ON INITIAL BOUNDARY VALUE PROBLEM FOR A SEMILINEAR REACTION DIFFUSION EQUATION AND NON-LINEAR SOURCE TERM WITH BLOW-UP

Abstract

In this paper, we consider the following initial-boundary value problem (P)⎧ ⎨ ⎩ ut(x,t)=εLu(x,t)+f(u(x,t)) in Ω×(0,T), ∂u(x,t) ∂N + b(x,t)g(u(x,t)) = 0 on ∂Ω×(0,T), u(x,0) = u0(x) in Ω , where Ω is a bounded domain in RN with smooth boundary ∂Ω,εis a positive parameter, L is an elliptic operator, b(x,t) ≥ 0 in∂Ω × R+, g ∈ C1(R), g(0)=0, f(s) is a positive, increasing, convex function for positive values of s and∞ ds f(s) < ∞. 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 the one of the solution of the following differential equation α(t)=f(α(t)), t > 0, α(0) = M, as ε tends to zero where M = sup x∈Ω u0(x). We also extend the above result to other classes of nonlinear parabolic equations with nonlinear boundary conditions. Finally, we give some numerical results to illustrate our analysis