A strongly singular parabolic problem on an unbounded domain

G. P. Trachanas; Nikolaos B. Zographopoulos

doi:10.3934/cpaa.2014.13.789

Article Contents

2014, Volume 13, Issue 2: 789-809. Doi: 10.3934/cpaa.2014.13.789

This issue Previous Article Positive solutions to involving Wolff potentials Next Article Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$

A strongly singular parabolic problem on an unbounded domain

G. P. Trachanas^1, and
Nikolaos B. Zographopoulos^2,

1.
Department of Mathematics, National Technical University of Athens, 15780, Athens
2.
Department of Mathematics & Engineering Sciences, Hellenic Army Academy, 16673, Athens

Received: March 31, 2013

Revised: August 31, 2013

Published: February 2014

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Abstract

We study the well-posedness and describe the asymptotic behavior of solutions of a strongly singular equation for the Cauchy problem on $R^N$. The strong singularity is exactly the critical case of the Caffarelli-Kohn-Nirenberg inequality. Moreover, we show the stabilization towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This equation is closely related to a heat equation with inverse-square potential, posed on $R^N$. In this case we have the appearance of the Hardy singularity energy.

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Mathematics Subject Classification: Primary: 35K67, 35P15; Secondary: 35A23.

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