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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.

MSC:

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