Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2020; 57(4): 1003-1031

Online first article July 9, 2020      Printed July 31, 2020

https://doi.org/10.4134/BKMS.b190720

Copyright © The Korean Mathematical Society.

Qualitative properties of weak solutions for $p$-Laplacian equations with nonlocal source and gradient absorption

Zakariya Chaouai, Abderrahmane El Hachimi

Mohammed V University, P.O. Box 1014; Mohammed V University, P.O. Box 1014

Abstract

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source $$ \dfrac{\partial u}{\partial t} -\operatorname{div}(|\nabla u|^{p-2}\nabla u) =\lambda u^{k}\int_{\Omega}u^{s}dx- \mu u^{l}|\nabla u|^{q}$$ 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.

Keywords: Parabolic $p$-Laplacian equation, global existence, blow-up, nonlocal source, gradient absorption

MSC numbers: Primary 35K55, 35B09, 35B51