The Hardy Inequality and the Asymptotic Behaviour of the Heat Equation with an Inverse-Square Potential

Abstract

We study the well-posedness and describe the asymptotic behavior of solutions of the heat equation with inverse-square potentials for the Cauchy–Dirichlet problem in a bounded domain and also for the Cauchy problem in RN. In the case of the bounded domain we use an improved form of the so-called Hardy–Poincaré inequality and prove the exponential stabilization towards a solution in separated variables. In RN we first establish a new weighted version of the Hardy–Poincaré inequality, and then show the stabilization towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This work complements and explains well-known work by Baras and Goldstein on the existence of global solutions and blow-up for these equations. In the present article the sign restriction on the data and solutions is removed, the functional framework for well-posedness is described, and the asymptotic rates calculated. Examples of non-uniqueness are also given.