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

✍ Scribed by Juan Luis Vazquez; Enrike Zuazua

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 304 KB
• Volume: 173
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3556

✦ Synopsis

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 R N . In the case of the bounded domain we use an improved form of the so-called Hardy Poincare inequality and prove the exponential stabilization towards a solution in separated variables. In R N we first establish a new weighted version of the Hardy Poincare 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.