Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data

Abstract

Let $\Omega\subseteq {\mathbb{R}^{N}}$ be a bounded open set, $N\geq 2$, and let $p>1$; we study the asymptotic behavior with respect to the time variable $t$ of the entropy solution of nonlinear parabolic problems whose model is $$ \begin{cases} u_{t}(x,t)-\Delta_{p} u(x,t)=\mu & \text{in}\ \Omega\times(0,T),\\ u(x,0)=u_{0}(x) & \text{in}\ \Omega, \end{cases} $$ where $T>0$ is any positive constant, $u_0 \in L^{1}(\Omega)$ a nonnegative function, and $\mu\in \mathcal{M}_{0}(Q)$ is a nonnegative measure with bounded variation over $Q=\Omega\times(0,T)$ which does not charge the sets of zero $p$-capacity; moreover, we consider $\mu$ that does not depend on time. In particular, we prove that solutions of such problems converge to stationary solutions.