A parabolic equation for the fractional Laplacian in the whole space: blow-up of nonnegative solutions

Abstract

UDC 517.9
The main aim of the present paper is to investigate under what conditions the nonnegative solutions blow-up for the parabolic problem $\dfrac{\partial u}{\partial t} = - (-\triangle)^{\frac{\alpha}{2}}u + \dfrac{c}{|x|^{\alpha}}u$ in $\mathbb{R}^{d}\times (0 , T),$ where $0<\alpha<\min(2,d),$ $(-\triangle)^{\frac{\alpha}{2}}$ is the fractional Laplacian on $\mathbb{R}^{d}$ and the initial condition $u_{0}$ is in $L^{2}(\mathbb{R}^{d}).$