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The Balance between Nonlinear Inwards and Outwards Boundary Flux for a Nonlinear Heat Equation

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Abstract:

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 tT for some finite T; (3) excepting the trivial solution when u0≡0, all solutions blow up as tT 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

Affiliations: 1: Departamento de Matemáticas, U. Autónoma de Madrid, Madrid, 28049, Spain 2: Departamento de Matemática, F.C.E y N., UBA, (1428) Buenos Aires, Argentina

Publication date: September 1, 2002

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