Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

We are concerned with the study of the following nonlocal diffusion problem where Ω is a bounded domain in RN with smooth boundary ∂Ω, J * u(x, t) = ∫RN J(x − y)u(y, t)dy, J : RN rightarrow R is a kernel which is nonnegative, symmetric (J(z) = J(−z)), bounded and ∫RN J(z)dz = 1 and f : (−1, b) rightarrow (0,∞) is a C1 convex, increasing function, lims rightarrow b f(s) = 1, ∫b 0 ds/f(s) ∞ with b a positive constant. The initial datum u0 Є C1(Ω) is nonnegative in Ω, with parallel u0 parallel ∞ = supxЄ Ω|u0(x)| b. Under some assumptions, we show that if ku0k1 is large enough, then the solution of the above problem quenches in a finite time, and its quenching time goes to that of the solution of the differential equation ά (t) = f(α(t)), t 0, α(0) = parallel u0 parallel ∞, as parallel u0 parallel ∞ tends to b. Finally, we give some numerical results to illustrate our analysis.