Existence and regularity results for solutions to nonlinear parabolic equations

Abstract

In this paper we prove some existence and regularity results for solutions to a class of nonlinear parabolic equations whose prototype is $$\left\{\begin{array}{lll} \displaystyle\frac{\partial u}{\partial t}-\Delta_p u=f(x,t) &\mbox{ in }Q,\cr u(x,0)=0 &\mbox{ in } \Omega,\cr u(x,t)=0 &\mbox{ on } \Gamma, \end{array}\right. $$ \noindent where $\Omega$ is a bounded open subset of ${{\mathbb R}^ N} $, $N\ge 2$, $Q$ is the cylinder $\Omega \times ]0,T[$, $T>0$, $\Gamma$ the lateral surface $\partial\Omega\times ]0,T[$, $\bigtriangleup _p$ is the so-called $p-$Laplace operator, $ p>1 $ and $f$ belongs to some space $ L^r (0,T;L^q(\Omega )),$ $r\geq 1$, $q\geq 1$.