\documentclass[twoside]{article} \usepackage{amssymb, graphicx} \pagestyle{myheadings} \markboth{\hfil Solutions of nonlinear parabolic equations \hfil EJDE--2001/60} {EJDE--2001/60\hfil L. Boccardo, T. Gallou\"et, \& J. L. Vazquez \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 60, pp. 1--20. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Solutions of nonlinear parabolic equations without growth restrictions on the data % \thanks{ {\em Mathematics Subject Classifications:} 35K55, 35K65. \hfil\break\indent {\em Key words:} Nonlinear parabolic equations, global existence, growth conditions. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted July 17, 2001. Published September 12, 2001.} } \date{} % \author{Lucio Boccardo, Thierry Gallou\"et, \& Juan Luis Vazquez \\ \quad\\ {\em \small Dedicated to the memory of our friend Philippe B\'{e}nilan} } \maketitle \begin{abstract} The purpose of this paper is to prove the existence of solutions for certain types of nonlinear parabolic partial differential equations posed in the whole space, when the data are assumed to be merely locally integrable functions, without any control of their behaviour at infinity. A simple representative example of such an equation is $$u_t-\Delta u + |u|^{s-1}u=f,$$ which admits a unique globally defined weak solution $u(x,t)$ if the initial function $u(x,0)$ is a locally integrable function in $\mathbb{R}^N$, $N\geq 1$, and the second member $f$ is a locally integrable function of $x\in\mathbb{R}^N$ and $t\in [0,T]$ whenever the exponent $s$ is larger than 1. The results extend to parabolic equations results obtained by Brezis and by the authors for elliptic equations. They have no equivalent for linear or sub-linear zero-order nonlinearities. \end{abstract} \newtheorem{theo}{Theorem}[section] \newtheorem{defin}[theo]{Definition} \newtheorem{coro}[theo]{Corollary} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction} In this paper we investigate the existence of solutions for a class of equations of the form $$u_t+L(u)+ h(x,t,u) =0, \label{eq1}$$ posed on the whole space $\mathbb{R}^N$, $N \geq 1$, with initial condition $u(x,0)= u_0(x)$. In the equation, $L$ is an elliptic differential operator in divergence form with some structure conditions, which include the Laplacian operator, $L(u)=-\Delta u$, or the $p$-Laplacian operator, $$L(u)= -\mathop{\rm div}(| Du|^{p-2} Du),$$ and $h$ is a function of the variables $x,t,u,$ which grows uniformly with $u$ at a sufficient rate as $| x|\to \infty$. The main novelty of the problem posed here is that the initial condition $u_0$ is a locally integrable function defined for $x\in \mathbb{R}^N$ without any control of its growth as $| x|\to \infty$, and so is the dependence of $h$ on $x$. We will prove existence of a solution of this problem by imposing the condition that the growth of $h$ with respect to $u$ is larger than the structural growth of the elliptic operator. In the case where $L$ is the Laplace operator, this basic growth assumption says that the initial-value problem for equation (\ref{eq1}) has a solution for every $u_0\in L^1_{{\rm loc}}(\mathbb{R}^N)$ if $$\label{form} h(x,t,u)u\ge c\,| u|^{s+1} -f(x,t)\, u,$$ where $s>1$ and $f\in L^1(0,T; L^1_{{\rm loc}}(\mathbb{R}^N))$. In the case of the $p$-Laplacian operator the above conditions take the form \ $s>p-1$. Such a condition