Blow-up problems for quasilinear reaction diffusion equations with weighted nonlocal source

Abstract

In this paper, we investigate the following quasilinear reaction diffusion equations $$ \begin{cases} \left(b(u)\right)_t =\nabla\cdot\left(\rho\left(|\nabla u|^2\right)\nabla u\right)+c(x)f(u) &\hbox{ in } \Omega\times(0,t^{*}),\\ \frac{\partial u}{\partial \nu}=0 &\hbox{ on } \partial\Omega\times(0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 & \hbox{ in } \overline{\Omega}. \end{cases} $$ Here $\Omega$ is a bounded domain in $\mathbb{R}^{n}\ (n\geq2)$ with smooth boundary $\partial\Omega$. Weighted nonlocal source satisfies $$ c(x)f(u(x,t))\leq a_1+a_2\left(u(x,t)\right)^{p}\left(\int_{\Omega}\left(u(x,t)\right)^{\alpha}{\rm d}x\right)^{m}, $$ where $a_2,p,\alpha$ are some positive constants and $a_1, m$ are some nonnegative constants. We make use of a differential inequality technique and Sobolev inequality to obtain a lower bound for the blow-up time of the solution. In addition, an upper bound for the blow-up time is also derived.

Keywords

• blow-up problems
• quasilinear reaction equation
• weighted nonlocal source