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Asymptotic behavior of solutions for nonlinear parabolic operators with natural growth term and measure data. (English) Zbl 1445.35051

Summary: 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 \[\begin{cases} u_t-\Delta_pu+g(u)|\nabla u|^p=\mu &\text{ in }(0,T)\times \Omega ,\\ u(0,x)=u_0(x)&\text{ in }\Omega ,\\ u(t,x)=0&\text{ on }(0,T)\times \partial \Omega \end{cases}\] 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.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
28A12 Contents, measures, outer measures, capacities
35B51 Comparison principles in context of PDEs
35R06 PDEs with measure
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