Blow-up sets for linear diffusion equations in one dimension

Author： Quirós Fernando  

Publisher： Springer Publishing Company

ISSN： 0044-2275

Source： Zeitschrift für angewandte Mathematik und Physik ZAMP, Vol.55, Iss.2, 2004-03, pp. : 357-362

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Abstract

We consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval [0,a], depending on the Dirichlet data, we prove that the effective blow-up set, that is, the set of points where the solution behaves like u(0,t), consists always only of the origin. As an application of our results we consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. We show that by prescribing the non-linearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.

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