The heat equation for the Dirichlet fractional Laplacian with negative potentials: Existence and blow-up of nonnegative solutions

Abstract

Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of R d are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.

Ground-state representation for fractional Laplacian on half-line

“…The subject of this paper is also strictly connected with Schrödinger perturbations of semigroups by Hardy potentials; see e.g. [1,2,11,25,5,17,15,18]. In particular, in [11] the authors studied the Schrödinger perturbations of the fractional Laplacian in R d by κ|x| −α and obtained sharp estimates of the perturbed semigroup.…”

Relativistic stable operators with critical potentials

“…Cholewa et al [26] studied the parabolic equation (also of order greater than 2) in the context of homogeneity. We also refer to BenAmor [7], Chen and Weth [22], Jakubowski and Maciocha [43] for the fractional Laplacian with Hardy potential on subsets of R d , and to Frank et al [34], where the operator jxj ˇ. . / ˛=2 C Äjxj ˛ 1/, for certain ˇ> 0, was treated.…”

Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

“…In [1,2] for κ > κ * := 2 α Γ((d+α)/4) 2 Γ((d−α)/4) 2 the phenomenon of instantaneous blow up of heat kernel was proven. In [5], the author gives the upper bound for the heat kernel of L with the Dirichlet conditions on bounded open subsets of R d . In the recent paper [8], the following sharp estimates for the heat kernelp(t, x, y) of L were obtained.…”