On the diffusion p(x)-Laplacian with logarithmic nonlinearity

Abstract

In this paper, we consider the following class of heat equation involving p(x)-Laplacian with logarithmic nonlinearity $$\begin{aligned}\left\{ \begin{array}{llc} u_{t}-\Delta _{p(x)}u=|u|^{s(x)-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} \partial \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset \mathbb {R}^{N}$$ ( $$N\ge 1$$ ) is a bounded domain with smooth boundary $$\partial \Omega $$ , $$p, s: \overline{\Omega }\rightarrow \mathbb {R}_{+}$$ are continuous functions that satisfy some technical conditions and $$-\Delta _{p(x)}$$ is the $$p(x)-$$ Laplacian, which generalizes the $$p-$$ Laplacian operator $$-\Delta _{p}$$ . The local existence will be done by using the Galerkin method. Then, by using the concavity method we prove that the local solutions blow-up in finite time under suitable conditions. In order to prove the global existence, we will use the potential well theory combined with the Pohozaev manifold that is a novelty for this type of problem. The difficulty here is the lack of logarithmic Sobolev inequality which seems there is no logarithmic Sobolev inequality concerning the p(x)-Laplacian yet.