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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The heat equation with a singular potential

Authors: Pierre Baras and Jerome A. Goldstein
Journal: Trans. Amer. Math. Soc. 284 (1984), 121-139
MSC: Primary 35K05; Secondary 60J65
MathSciNet review: 742415
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Abstract: Of concern is the singular problem $ \partial u/\partial t = \Delta u + (c/\vert x{\vert^2})\,u + f(t,x), u(x,0) = u_{0}(x)$, and its generalizations. Here $ c \geqslant 0,x \in {{\mathbf{R}}^N},t > 0$, and $ f$ and $ {u_0}$ are nonnegative and not both identically zero. There is a dimension dependent constant $ {C_{\ast} }(N)$ such that the problem has no solution for $ c > {C_{\ast} }(N)$. For $ c \leqslant {C_{\ast} }(N)$ necessary and sufficient conditions are found for $ f$ and $ {u_0}$ so that a nonnegative solution exists.

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Keywords: Heat equation, singular potential, inverse square potential, nonexistence, Feynman-Kac formula
Article copyright: © Copyright 1984 American Mathematical Society