Existence of solution for a class of heat equation in whole $ \mathbb{R}^N $

Claudianor O. Alves; Tahir Boudjeriou

doi:10.3934/dcds.2021031

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2021, Volume 41, Issue 9: 4125-4144. Doi: 10.3934/dcds.2021031

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Existence of solution for a class of heat equation in whole $ \mathbb{R}^N $

Claudianor O. Alves^1, and
Tahir Boudjeriou^2, ,

1.
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB, Brazil
2.
Department of Mathematics, Faculty of Exact Sciences, Lab. of Applied Mathematics, University of Bejaia, Bejaia, 06000, Algeria

^* Corresponding author: Tahir Boudjeriou
^* Corresponding author: Tahir Boudjeriou

Received: March 2020

Revised: December 2020

Early access: February 2021

Published: September 2021

C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7

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Abstract

In this paper we study the local and global existence of solutions for a class of heat equation in whole $ \mathbb{R}^{N} $ where the nonlinearity has a critical growth for $ N \geq 2 $. In order to prove the global existence, we will use the potential well theory combined with the Nehari manifold, and also with the Pohozaev manifold that is a novelty for this type of problem. Moreover, the blow-up phenomena of local solutions is investigated by combining the subdifferential approach with the concavity method.

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Mathematics Subject Classification: 35K60, 34B10, 35J15.

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