Asymptotic behavior of solutions for nonlinear parabolic operators with natural growth term and measure data

Abstract

We are interested in the asymptotic behavior, as t tends to $$+\infty $$ , of finite energy solutions and entropy solutions $$u_{n}$$ of nonlinear parabolic problems whose model is 0.1 $$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-\Delta _{p}u+g(u)|\nabla u|^{p}=\mu &{}\text { in }(0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)&{}\text { in }\Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega \end{array}\right. } \end{aligned}$$ where $$\Omega \subseteq \mathbb {R}^{N}$$ is a bounded open set, $$N\ge 3$$ , $$u_{0}\in L^{1}(\Omega )$$ is a nonnegative initial data, while $$g:\mathbb {R}\mapsto \mathbb {R}$$ is a real function in $$C^{1}(\mathbb {R})$$ which satisfies sign condition with positive derivative and $$\mu $$ is a nonnegative measure independent on time which does not charge sets of null p-capacity.