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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
DOI: https://doi.org/10.1090/S0002-9947-1984-0742415-3
MathSciNet review: 742415
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Abstract: Of concern is the singular problem $\partial u/\partial t = \Delta u + (c/|x{|^2}) u + f(t,x), u(x,0) = u_{0}(x)$, 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