has been shown to be necessary under certain assumptions. The main idea behind the result is the existence of a priori estimates of local type under the stated conditions, which are first established for a sequence of approximate problems and then preserved in the limit. There is a similar phenomenon for elliptic equations, which has been studied by Brezis \cite{Br} {\sl in the semilinear case } (i.e., when $L$ is the Laplacian), and by the present authors, \cite{BGV} for general operators (see also \cite{Le}). The main steps of the proof consist of obtaining local estimates for suitable approximate problems and then passing to the limit. There are essentially two difficulties introduced in treating nonlinear elliptic operators $L$ instead of the Laplacian. The first one is to obtain local estimates on the solutions $u$ and the gradients $Du$, since {\sl we can only integrate once by parts} in $x$. It is at this stage that the growth condition on $h$ is needed, even when the operator $L$ is the Laplacian. The second difficulty is to pass to the limit when the nonlinearity of $L$ depends also on $Du$. Though the general principle is similar to the elliptic case as is usual in many parabolic problems, the time variable makes for a shift in the results and critical exponents that we describe in some detail. We also show a second phenomenon: for large growth rates of $h$ with respect to $u$ (i.e., large values of $s$ in (\ref{form})) the solutions have {\sl better regularity} both for $u$ and $Du$ than the one expected from the standard theory. The paper is organized as follows. We present the problem in Section \ref{sect-struct} and state the basic existence and regularity result in Section \ref{bexreg}, while the improved regularity obtainable for strongly superlinear lower-order terms is stated in Section \ref{imprreg}. The proof of these results occupies the following three sections: Section \ref{sect-local} contains the basic local estimates, Section \ref{sect-local2} completes the derivation of a priori estimates, and Section \ref{const} contains the construction of the solutions. The case $p>N$ and the variational approach are reviewed in the next section. Finally, Section \ref{uniq} deals with the question of uniqueness of the local solutions, which remains an open problem in general. However, the constructed solutions have the standard good properties of parabolic problems, and we show at the end of the paper how to construct classes of unique solutions which enjoy the Maximum Principle. \section{Statement of the problem. Conditions} \label{sect-struct} In subsequent sections we will study the class of equations that we write for convenience in the form $$u_t-\mathop{\rm div} A(x,t,Du) +g(x,t,u)= f(x,t)\,. \label{eq2}$$ They are posed in $Q=\{(x,t): x\in \mathbb{R}^N, \ 01$ and $c>0$ such that for all $\xi$ and a.e. $(x,t)$ $$A(x,t,\xi)\cdot\xi \ge c|\xi|^p\,,$$ where the dot is used to denote scalar product of vectors in $\mathbb{R}^N$. \item [(A3)] There exist functions $b(x,t)\in L^{p'}_{{\rm loc}}(\mathbb{R}^N\times[0,T))$ ($p'=p/(p-1)$), and $d(x,t)$, locally bounded in $\mathbb{R}^N\times[0,T)$, such that for all $\xi$ and a.e. $(x,t)$ $$|A(x,t,\xi)| \le b(x,t)+d(x,t) |\xi|^{p-1}\,.$$ \item [(A4)] For a.e. $(x,t)\in Q$ and all $(\xi,\eta)\in \mathbb{R}^N\times\mathbb{R}^N$, $\xi\ne \eta,$ we have $$(A(x,t,\xi)-A(x,t,\eta))\cdot(\xi-\eta) > 0.