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January/February 2022 On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior
Gael Diebou Yomgne
Differential Integral Equations 35(1/2): 57-88 (January/February 2022).

Abstract

This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\mathbb R^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\mathbb R^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution $u\equiv 0$ is shown to be asymptotically stable in $L^{q_c}(\mathbb R^{n})$ -- it is the only self-similar solution which is initially small in $L^{q_c}(\mathbb R^{n})$. Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.

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Gael Diebou Yomgne. "On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behavior." Differential Integral Equations 35 (1/2) 57 - 88, January/February 2022.

Information

Published: January/February 2022
First available in Project Euclid: 6 January 2022

Subjects:
Primary: 35B35 , 35B44 , 35C06 , 35G25

Rights: Copyright © 2022 Khayyam Publishing, Inc.

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Vol.35 • No. 1/2 • January/February 2022
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