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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
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.

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

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