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SINGULAR SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR ABSORPTION AND HARDY POTENTIALS

VITALI LISKEVICH

Department of Mathematics, University of Swansea, Swansea SA2 8PP, UK

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ANDREY SHISHKOV

Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine

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ZEEV SOBOL

Department of Mathematics, University of Swansea, Swansea SA2 8PP, UK

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https://doi.org/10.1142/S0219199712500137Cited by:5 (Source: Crossref)

Abstract

We study the existence and nonexistence of singular solutions to the equation , p > 1, in ℝ^N × [0, ∞), N ≥ 3, with a singularity at the point (0, 0), that is, nonnegative solutions satisfying u(x, 0) = 0 for x ≠ 0, assuming that α > -2 and . The problem is transferred to the one for a weighted Laplace–Beltrami operator with a nonlinear absorption, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a source solution with a multiple of the Dirac mass as initial datum, or as a unique very singular solution, leads to a complete classification of singular solutions to the original problem, which exist if and only if .

Keywords:

AMSC: 35K15, 35K58, 35K90, 35B25

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