Blow-up under oscillatory boundary conditions

Abstract

Consider the problem

(1) u_t=u_xx+δe^u,   if 0<x<1,t>0,

(2) u(0,t)=asin ωt,  u(1,t)=0 for t≥0,

(3) u(x,0)=u₀(x),

where a>0,ω>0, and u₀(x) is a continuous and bounded real function. If we replace the first condition in (2) by u(0,t)=0, it is known that the corresponding Cauchy-Dirichlet problem (P₀) is characterized by the existence of a critical parameter δ_FK such that (i) If δ>δ_FK, every solution of (P₀) blows up in finite time, and (ii) If δ<δ_FK there exist global solutions of (P₀) for some choices of u₀(x). A similar parameter δ_c=δ_c(a,ω) exists for problem (1)-(3) and is such that δ_c≤δ_FK. We obtain here an asymptotic formula for δ_c(a,ω) when a or ω (or both) are arbitrarily large.