The Balance between Nonlinear Inwards and Outwards Boundary Flux for a Nonlinear Heat Equation
We study nonnegative solutions ofut=(um)xx( x,t)∈(0,L)×(0,T),-(um) x(0,t)=up(0,t),t∈(0,T),(um)x(L,t)=-uq(L,t),t∈(0,T),u(x,0)=u0(x),x ∈(0,L),where m, p, q, and L are positive parameters. For different values of the parameters three situations may occur: (1) all solutions of this problem exist for all t>0; (2) for certain initial data functions the solution exists for all t>0 while for others the solution blows up as t↗T for some finite T; (3) excepting the trivial solution when u0≡0, all solutions blow up as t↗T for some finite T. We identify in terms of the parameters which of them actually happens. For solutions which blow up we find the blow-up rate and the blow-up set. © 2002 Elsevier Science (USA).
Document Type: Research Article
Departamento de Matemáticas, U. Autónoma de Madrid, Madrid, 28049, Spain
Departamento de Matemática, F.C.E y N., UBA, (1428) Buenos Aires, Argentina
Publication date: September 1, 2002