Existence and regularity results for nonlinear parabolic equations with quadratic growth with respect to the gradient

Abstract

In this paper we study the existence and regularity of solutions of nonlinear parabolic equations of the type

$$\begin{aligned} \left\{ \begin{aligned}&{\partial u \over \partial t}- {{\mathrm{div}}}\Big ((a(x, t)+|u|^q) \nabla u\Big )+b(x, t)u|u|^{p-1}|\nabla u|^2= f\ \ {{\mathrm{in}}}\ Q,\\&u(t=0)= 0\ \ {{\mathrm{in}}}\ \Omega ,\\&u=0\ \ {{\mathrm{on}}}\ {\partial \Omega }\times (0,T),\\ \end{aligned} \right. \end{aligned}$$

where $Q= \Omega \times (0, T)$, $\Omega$ is a bounded open subset of ${\mathbb {R}}^N (N > 2)$, a(x, t), b(x, t) are measurable positive functions, $p, q > 0$ and f belongs to $L^m(Q)$ for some $m \ge 1$.