$$ \end{itemize} Hypotheses (A1)--(A4) are classical in the study of nonlinear operators in divergence form, see \cite{L}. Moreover, unless mention to the contrary the structural exponent $p$ will be taken over a critical value, $p> p_1= (2N+1)/(N+1)$. This is done to avoid the functional difficulties related to the definition of the gradient $D u$ which may arise when dealing with $L^{1}$ data, cf. \cite{B6} for a complete discussion of the elliptic problem in the context of entropy solutions, and \cite{A4} for the parabolic case. See also \cite{BM} for the same problem in the framework of renormalized solutions. The model example of a function satisfying (A1)-(A4) is of course $A(x,t,\xi)= |\xi|^{p-2}\xi$, which for $p=2$ leads to the Laplace operator. The assumptions on $g$ are the following: \begin{itemize} \item [(G1)] $g(x,t,\sigma):\mathbb{R}^N\times \mathbb{R}\to\mathbb{R}$ is measurable in $x\in\mathbb{R}^N$, $t\in (0,T)$ for any fixed $\sigma\in\mathbb{R}$ and continuous in $\sigma$ for a.e. $(x,t)$. \item [(G2)] There exists an exponent $s>0$ and a constant $c_2>0$, such that for all $\sigma$ and almost every $(x,t)$ $$g(x,t,\sigma)\,\sigma \ge c_2|\sigma|^{s+1}\,.$$ \item [(G3)] For all $k>0$ the function $$G_k(x,t)= \sup_{|\sigma| \le k}|g(x,t,\sigma)|$$ is locally integrable over $\mathbb{R}^N\times [0,T]$. \end{itemize} Let us remark that for $s$ large enough we relax the condition $p>p_1$ and approach $p=1$ without getting out of the standard functional framework. Indeed, we may replace it by $$p>\frac {s+1}s$$ which is better than $p>p_1$ if $s>(N+1)/N$. Finally, we assume that $f\in L^1(0,T; L^1_{{\rm loc}}(\mathbb{R}^N))$ and the initial data $u_0\in L^1_{{\rm loc}}(\mathbb{R}^N)$. We look for a global weak solution to (\ref{eq2}), i.e. a function $u\in L^1(0,T; W^{1,1}_{{\rm loc}}(\mathbb{R}^N))$ such that both $A(x,t,Du)$ and $g(x,t,u)$ are well defined in $L^1_{{\rm loc}}(\mathbb{R}^N)$ and (\ref{eq2}) is satisfied in ${\cal D}'(Q)$ and the initial condition is also satisfied in the precise sense that we state next. \begin {defin} \rm A function $u\in L^1(0,T; W^{1,1}_{{\rm loc}}(\mathbb{R}^N))$ is said to be a {\sl weak solution} of problem (\ref{eq2})-(\ref{ic}) if $| Du|^{p-1}$ and $g(x,t,u(x,t)) \in L^1_{{\rm loc}}(\mathbb{R}^N\times[0,T))$, and \begin{eqnarray} \label{icecream} \lefteqn{\int_0^T\int_{\mathbb{R}^N} ( A(x,t,Du)\cdot D\phi - u\,\phi_t)\,dx\,dt + \int_0^T\int_{\mathbb{R}^N} g(x,t,u)\phi\,dx\,dt}\\ &=& \int_0^T\int_{\mathbb{R}^N} f(x,t)\phi(x,t)\,dx\,dt + \int_{\mathbb{R}^N} u_0(x)\,\phi(x,0)\,dx \hspace{2cm}\nonumber \end{eqnarray} holds for every test function $\phi\in C^1_c(\mathbb{R}^N\times [0,T))$, the $C^1$ functions with compact support. \end{defin} We will find weak solutions such that $u\in L^\infty(0,T; L^1_{{\rm loc}}(\mathbb{R}^N))$. Let us remark that under these conditions $u$ satisfies the initial condition in the following sense $$\frac 1 {\tau} \int_0^{\tau} \int_\mathbb{R}^N u(x,t)\,\phi(x)\,dx\,dt \to \int_{\mathbb{R}^N} u_0(x)\,\phi(x)\,dx \label{ice}$$ for every continuous function $\phi$ with compact support from $\mathbb{R}^N$ to $\mathbb{R}$. Indeed, it is easy to prove (\ref{ice}) for $\phi \in C^1(\mathbb{R}^N,\mathbb{R})$ with compact support, taking $\phi(x)(\tau-t)^+$ as test functions in (\ref{icecream}) (such test functions are