A reaction-diffusion model for a class of nonlinear parabolic equations with moving boundaries: Existence, uniqueness, exponential decay and simulation

The aim of this paper is to establish the existence, uniqueness and asymptotic behaviour of a strong regular solution for a class of nonlinear equations of reaction-diffusion nonlocal type with moving boundaries: {u(t) - a (integral(Omega t) u(x, t)dx)u(xx) = f (x, t), (x, t) is an element of Q(t), u(alpha(t), t) = u(beta(t), t) = 0, t > 0, u(x, 0) = u(0)(x), x is an element of Omega(0) =]alpha(0),beta(0)[, where Q(t) is a bounded non-cylindrical domain defined by Q(t) = {(x, t) is an element of R-2 : alpha(t) < x < beta(t), for all 0 < t < T}. Moreover, we study the properties of the solution and implement a numerical algorithm based on the Moving Finite Element Method (MFEM) with polynomial approximations of any degree, to solve this class of problems. Some numerical tests are investigated to evaluate the performance of our Matlab code based on the MFEM and illustrate the exponential decay of the solution. (C) 2014 Elsevier Inc. All rights reserved.