Some Nonlinear Parabolic Problems with Singular Natural Growth Term

Abstract

In this paper, we study the existence and regularity results for nonlinear parabolic problems with singular natural growth gradient terms

$$\begin{aligned} \left\{ \begin{array}{lll} \frac{\partial u}{\partial t}-{\varDelta }_{p}u+b(x,t)\frac{|\nabla u|^{p}}{u^{\theta }}=f &{} \text{ in } &{} Q,\\ u(x,t)=0 &{} \text{ on } &{} {\varGamma },\\ u(x,0)=0 &{} \text{ in } &{} {\varOmega }, \end{array} \right. \end{aligned}$$

where ${\varOmega }$ is a bounded open subset of ${\mathbb {R}}^{N},$ $N\ge 2,$ Q is the cylinder ${\varOmega }\times (0,T),$ $T>0,$ ${\varGamma }$ the lateral surface $\partial {\varOmega }\times (0,T),$ ${\varDelta }_{p}$ is the so-called $p-$Laplace operator, ${\varDelta }_{p}u=\text{ div }(|\nabla u|^{p-2}\nabla u)$ with $2\le p<N,$ b is a positive measurable bounded function, $0<\theta <1,$ and f belongs to Lebesgue space $L^{m}(Q),$ $m\ge 1$.