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