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

We consider the Cauchy problem for the Hardy parabolic equation ∂tu−Δu=|x|−γup with initial data u0 singular at some point z. Our main results show that, if z≠0, then the optimal strength of the admissible singularity of u0 at z for the solvability of the equation is the same as that of the Fujita e...

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Bibliographic Details
Published in:Journal of Differential Equations 2021-09, Vol.296, p.822-848
Main Authors: Hisa, Kotaro, Takahashi, Jin
Format: Article
Language:English
Subjects:
Hardy parabolic equation
Optimal singularity
Solvability
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Summary:We consider the Cauchy problem for the Hardy parabolic equation ∂tu−Δu=|x|−γup with initial data u0 singular at some point z. Our main results show that, if z≠0, then the optimal strength of the admissible singularity of u0 at z for the solvability of the equation is the same as that of the Fujita equation ∂tu−Δu=up. 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 ∂tu+(−Δ)θ/2u=|x|−γup with 0
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2021.06.011