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

By Juan Luis V'azquez, E. Zuazua. 2000

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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–Poincare inequality and prove the exponential stabilization towards a solution in separated variables. In RN 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/33ca02789d8ece36ff497c679c9b0c45b89e7b7d

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Citations in the corpus

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains (2006, methodology): “e closure of C∞ 0 (Ω) in the norm u → ||u||H(Ω) := ˆZ Ω |∇u|2 − (N −2)2 4|x|2 u2 dx ˙1 2 . (1.5) Let us recall that H(Ω) ⊂ W1,q 0 (Ω) for any 1 ≤ q < 2, due to the results in Subsection 4.1 of [VZ]. It follows, using Hardy inequality, that K is a nonnegative self-adjoint operator on L2(Ω) such ”
Caffarelli-Kohn-Nirenberg inequalities with remainder terms (2003, methodology): “Our approach is quite different from that in [5,14], in some sense simpler and easier to be adapted for the weighted versions.”
Nonexistence of local solutions to semilinear partial differential inequalities (2004, background): “4) and semilin problems involving an inverse square potential (in particular, see [9,22,23]).”
Fractional Laplacian with Hardy potential (2017, methodology): “[36] J. L. Vazquez and E. Zuazua.”
The extremal solution of a boundary reaction problem (2008, result): “We will give here a di fferent proof of this result which o ffers a sharper version, analogous to improvements of (8) found by Brezis and V ázquez [4] or V́azquez and Zuazua [24] (see also [2, 4, 13, 19, 24] for other improved versions of the Hardy inequality (8)) : Theorem 1.”
Critical Hardy–Sobolev inequalities (2006, methodology): “local estimates in a neighborhood of K, see Theorem 5.1. For other directions in improving Hardy inequalities we refer to [AE], [BV], [BFT], [BM], [CM], [DELV], [GP], [HHL], [M], [MMP], [Ti1], [Ti2], [VZ] and references therein. The paper is organized as follows. In section 2 we establish auxiliary ”
Improved Hardy and Rellich inequalities on Riemannian manifolds (2007, unknown): “[BG], [BV], [GP], [ VZ ], [CM], [GZ], [D], [K1] and references therein.”
Hardy-type inequalities related to degenerate elliptic differential operators (2006, unknown): “[9, 12, 28, 32, 45, 46, 49].”
Optimizing Improved Hardy Inequalities (2002, methodology): “For instance, they have been useful in the study of the stability of solutions of semilinear elliptic and parabolic equations (cf [BV], [CM1] [PV], [V]), in the existence and asymptotic behavior of the heat equation with singular potentials, (cf [CM2], [ VZ ]), as well as in the study of the stabili”
Some improved Caffarelli-Kohn-Nirenberg inequalities (2005, background): “V´azquez-Zuazua in [ 15 ], among other results, improve the previous inequality by showing that if 1 0 such that,”