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Blow-up time for a semilinear parabolic equation with variable reaction

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In this paper, we address the solution of a semilinear heat equation with variable reaction subject to Dirichlet boundary conditions and nonnegative initial datum. Under some assumptions, we show that the solution of the above problem blows up in a finite time, and its blow-up time goes to that of the solution of a certain differential equation. Finally, we give some numerical results to illustrate our analysis. | Turk J Math 35 (2011) , 87 – 100. ¨ ITAK ˙ c TUB doi:10.3906/mat-0903-47 Blow-up time for a semilinear parabolic equation with variable reaction Th´eodore Kouassi Boni and Remi Kouadio Kouakou Abstract In this paper, we address the solution of a semilinear heat equation with variable reaction subject to Dirichlet boundary conditions and nonnegative initial datum. Under some assumptions, we show that the solution of the above problem blows up in a finite time, and its blow-up time goes to that of the solution of a certain differential equation. Finally, we give some numerical results to illustrate our analysis. Key Words: Semilinear parabolic equation, blow-up, numerical blow-up time. 1. Introduction Let Ω be a bounded domain in RN with smooth boundary ∂Ω. Consider the initial-boundary value problem for a semilinear parabolic equation with variable reaction subject to Dirichlet boundary conditions of the form ut (x, t) = εΔu(x, t) + ep(x)u(x,t) u(x, t) = 0 on in Ω × (0, T ), ∂Ω × (0, T ), u(x, 0) = u0 (x) ≥ 0 in Ω, (1.1) (1.2) (1.3) where p ∈ C1 (Ω), supx∈Ω p(x) = p0 > 0 , Δ is the Laplacian and ε a positive parameter. The initial datum u0 ∈ C 1 (Ω) and u0 (x) is nonnegative in Ω. Here, (0, T ) is the maximal time interval on which the solution u exists. The time T may be finite or infinite. When T is infinite, then we say that the solution u exists globally. When T is finite, then the solution u develops a singularity in a finite time, namely, lim u(·, t) ∞ = ∞, t→T where u(·, t) ∞ = supx∈Ω |u(x, t)| . In this last case, we say that the solution u blows up in a finite time, and the time T is called the blow-up time of the solution u . 2000 AMS Mathematics Subject Classification: 35B40, 35B50, 35K60, 65M06. 87 BONI, KOUAKOU Throughout this paper, we suppose that there exists a ∈ Ω such that M = sup u0 (x) = u0 (a) and p0 = sup p(x) = p(a). x∈Ω x∈Ω For our problem described in (1.1)–(1.3), it is well known that the local in time existence and .