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Time-isolated Singularities of Temperatures

Part of: Partial differential equations Parabolic equations and systems Representations of solutions Higher-dimensional theory

Published online by Cambridge University Press:  09 April 2009

Neil A. Watson
Neil A. Watson
Affiliation:
Department of Mathematics and Statistics University of CanterburyChristchurchNew Zealand
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Abstract

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We study singularities of solutions of the heat equation, that are not necessarily isolated but occur only in a single characteristic hyperplane. We prove a decomposition theorem for certain solutions on D+ = D ∩ (Rn × ]0. ∞[), for a suitable open set D, with singularities at compact subset K of Rn × {0}, in terms of Gauss-Weierstrass integrals. We use this to prove a representation theorem for certain solutions on D+, with singularities at K, as the sums of potentials and Dirichlet solutions. We also give conditions under which K is removable for solutions on D∖K.

Keywords

TemperaturesingularityGauss-Weierstrass integralpotentialDirichlet solution

MSC classification

Secondary: 35K05: Heat equation 35B30: Dependence of solutions on initial and boundary data, parameters 35B60: Continuation and prolongation of solutions 35C05: Solutions in closed form 31B35: Connections with differential equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 416 - 429
Copyright
Copyright © Australian Mathematical Society 1998

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