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Proceedings of the American Mathematical Society

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An approximation theorem of Runge type for the heat equation


Author: B. Frank Jones
Journal: Proc. Amer. Math. Soc. 52 (1975), 289-292
MSC: Primary 35K05
DOI: https://doi.org/10.1090/S0002-9939-1975-0387815-9
MathSciNet review: 0387815
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Abstract: If $\Omega$ is an open subset of ${{\mathbf {R}}^{n + 1}}$, the approximation problem is to decide whether every solution of the heat equation on $\Omega$ can be approximated by solutions defined on all of ${{\mathbf {R}}^{n + 1}}$. The necessary and sufficient condition on $\Omega$ which insures this type of approximation is that every section of $\Omega$ taken by hyperplanes orthogonal to the $t$-axis be an open set without “holes,” i.e., whose complement has no compact component. Part of the proof involves the Tychonoff counterexample for the initial value problem.


References [Enhancements On Off] (What's this?)

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Keywords: Nonuniqueness for initial value problem, simply connected, exponential-polynomial solutions
Article copyright: © Copyright 1975 American Mathematical Society