avalaible by regularization). Then (\ref{ice}) holds for every $\phi \in C(\mathbb{R}^N,\mathbb{R})$ with compact support since the family $(\frac 1 \tau \int_0^{\tau}u(.,t)\,dt)_{0 < \tau \le T}$ is bounded in $L^1(\Omega)$ for any bounded subset $\Omega$ of $\mathbb{R}^N$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Basic existence and regularity results} \label{bexreg} The main existence and regularity result is the following \begin{theo}\label{theo1} Let $p > p_1=(2N+1)/(N+1)$ and $s>p-1$. Then for every $u_0\in L^1_{{\rm loc}}(\mathbb{R}^N)$ and every $f\in L^1(0,T; L^1_{{\rm loc}}(\mathbb{R}^N))$ there exists a weak solution of the Cauchy problem (\ref{eq2})-(\ref{ic}) with $u\in L^\infty(0,T; L^1_{{\rm loc}}(\mathbb{R}^N))\cap L^s(0,T; L^s_{{\rm loc}} (\mathbb{R}^N))$ and also $$u\in L^r(0,T; W^{1,q}_{{\rm loc}}(\mathbb{R}^N)),$$ under the restrictions $1\le r, q N+1. Furthermore, if$u_0\ge 0$and$f\ge 0$then we construct a solution such that$u\ge 0$. \end{theo} In the preceding theorem, we can be more precise about the Maximum Principe (or M.P. in short). Indeed, if$g$is nondecreasing with respect to$u$, then, if$u_0 \ge v_0$a.e. and$f \ge g$a.e., we can construct corresponding solutions,$u$and$v$, satisfying$u \ge v$a.e. (where$v$is the constructed solution of (\ref{eq2})-(\ref{ic}) with$v_0$and$g$instead of$u_0$and$f$). If one has uniqueness of the solution of (\ref{eq2})-(\ref{ic}) we then obtain the so called Maximum Principle. See more precise results in the final remark of Section \ref{uniq}. \medskip \noindent {\sc Remarks on the conditions.} The lower bound on$p$,$p>p_1$, is not essential. It is due to the fact that we do not want to get out of the classical weak formulation where the gradient is a locally integrable function. Exponents$1< p \leq p_1$can be handled by using a proper definition of gradient (see \cite {B6}) but we will not include the calculations here in order to avoid complicating too much the presentation. Note that the lower limit in the elliptic theory is$p_0p-1$(which, in the case of the Laplace operator, reads$s>1$and makes the lower order term$g$superlinear") is the essential ingredient in the existence of a priori estimates that allows for the whole local theory. As for the conditions on$q$, the limit$q1$for$p > p_1$. \smallskip \noindent$\bullet$For$p>N $the point$B$is not valid because it violates the conditions$qN$and we get locally bounded solutions in space, not far from the variational formulation. We will devote Section \ref{variat} to discuss this issue. \smallskip \noindent {\sc Regularity for$u$}. We shall now look briefly at the regularity$u\in L^r(0,T;L^m_{{\rm loc}}(\mathbb{R}^N))$obtained from the previous one by use of the standard Sobolev embeddings. Using the rule$m=qN/(N-q)$for$q N. From this we get for $2\le p < N$ the extremal point $r=p-1$ , $m=m_e$ with $m_e= N(p-1)/(N-p)$, which tends to infinity as $p\to N$. In the balanced case $r=m$ we get the admissible values $$1\le r=m < p-1 + \frac pN,$$ which is a better space than $L^s(0,T; L^s_{{\rm loc}}(\mathbb{R}^N))$ only if $s+1< p(N+1)/N$. \medskip \begin{figure}[ht] \begin{center} \includegraphics[width=0.4\textwidth]{fig1.eps} \quad \includegraphics[width=0.4\textwidth]{fig2.eps} \end{center} \caption{Admissible region for $2 < p < N$ and for $p<2$} \end{figure} \section{Improved Regularity}\label{imprreg} In