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

Abstract

<p style='text-indent:20px;'>In this paper we study the local and global existence of solutions for a class of heat equation in whole <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula> where the nonlinearity has a critical growth for <inline-formula><tex-math id="M2">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula>. 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.