Blow-up under oscillatory boundary conditions

Abstract

The object of this paper is the study of blowing-up phenomena for the initial-boundary value problem (Pa): ut=uxx+δeu for (x,t)∈(0,1)×(0,+∞), u(0,t)=asinωt and u(1,t)=0 for t∈[0,+∞), u(x,0)=u0(x) for x∈(0,1), where u0(x) is a continuous and bounded function, and a>0, ω>0 are real constants. It is known that if the amplitude a=0 in the oscillatory boundary condition above then there exists a critical parameter δFK (the so-called Frank-Kamenetskiĭ parameter) such that if δ<δFK the corresponding Cauchy-Dirichlet problem (P0) is globally solvable for suitable choices of u0(x), and each solution of (P0) blows up in a finite time if δ>δFK. The authors prove existence of a parameter δ(a,ω)≤δFK with similar critical properties. The essential part of the paper is devoted to the study of the asymptotic behavior of δ(a,ω) with respect to a and ω. For example, δ(a,ω)∼δFK as a→0 uniformly in ω. Further, the exact dependence of δ(a,ω) on the data in (Pa) is shown in the remaining limiting cases for a and ω.