Existence of sign-changing solutions for the Hénon type parabolic equation with singular initial data

Qiuping Geng; Jun Wang

doi:10.3934/cpaa.2024005

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2024, Volume 23, Issue 2: 165-194. Doi: 10.3934/cpaa.2024005

This issue Previous Article Multi-parameter perturbations for the space-periodic heat equation Next Article Multiplicity results of periodic solutions for a coupled system of wave equations

Existence of sign-changing solutions for the Hénon type parabolic equation with singular initial data

Qiuping Geng and
Jun Wang^,

School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu, 212013, China

^*Corresponding author: Jun Wang
^*Corresponding author: Jun Wang

Received: April 2023

Revised: December 2023

Early access: January 25, 2024

Published online: January 25, 2024

The first author is supported by the National Natural Science Foundation of China (grant No.12301134). The second author is supported by National Key R&D Program of China (2022YFA1005601) and the National Natural Science Foundation of China (grant Nos.11971202 and 12071185).

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Abstract

In the present paper, we study the Hénon type parabolic equation $ \phi_t-\Delta \phi = |x|^\beta |\phi|^{\alpha-1}\phi, x\in\mathbb{R}^{N} $ with singular initial data $ \eta|x|^{-\frac{2+\beta}{\alpha-1}} $, where $ \eta, \beta>0, \alpha>1 $. We prove the existence of infinitely many sign-changing self-similar solutions by analyzing the related inverted profile equation for $ N\geq1 $. This can be seen as an extension of the recent work of T. Cazenave at al.(Amer. J. Math. 142 (2020), 1439-1495) to the Hénon type parabolic equation.

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Mathematics Subject Classification: Primary: 35K15, 35J15, 34A12; Secondary: 35C06.

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