accordance with the last observation we may expect a better regularity when $s$ is large enough by taking into account the bound on $u$ in $L^s(L^s_{{\rm loc}})$. Indeed, there is an improvement of the previous regularity results for $s\gg 1$ which is reflected in the following result. \begin{theo}\label{theo2} Let $p>1$ and $s>p-1$, $s>1/(p-1)$. Then we can construct a weak solutions as before with the additional regularity $$\label{impr} u\in L^r(0,T; W^{1,q}_{{\rm loc}}(\mathbb{R}^N))$$ for every $r,q\ge 1$ such that $r s+1. \end{theo} The new condition$s(p-1)>1$is necessary for small values of$p$in order for the admissible set of exponents to be non-empty. Another extremal point$C$is now given by the coordinates$q=r=ps/(s+1)$. Taking into account the common point$A$and comparing the slopes of the critical lines implies that an improvement of the admissibility region takes place if$(N(p-2)+p)/N p\frac{N+1}{N}, the admissible $(q,r)$ region of Theorem \ref{theo1} is extended. The maximal $q$-regularity is improved when $N(p-1)/(N-1)\frac{N(p-1)}{N-p}, and in that case the new admissible region completely contains the previous one. \end{coro} The latter result is natural since the bound for$s$is just the Sobolev conjugate exponent of$q_e= N(p-1)/(N-1)$. In any case, it is clear that as$s\to\infty$we approach the variational regularity$q=p$. Note also that for the Laplacian case the improvement of regularity starts from the exponent$s=(N+2)/N$. One further improvement concerns the lower bound for$p$. We remark that whenever$p > (s+1)/s$we have estimates that allow to construct a weak solution in$ L^r(0,T;W^{1,q}_{{\rm loc}}(\mathbb{R}^N))$for some$r,q\ge 1$. This means that for large enough$s$the restriction$p > p_1$is overcome inside the framework of weak solutions. \section{Basic local estimates} \label{sect-local} \noindent {\sc Model equation. Basic Step.} In order to present the main ideas without undue complications we begin with the model example $$\label{plp} u_t-\mathop{\rm div}(| Du|^{p-2} Du) + | u|^{s-1} u = f.$$ We perform the analysis on the solutions of the Cauchy-Dirichlet problems posed in balls$B_R(0)$with zero Dirichlet conditions on the lateral boundary$| x|=R$. Both$u_0$and$f$are suitably cut into bounded functions defined for$x\in B_R$,$0\le t\le T$, preserving their$L^1_{{\rm loc}}$bounds. It follows from standard theory \cite{L} that there exists a solution$u=u_R$of the approximate problem, and $$u\in L^p(0,T; W^{1,p}_0(B_R))\cap C([0,T];L^2(B_R)),$$ $$u_t\in L^{p'}(0,T; W^{-1,p'}(B_R)), \quad | u|^s\in L^1(B_R\times (0,T)).$$ Moreover, the Maximum Principle holds and, in particular,$u_0\ge 0$and$f\ge 0$imply$u\ge 0$. We want to obtain local estimates of the size of$u$and its gradient$Du$which are uniform in$R$. In order to do that we take a cutoff function in the space variables$\theta(x)$, which is smooth, supported in a ball$B_{2\rho}(0)$, with$0<2\rho0$by $$\phi(u)= \int_0^u \frac{1}{(1+ s)^{m+1}}\,ds= \frac 1m-\frac 1{m(1+u)^m}, \label{phi}$$ and by$\phi(u)=-\phi(-u)$for$u<0$. It is a bounded monotone function. We also introduce $$\psi(u)= \int_0^u \phi(s)\,ds. \label{psi}$$ Now we multiply (\ref{plp}) by$\phi(u(x,t))\,\theta^\gamma(x)$,$\gamma>1$, and integrate by parts to obtain \begin{eqnarray*} \lefteqn{\int_{B_R} \psi(u(x,T))\theta(x)^\gamma\,dx + \int_0^T \int_{B_R} | Du|^p\phi'(u)\theta^\gamma\,dx\,dt }\\ \lefteqn{+\int_0^T \int_{B_R} | u|^{s-1} u \phi(u)\theta^\gamma\,dx\,dt +\gamma \int_0^T \int_{B_R} (Du\cdot D\theta) | Du|^{p-2} \,\phi(u)\theta^{\gamma-1}\,dx\,dt} \\ &=& \int_{B_R} \psi(u_0(x))\theta(x)^\gamma\,dx + \int_0^T \int_{B_R} f\phi(u)\theta^\gamma\,dx\,dt. \hspace{35mm} \end{eqnarray*} We remark that \begin{eqnarray*} | Du|^{p-1}| D\theta |\,\phi(u)\theta^{\gamma-1} &\le& \frac 1 {2 \gamma} | Du|^p\phi'(u)\theta^\gamma + C(p,\gamma) | D\theta|^p\theta^{\gamma-p} \frac{\phi(u)^p}{\phi'(u)^{p-1}}\\ &\le& \frac 1 {2 \gamma} | Du|^p\phi'(u)\theta^\gamma + C_1 (1+ | u| )^{(m+1)(p-1)}\theta^{\gamma-p}, \end{eqnarray*} where$C_1>0$depends on$p$,$m$,$\gamma>1$and$\rho>0$. We now choose$m$so that$(m+1)(p-1)0$since$s>p-1$by assumption. Using Young's inequality we get $$C_1 (1+ | u| )^{(m+1)(p-1)}\theta^{\gamma-p} \le \frac 12 | u|^{s-1} u \phi(u)\theta^\gamma + C_2 (1 + \theta^{\gamma-\frac{ps}{s-(m+1)(p-1)}}). \label{coucouclock}$$ In view of the last exponent we choose$\gamma> ps/(s-(m+1)(p-1))$and then \begin{eqnarray}\label{3est} \lefteqn{\int_{B_R} \psi(u(x,T))\theta(x)^\gamma\,dx + \frac 12\int_0^T \int_{B_R} | Du|^p\phi'(u)\theta^\gamma\,dx\,dt }\nonumber \\ \lefteqn{ +\frac 12 \int_0^T \int_{B_R} | u|^{s-1} u \phi(u)\theta^\gamma\,dx\,dt } \\ &\le& \int_{B_{2\rho}} \psi(u_0(x))\,dx + C_3\int_0^T \int_{B_{2\rho}} | f|\,dx\,dt + C_3\,T| B_{2\rho}|, \nonumber \end{eqnarray} where$| B_{2\rho}|$denotes the volume of$ B_{2\rho}$in$\mathbb{R}^N$. In view of our assumptions on the data we conclude that $$\label{C4} \int_0^T \int_{B_\rho} \frac{| Du|^p}{(1+| u|)^{m+1}}\,dx\,dt \le C_4, \quad \int_0^T \int_{B_\rho} | u|^{s}\,dx\,dt\le C_4,$$ where$C_4$depends on$\rho$,$p$,$s$and$T$and not on$R$. The dependence on$\rho$takes place through the local norms of$u_0$and$f$. The main point is that the different constants$C_i$appearing in this calculation do not depend on$R$. Besides, in view of the form of$\psi$, the first term in (\ref{3est}) (replacing$T$by$t$, which is possible if$0 \le t \le T$) gives, for every$00$so that the exponent $$\alpha=\frac{(m+1)q}{p-q}$$ is equal or less than 1, and then the right-hand side is bounded and so is the left-hand side. This leads to a bound in$L^r(0,T; W^{1,q}(B_{\rho}))$with$1\le q

\frac{(\alpha-1)(p-q)r}{(q^\star-1)q(p-r)}. In order to fulfill this inequality we may choose a small value of $\alpha$, always larger than $q/(p-q)$, by taking $m>0$ very small. Therefore, we are reduced to check the limit case $\alpha-1=(2q-p)/(p-q)$ which gives $$\frac{q^\star-1}{q^\star}>\frac{2q-p}{q(p-r)}.$$ Working out this formula gives the relation $$\frac{(p-2)N+p}r+\frac Nq> N+1.$$ This relation is complemented by the restrictions on $q$: $1\le q0$ $$\int_{B_\rho} | Du|^q\,dx\le \Big(\int_{B_\rho} \frac{| Du|^p}{(1+| u|)^{m+1}}\,dx\Big)^{q/p} \Big(\int_{B_\rho} (1+| u|)^{\alpha}dx\Big)^{(p-q)/p},$$ where $\alpha=(m+1)q/(p-q)$ as before. Rising to the power $r/q$ with $r0$ so that $1< \alpha s+1. Note that this inequality is satisfied for all$q

