Blow-Up for Discretization of a Localized Semilinear Heat Equation

Abstract

Abstract. This paper concerns the study of the numerical approximation for the following initial-boundary value problem:and γ is a positive parameter. Under some assumptions, we prove that the solution of a discrete form of the above problem blows up in a finite time and estimate its numerical blow-up time. We also show that the numerical blow-up time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.

Blow-Up for Discretizations of Some Reaction-Diffusion Equations With a Nonlinear Convection Term

“…The lemma below reveals the positivity of the discrete solution. h , n ≥ 0, be the solution of the discrete problem (7)- (10). Then…”

“…be the solution of the discrete problem (7)- (10). Suppose that there exists a positive integer λ ∈ (0, 1) such that…”

“…In Figures 3, 4, 5 and 6, we see that the blow-up is local when the initial data is not a constant. Figures 7,8,9,10,11 and 12 show the effect of the convection term on the evolution of the solution. In Figures 13, 14, 15, 16, 17 and 18, we observe that the solution of our problem blows up in a finite time, when the initial data is (…”

“…We also show that the numerical blow-up time converges to the theoretical one when the mesh size goes to zero. A similar study has been undertaken (see [10], [22]).…”