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

Abstract

We consider the Cauchy problem for the Hardy parabolic equation ∂ t u − ∆u = |x| −γ u p with initial data u 0 singular at some point z. Our main results show that, if z = 0, then the optimal strength of the singularity of u 0 at z for the solvability of the equation is the same as that of the Fujita equation ∂ t u − ∆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 ∂ t u + (−∆) θ/2 u = |x| −γ u p with 0 < θ < 2.

Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces

“…Concerning the global dynamics and asymptotic behaviors, we refer to [4,13,14] for the Fujita and Hardy cases of (1.1) with Sobolevcritical exponents. Articles [10,11] give definitive results on the optimal singularity of initial data to assure the solvability for γ ≤ 0. In [22], unconditional uniqueness has been established for the Hardy case γ < 0.…”