0$the following sequences are bounded uniformly in$n>2\rho$: $$\begin{array}{c} \{u_n\} \quad\mbox{in } L^r(0,T; W^{1,q}(B_\rho(0))), \\[3pt] \{g(x,t,u_n)\} \quad \mbox{in } L^1(B_\rho(0)\times(0,T)), \\[3pt] \{u'_{n}\} \quad \mbox{in } L^1(0,T; W^{-1,\delta}(B_\rho(0)))+ L^1(0,T; L^{1}(B_\rho(0))), \end{array}$$ for some$\delta >1$. Moreover, since one obtains that the sequence$\{u'_n\}$is bounded in$L^1(0,T;W^{-1,1}(B_\rho(0)))$, using compactness arguments (see \cite{Si}) it is easy to see that the sequence$\{u_n\}$is relatively compact in$L^1(Q_\rho)$. By a diagonal process we may select a subsequence, also denoted by$\{u_n\}$, such that $$u_n\to u \quad \mbox{a.e. and in } \ L^1(0,T; L^1_{{\rm loc}}(\mathbb{R}^N)),$$ and also$u_n\to u$weakly in$L^r(0,T; W^{1,q}_{{\rm loc}}(\mathbb{R}^N))$for$q,r$as in Theorem \ref{theo2} and Corollary \ref{cor4}. We want to pass to the limit in the equation in order to get a solution of the original problem. We need to prove first the convergence of the sequence$\{g(x,t,u_n)\}$, and also the$a.e.$convergence (up to a subsequence) of the gradients of$\{u_n\}$, which will imply the convergence of$\{Du_n\}$to$Du$in$L^r(0,T; L^{q}_{{\rm loc}}(B_R))$, for any$R>0$. Let us prove the result about$g(x,t,u_n)$, which is based on an argument of local equi-integrability. We resume the notations and calculations of Section \ref{sect-local} with slight variations. We consider the function$\phi$defined in (\ref{phi}) and we displace it by an amount$t>0$to get $$\phi^t(s)=\left\{ \begin{array}{ll} \phi(s-t),\quad & s\geq t\\ 0, & |s|1. Dispensing with the superscripts t for \phi and \psi and much as in (\ref{3est}) we have \begin{eqnarray*} \lefteqn{\int_{B_R} \psi(u_n(x,T))\theta(x)^\gamma\,dx + c \int_0^T \int_{B_R} | Du_n|^p\phi'(u_n) \theta^\gamma\,dx\,dt }\\ \lefteqn{+\int_0^T \int_{B_R} g(x,t,u_n) \phi(u_n)\theta^\gamma\,dx\,dt + \gamma \int_0^T \int_{B_R} (A(Du_n)\cdot D\theta) \phi(u_n)\theta^{\gamma-1}\,dx\,dt }\\ &\le& \int_{B_R} \psi(u_{0n}(x))\theta(x)^\gamma\,dx + \int_0^T \int_{B_R} f_n\phi(u_n)\theta^\gamma\,dx\,dt. \hspace{3cm} \end{eqnarray*} Now we use the following inequalities for some \varepsilon>0 (thanks to the boundedness of \phi and Young's inequality as in (\ref{coucouclock}), recall that \frac{s}{(m+1)(p-1)}>1) \begin{eqnarray*} \lefteqn{|Du_n|^{p-1}|D\theta|\phi(u_n)\theta^{\gamma -1}}\\ &\leq& \varepsilon |Du_n|^{p}\phi'(u_n)\theta^{\gamma } + c(p,\varepsilon) |D\theta|^p \theta^{\gamma -p} \frac{\phi(u_n)^p}{\phi'(u_n)^{p-1}} \\ &\le& \varepsilon |Du_n|^{p}\phi'(u_n)\theta^{\gamma } + \varepsilon |u_n|^{s-1}u_n \phi(u_n) \theta^\gamma + c(\theta, \varepsilon, p, s) \phi(u_n)^{p}. \end{eqnarray*} Choosing \varepsilon >0 small enough and working as in (\ref{3est}) this leads to \begin{eqnarray} \label{3est.b} \lefteqn{\int_{B_R} \psi(u_n(x,T))\theta(x)^\gamma\,dx + \frac c 2 \int_0^T \int_{B_R}| Du_n|^p\phi'(u_n)\theta^\gamma\,dx\,dt }\nonumber\\ \lefteqn{ + \frac 1 2 \int_0^T\int_{B_R} g(x,t,u_n)\phi(u_n)\theta^\gamma\,dx\,dt }\\ &\le& \int_{B_{2R}} \psi(u_0(x))\theta^\gamma \,dx + C_3\int_0^T \int_{B_{2R}} | f|\,\phi(u_n)\,dx\,dt \nonumber\\ && + C_3\int_0^T \int_{B_{2R}} \phi^p(u_n)\,dx\,dt. \nonumber \end{eqnarray} We recall that \phi stands for \phi^t and \psi for \psi^t. As t\to\infty the right-hand side of the last displayed formula tends to zero, hence instead of the estimates (\ref{C4}) we now conclude that$$ \int_0^T \int_{B_R} g(x,t,u_n)\phi(u_n)\theta^\gamma\,dx\,dt\to 0 $$as t\to\infty, uniformly in n. Since, for \tau large \phi^\tau (u_n)\geq c_0\chi_{\{ |u_n|>2\tau\}}, we get$$ \lim\limits_{\tau\to\infty}\sup\limits_{n} \int_0^T \int_{B_R} g(x,t,u_n) \theta^\gamma\chi_{\{ |u_n|>2\tau\}}\,dx\,dt\to 