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Qualitative properties of weak solutions for \(p\)-Laplacian equations with nonlocal source and gradient absorption. (English) Zbl 1456.35118

Summary: We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source \[\dfrac{\partial u}{\partial t} -\operatorname{div}(|\nabla u|^{p-2}\nabla u) =\lambda u^k\int_{\Omega}u^sdx- \mu u^l|\nabla u|^q\] in a bounded domain \(\Omega\subset\mathbb{R}^N \), where \(p>1\), the parameters \(k,s,l,q,\lambda>0\) and \(\mu\geq 0\). Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on \(|\nabla u|\). Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.

MSC:

35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35K20 Initial-boundary value problems for second-order parabolic equations
35B09 Positive solutions to PDEs
35B51 Comparison principles in context of PDEs
35B44 Blow-up in context of PDEs
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