q'$, we get the conclusion of (i).

(ii) When $(x_0,t_0), (x_0,t_1)\in Q$ with $0<t_0<t_1< t_0+ \sigma $, 
we choose $\rho ^*$ as in \eqref{e2.6}. We  define
\begin{gather*}
N^*=\{ x\in \mathbb{R}^n :|x-x_0|< \rho ^*\} \times (t_0,t_1]\subset 
\mathbb{R}^n \times (0,T], \\
w^{\pm } (x,t) = v^{\pm }(x,t)+ D(t_1-t_0)^{(1+\beta )/2} \text{ in } N^*\cap Q, \\
s= \sup _{t_0\le t\le t_1, x \in \overline{\Omega }}|u(x,t)-u(x,t_0)|.
\end{gather*}
By a similar argument as in the proof of (i), we have
\begin{gather*} 
L w^{\pm } \le 0 \quad \text{in } N^*\cap Q,\\
w^{\pm } \ge 0 \quad \text{on the parabolic boundary of }N^* \cap Q.
 \end{gather*}
If we choose $\mu = \max \{ \mu _1, \mu _2, \mu _2C_ 2+ \mu _3, 
D \sigma ^{(1+\beta )/2}\}$, from a similar argument as in (i) we can 
get the conclusion of (ii).

\subsection*{Acknowledgments} 
We would like to thank the anonymous referee for his or her very kind  
advice about a previous version of this article.

\begin{thebibliography}{0}

\bibitem{Ar} J.~Aramaki; 
\emph{Existence and uniqueness of a classical solution for a quasilinear
parabolic type  equation associated with the Maxwell equation}, to appear.

\bibitem{Aron} D.~G.~Aronson, L.~A.~Caffarelli; 
\emph{Optimal regularity for one-dimensional porous medium flow},
Rev. Mat. Iberoamericana 2, (1986), 357-366.

\bibitem{Ben} P.~B\'enilan; 
\emph{A strong $L^p$ regularity for solutions of the porous media equation}, 
in: Contributions to Nonlinear Partial Differential Equations 
(Eds. C.~Bardos et al.), Reserch Notes in Mathematics 89, 
Pitman, London, (1983), 39-58.

\bibitem{Br} A.~Brandt;
 \emph{Interior Schauder estimates for parabolic differential- (or difference-) 
equations via the maximum principle}, Israel J. Math. 7, (1969), 254-262.

\bibitem{Fr} A.~Friedman; 
\emph{Partial Differential Equations of Parabolic Type}, Princeton Hall Inc.,
 Englewood Cliffs, NJ, (1964).

\bibitem{Gil} B.~H.~Gilding; 
\emph{H\"older continuity of solutions of parabolic equations},
 J. London Math. Soc., Vol. 13(2), (1976), 103-106.

\bibitem{Kne} B.~F.~Knerr; 
\emph{Parabolic interior Schauder estimates by the maximum pronciple}, 
Arch. Rational Mech. Anal. 75, (1980), 51-58.

\bibitem{LSU} O.~A.~Ladyzhenskaja, V.~A.~Solonnikov, N.~N.~Ural'ceva; 
\emph{Linear and Quasi-linear Equations of Parabolic Type}, Translations 
of Mathematical Monographs 23 AMS, Providence, RI, (1968).

\bibitem{Li} G.~M.~Lieberman; 
\emph{Second Order Parabolic Differential Equations}, World Scientific, (2005).

\end{thebibliography}

\end{document}