Remarks on the quenching estimate for a nonlocal diffusion problem with a reaction term

Abstract

In this paper, we address the following initial value problem\[\begin{array}{ll}\hbox{\(u_t=\int_{\Omega}J(x-y)(u(y, t)-u(x, t)){\rm d}y+f(u(x, t))\quad \mbox{in}\quad \overline{\Omega}\times(0,T)\),} \\\hbox{\(u(x,0)=u_{0}(x)\geq 0\quad \mbox{in}\quad \overline{\Omega}\),} \\\end{array}\]where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(f: (-\infty, b)\rightarrow (0, \infty)\) is a \(C^1\) convex nondecreasing function, \(\lim_{s\rightarrow b^{-}}f(s)=\infty\), \(\int^{\infty}\tfrac{{\rm d}\sigma}{f(\sigma)}<\infty\), with \(b\) a positive constant, \(J:\mathbb{R}^N\rightarrow \mathbb{R}\) is a kernel which is measurable, nonnegative and bounded in \(\mathbb{R}^N\). Under some conditions, we show that the solution of a perturbed form of the above problem quenches in a finite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of the initial datum. Finally, we give some numerical results to illustrate our analysis.