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

Abstract

In this paper we study the local and global existence of solutions for a class of heat equation in whole \\begin{document}$ \\mathbb{R}^{N} $\\end{document} where the nonlinearity has a critical growth for \\begin{document}$ N \\geq 2 $\\end{document} . 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.