Existence of Nonnegative Solutions for a Fractional Parabolic Equation in the Entire Space

Abstract

We study the existence of nonnegative solutions of a parabolic problem $$ \frac{\partial u}{\partial t}=-{\left(-\Delta \right)}^{\frac{\alpha }{2}}u+\frac{c}{{\left|x\right|}^{\alpha }}u $$ in ℝd × (0, T). Here, 0 < α < min(2, d), $$ {\left(-\Delta \right)}^{\frac{\alpha }{2}} $$ is the fractional Laplacian on ℝd and u0 ∈ L2(ℝd).