Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data. (English) Zbl 1152.35323

Summary: 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 \[ \begin{aligned} u_t(x,t)- \Delta_p u(x,t)= \mu\quad &\text{in }\Omega\times (0,T),\\ u(x,0)= u_0(x)\quad &\text{in }\Omega,\end{aligned} \] 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.


35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data