Quenching Time for Some Semilinear Equations with A Potential

This paper concerns the study of a semilinear parabolic equation subject to Neumann boundary conditions, with a potential and positive initial datum. Under some assumptions, we show that the solution of the above problem quenches in a finite time and estimate its quenching time. Finally, we give some numerical results to illustrate our analysis. Key-words: Quenching, semilinear parabolic equation, numerical quenching time. AMS subject classification (2010): 35B40, 35B50, 35K60, 65M06. 1-Introduction Let Ω be a bounded domain in Rwith smooth boundary ∂Ω. Consider the following initial-boundary value problem ut = ∆u − c x, t u −p x in Ω × 0, T , (1) ∂u ∂ν = 0 on ∂Ω × 0, T , (2) u x, 0 = u0 x > 0 in Ω , (3) where ∆ is the Laplacian, ν the exterior normal unit vector on ∂Ω. We suppose that the initial datum u0 ∈ C (Ω ) and u0(x) > 0 in Ω . Here the potential c(x, t) is a nonnegative locally Hölder continuous function defined for x ∈ Ω and t ≥ 0. The exponent p ∈ C(Ω), 0 < p0 = infx∈Ω p x < supx∈Ω p x = p+. Here (0, T) is the maximal time interval of existence of the solution u, and by a solution, we mean the following. Definition1.1. A solution of (1)-(3) is a function u(x, t)continuous in Ω × 0,T , u x, t > 0 in Ω × 0,T , and twice continuously differentiable in x and once in t in Ω × (0,T). 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 quenching in a finite time, namely limt⟶Tumin (t) = 0, where umin t = minx∈Ω u x, t . In this last case, we say that the solution u quenches in a finite time and the time T is called the quenching time of the solution u. Since the pioneering work of Kawarada in 1975 (see, [25]), the study of the phenomenon of quenching for semilinear heat equations has attracted a considerable attention (see, for example [3]-[4], [6]-[8], [11], [14], [24], [26], [28]-[30], [36] and the references cited therein). More precisely, in [7] Boni has studied the problem (1)-(3) for the phenomenon of blow-up. He has given some sufficient conditions under which solutions to such equation tend to zero or blow up in a finite time. In the same way, some authors have proved the existence and uniqueness of solution (see, [16], [27]). In [8], Boni and Kouakou have treated a similar problem with variable exponent. They have estimated the quenching time and studied its continuity as a function of the initial datum u0. The originality of this work is that it is the first attempt of studying the phenomenon of quenching with variable exponent and a potential depending both on space and time. Using standard methods, the local in time existence and uniqueness of solutions can be easily proved (see, [7], [16]). 1 Université Nangui Abrogoua, UFR-SFA, Laboratoire de Mathématiques et Informatique, 02 BP 801 Abidjan 02, (Côte d’Ivoire), E-mail: krkouakou@yahoo.fr 2 Université Péléforo Gon Coulibaly de Korhogo, BP 1328 Korhogo, (Côte d’Ivoire), E-mail: firmingoh@yahoo.fr 4 American Review of Mathematics and Statistics, Vol. 7, No. 2, December 2019 Our aim in this paper consists in showing that, under some hypotheses, the solution of (1)-(3) quenches in a finite time. If we set g(x,u) = c(x, t)u, then we observe that the function g is continuous in both variables and locally Lipschitz in the second one. Let us notice that, because the initial datum of the problem considered is sufficiently regular, the solution of this problem exists and is regular. In addition, we note that the regularity of solution is as important as the regularity of the initial data, and the maximum principle holds (see, [16], [27], [33]). This paper is structured as follows. In the following section, we show that under some assumptions, the solution u of (1)(3) quenches in a finite time and estimate its quenching time and finally, in the last section, we give some numerical results to illustrate our analysis.