0$$This means that$\{g(u_n)\}$is equi-integrable. By Vitali's Lemma$g(u_n)\to g(u)$in$L^1(0,T; L^1_{{\rm loc}}(\mathbb{R}^N)$and a.e. The last step is the a.e. convergence of the sequence$\{Du_n\}$and this is done by means of the concept of convergence in measure as in the papers \cite{BG}, \cite{B6}. We will prove that, for any$\rho>0$, the sequence$\{Du_n\}$is a Cauchy sequence in measure on$B_\rho\times [0,T]$. To prove this result, let$\rho>0$, let$\varepsilon >0$and$T_{\varepsilon}(s)=\max\{-\varepsilon,\min\{\varepsilon,s\}\}$. Define${Q_\varepsilon}=\{(x,t)\in B_\rho(0) : |u_n(x,t)-u_m(x,t)|\leq \varepsilon \}$. Taking a cutoff function$\theta$equal to$1$on$B_\rho$and$0$outside$B_{2\rho}$and$T_\varepsilon (u_n-u_m) \theta$as test function in the equations satisfied by$u_n$and$u_m$lead to (neglecting the positive contribution of the term with time derivative) \begin{eqnarray} \label{teps} \lefteqn{\int \int_{Q_\varepsilon} (A(x,t,Du_n)-A(x,t,Du_m))(Du_n-Du_m)dx dt}\\ &\le& \varepsilon \int_0^T \int_{B_{2\rho}} (| F_n | + | F_m |)dx dt, \hspace{35mm} \nonumber \end{eqnarray} where$\{F_n\}$is a bounded sequence in$L^1(B_{2\rho} \times (0,T))$(its bound depends on bounds already obtained for the sequences$\{u_n\}$and$\{Du_n\}$). From now on, we can follow the proof of \cite{B6} and we can show that the sequence$\{Du_n\}$is a Cauchy sequence in measure on$B_\rho \times [0,T]$. Thus, up to a subsequence, the sequence$\{Du_n\}$converges a.e. to$Du$. The passage to the limit is now standard. The improved regularity results of Theorem \ref{theo2} and Corollary \ref{cor4} follow in the same way from the estimates we derived in the last part of Section \ref{sect-local}. \section{Variational framework for the case$p > N$} \label{variat} When$p > N$we expect better regularity, as happens in the elliptic case. Indeed, it has been observed in \cite{BGV} that an estimate for$|Du|$in$L^q_{{\rm loc}}(\mathbb{R}^N)$together with an estimate for$u$in$L^s_{{\rm loc}}(\mathbb{R}^N)$implies an estimate for$u$in$L^\infty_{{\rm loc}}(\mathbb{R}^N)$for the solutions of the elliptic equation (solutions of our problem with no time dependence), and then we get the regularity$|Du|\in L^p_{{\rm loc}}(\mathbb{R}^N)$, which makes the theory with right-hand side in$L^1_{{\rm loc}}(\mathbb{R}^N)$fall into the usual variational framework. However, in the parabolic case there is an extra complication due to the presence of the time variable, so that the regularity of the gradient$|Du|\in L^1(0,T; L_{{\rm loc}}^q(\mathbb{R}^N))$,$q>N$, is not enough to guarantee the local boundedness of$u$, and the corresponding statement is not so clear. Indeed, we have the following improvement of Theorem \ref{theo1}. \begin{theo} For$s>p-1$and$p>N$we also get the regularity$u\in L^r(0,T; W^{1,q}_{{\rm loc}}(\mathbb{R}^N))\cap L^r(0,T; L^\infty_{{\rm loc}}(\mathbb{R}^N))$under the conditions$NN$. In this case we may select$q$in the range$N < q N$implies by Morrey's inequality, cf. \cite[Theorem 4.5.3.3]{EG}, that we have an estimate of the form $$\label{sobol} \|u\|_{L^\infty(B_{\rho})} \le C + C\Big(\int_{B_\rho} | Du|^q\,dx\Big)^{1/q},$$ where$C$depends only on$\rho, q$and the$L^1$norm of$u$in$B_\rho$, which is uniformly bounded independently of$R$thanks to the previous estimate on$u$in$L^\infty(0,T; L^1_{{\rm loc}}(\mathbb{R}^N))$. Continuing as before, we can write for$r