Blow-up under oscillatory boundary conditions

Article type: Research Article

Authors: Herrero, M.A.^{; ;} | Lacey, A.A.^; | Velazquez, J.J.L.^;

Affiliations: Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain | Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, United Kingdom

Note: [] Correspondence to: M.A. Herrero, Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain.

Note: [] Partially supported by ClCYT Grant PB90-0235, EEC Contract SC1-0019-C, and SERC Grant GR/D/73096.

Note: [] Partially supported by EEC Contract SC1-0019-C.

Note: [] Partially supported by ClCYT Grant PB90-0235, EEC Contract SC1-0019-C, and SERC Grant GR/D/73096.

Abstract: Consider the problem (1) ut=uxx+δeu,   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 δFK such that (i) If δ>δFK, every solution of (P0) blows up in finite time, and (ii) If δ<δFK there exist global solutions of (P0) for some choices of u0(x). A similar parameter δc=δc(a,ω) exists for problem (1)-(3) and is such that δc≤δFK. We obtain here an asymptotic formula for δc(a,ω) when a or ω (or both) are arbitrarily large.

DOI: 10.3233/ASY-1994-9101

Journal: Asymptotic Analysis, vol. 9, no. 1, pp. 1-22, 1994