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The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. (English) Zbl 0953.35053
The paper is concerned with the well-posedness and the asymptotic behavior of solutions of the heat equation with the potential: \(u_t=\Delta u +\lambda |x|^{-2} u\) for the Cauchy-Dirichlet problem in a bounded domain as well as for the Cauchy problem in \({\mathbb{R}}^N\). In the case of the bounded domain, the use of an improved form of the so-called Hardy-Poincaré inequality allows the authors to prove the exponential stabilisation towards a solution in separated variables. In \({\mathbb{R}}^N\), they first establish a new version of the Hardy-Poincaré inequality, and next, they show the stabilisation towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This work complements and explains the paper by P. Baras and J. A. Goldstein [Trans. Am. Math. Soc. 284, 121-139 (1984; Zbl 0556.35063)] on the existence of global solutions and blow-up for this equation. In the present article, the sign restriction on the data and solutions is removed, the functional framework for the well-posedness is described, and the asymptotic rates are calculated. Examples of non-uniqueness are also given.

MSC:
35K10 Second-order parabolic equations
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
Citations:
Zbl 0556.35063
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References:
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