On the blow-up semidiscretizations in time of some non-local parabolic problems with Neumann boundary conditions

Abstract

In this paper, we address the following initial value problem \[\begin{align} u_t &= \int_\Omega J(x-y)(u(y,t) - u(x,t))dy + f(u) \quad \text{in } \overline{\Omega} \times (0,T), \\ u(x,0) &= \varphi(x) \geq 0 \quad \text{in } \Omega, \end{align}\]

 where \(f : [0,\infty) \to [0,\infty)\) is a \(C^1\) nondecreasing function, \(\int^\infty \frac{ds}{f(s)} < \infty\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega, J : \mathbb{R}^N \to \mathbb{R}\) is a kernel which is nonnegative and bounded in \(\mathbb{R}^N\). Under some conditions, we show that the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove that the semidiscrete blow-up time converges to the real one when the mesh size goes to zero. Finally, we give some numerical results to illustrate our analysis.