A strongly singular parabolic problem on an unbounded domain

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.