ASYMPTOTIC BEHAVIOUR OF THE BLOW-UP TIME FOR NONLOCAL DIFFUSION PROBLEM WITH SOME PARAMETERS

This paper concerns the study of the following parabolic equation $$(P)\\\left\{%\begin{array}{ll}\hbox{$u_t-\varepsilon(J*u-u)= \gamma f(u)\quad \mbox{in}\quad \Omega\times(0,T),$} \\\hbox{$u=0\quad \mbox{on}\quad (\mathbb{R}^N-\Omega)\times(0,T),$} \\\hbox{$u(x,0)=u_{0}(x)>0,\quad \mbox{on}\quad \Omega$,}\end{array}%\right. $$ where $J*u(x,t)=\int_{\mathbb{R}^N}J(x-y)u(y,t)dy,$ J:$\mathbb{R}^N\longrightarrow\mathbb{R}_+$ is nonnegative, and symmetric function (J(z)=J(-z)) bounded and $\int_{\mathbb{R}^N}J(z)dz=1,$ the initial data $u_0(x)\in C^0(\Omega),$ and $\gamma$ is a positive parameter. We find some conditions under which the solution of semidiscrete form of the above problem blows up in a finite time and estimate its semi-discrete blow-up time. We also prove the convergence of the semidiscrete form blow-up time to the real one when the mesh size tends to zero. Finally, we give some numerical results to illustrate our analysis.