Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Global solution to a non-classical heat problem in the semi-space $ \mathbb{R}^{+}\times\mathbb{R}^{n-1}$

Authors: Mahdi Boukrouche and Domingo A. Tarzia
Journal: Quart. Appl. Math. 72 (2014), 347-361
MSC (2010): Primary 35C15, 35K05, 35K20, 35K60, 80A20
DOI: https://doi.org/10.1090/S0033-569X-2014-01344-1
Published electronically: March 14, 2014
MathSciNet review: 3186241
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Abstract: We consider the non-classical heat equation in the $ n$-dimensional domain $ D=\mathbb{R}^{+}\times \mathbb{R}^{n-1}$ for which the internal energy supply depends on the heat flux on the boundary $ S=\partial D$. The problem is motivated by the modeling of temperature regulation in the medium. Using the Green function for the domain $ D$, the solution is found for an integral representation depending on the heat flux $ V$ on $ S$ which is an additional unknown of the problem. We obtain that $ V$ must satisfy a Volterra integral equation of second kind at time $ t$ with a parameter in $ \mathbb{R}^{n-1}$. Under some conditions on data, we show that there exists a unique local solution which can be extended globally in time. This work generalizes the results obtained in the one-dimensional case.

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Mahdi Boukrouche
Affiliation: Lyon University, F-42023 Saint-Etienne, Institut Camille Jordan CNRS UMR 5208, 23 rue Paul Michelon 42023 Saint-Etienne Cedex 2, France
Email: Mahdi.Boukrouche@univ-st-etienne.fr

Domingo A. Tarzia
Affiliation: Departamento de Matemática-CONICET, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina
Email: DTarzia@austral.edu.ar

DOI: https://doi.org/10.1090/S0033-569X-2014-01344-1
Keywords: Non-classical $n$-dimensional heat equation, Volterra integral equation, existence and uniqueness of solution, integral representation of the solution
Received by editor(s): July 19, 2012
Published electronically: March 14, 2014
Article copyright: © Copyright 2014 Brown University

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