A Parabolic Equation for the Fractional Laplacian in the Entire Space: Blow-Up of Nonnegative Solutions

Abstract

The main aim of the present paper is to investigate the conditions under which the nonnegative solutions blow-up for the parabolic problem $$ \frac{\partial u}{\partial t}=-{\left(-\Delta \right)}^{\frac{\alpha }{2}}u+\frac{c}{{\left|x\right|}^{\alpha }}u\kern1em \mathrm{in}\kern1em {\mathrm{\mathbb{R}}}^{\mathrm{d}}\times \left(0,T\right), $$ where 0 < α < min(2, d), $$ {\left(-\Delta \right)}^{\frac{\alpha }{2}} $$is the fractional Laplacian on ℝd and the initial condition u0 is in L2(ℝd).