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2003 The Cauchy problem for the heat equation with a singular potential
Claudio Marchi
Differential Integral Equations 16(9): 1065-1081 (2003).


The aim of this paper is to investigate the well-posedness of the Cauchy problem $$ \begin{cases} \frac {\partial u}{\partial t} = \Delta u +V(x)u & \qquad \text {in }\mathbb R^N\times (0,T),\quad N\geq 3,\\ u(x,0) = u_0 (x), & \qquad \text {on }\mathbb R^N \end{cases} $$ where the potential $V$ is defined by $V=V(x):=\lambda/|x|^2$, $0\leq \lambda < (N-2)^2/4$. Roughly speaking, we prove that a sufficient condition for existence and uniqueness of the solution is to restrict the growths of the solution $u$ and of the initial datum $u_0$ as $ \vert x \vert \rightarrow \infty $ (at most like $e^{c|x|^2}$, with $c\in \mathbb R_+$) and near the origin (at most like $k|x|^\alpha$, with $k\in \mathbb R_+$, while $\alpha$ is a parameter depending on $\lambda$). For $\lambda >0$, the solution shall present a lack of regularity in the origin which is due only to the presence of the singular potential.


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Claudio Marchi. "The Cauchy problem for the heat equation with a singular potential." Differential Integral Equations 16 (9) 1065 - 1081, 2003.


Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1038.35022
MathSciNet: MR1989541

Primary: 35K15
Secondary: 35B30

Rights: Copyright © 2003 Khayyam Publishing, Inc.


Vol.16 • No. 9 • 2003
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