Optimal singularities of initial data for solvability of the Hardy parabolic equation

[摘要] We consider the Cauchy problem for the Hardy parabolic equation partial derivative(t)u - Delta u = vertical bar x vertical bar(-gamma) u(p) with initial data u(0) singular at some point z. Our main results show that, if z not equal 0, then the optimal strength of the admissible singularity of u(0) at zfor the solvability of the equation is the same as that of the Fujita equation partial derivative(t)u - Delta u = u(p). Moreover, if z = 0, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case partial derivative(t)u +(-Delta)(theta/2)u = vertical bar x vertical bar(-gamma) upwith 0 < theta < 2. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-09-25

[关键词] Hardy parabolic equation;Solvability;Optimal singularity