The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential

We study the well-posedness and describe the asymptotic behavior of solutio ns of the heat equation with inverse-square potentials for the Cauchy-Diric hlet problem in a bounded domain and also for the Cauchy problem in R-N. In the case of the bounded domain we use an improved form of the so-called Ha rdy-Poincare inequality and prove the exponential stabilization towards a s olution in separated variables. In R-N we first establish a new weighted ve rsion of the Hardy-Poincare inequality, and then show the stabilization tow ards a radially symmetric solution in self-similar variables with a polynom ial decay rare. This work complements and explains well-known work by Baras and Goldstein on the existence of global solutions and blow-up for these e quations. In the present article the sign restriction on the data and solut ions is removed, the functional framework For well-posedness is described, and the asymptotic rates calculated. Examples of non-uniqueness are also gi ven. (C) 2000 Academic Press.