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

2021/02/08 by Kotaro Hisa, Hisa, Kotaro, Jin Takahashi +1 · 2 citations
Engineering · Mathematics · #35K15 #35K58 (Primary) 35A01 #35K67 (Secondary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2102.04618

openalex publication_date 2021/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem for the Hardy parabolic equation \∂_tν-\Δ u=|x|^-\γu^p with initial data u₀ singular at some point\nz. Our main results show that, if z\≠ 0, then the optimal strength of the\nsingularity of u₀ at z for the solvability of the equation is the same as\nthat of the Fujita equation \∂_t u-\Δ u=u^p. Moreover, if z=0,\nthen the optimal singularity for the Hardy parabolic equation is weaker than\nthat of the Fujita equation. We also obtain analogous results for a fractional\ncase \∂_t u+(-\Δ)^\θ/2 u=|x|^-\γu^p with 0<\θ<2.\n

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