Differential and Integral Equations

The Cauchy problem for the heat equation with a singular potential

Claudio Marchi

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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.

Article information

Differential Integral Equations, Volume 16, Number 9 (2003), 1065-1081.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K15: Initial value problems for second-order parabolic equations
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]


Marchi, Claudio. The Cauchy problem for the heat equation with a singular potential. Differential Integral Equations 16 (2003), no. 9, 1065--1081. https://projecteuclid.org/euclid.die/1356060557

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