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

In this paper, we consider the following class of heat equation involvingp(x)-Laplacian with logarithmic nonlinearity {u(t) - Delta(p(x))u = vertical bar u vertical bar(s(x)-2) u log(vertical bar u vertical bar) in Omega, t > 0, u = 0 in partial derivative Omega, t>0, u(x, 0) = u(0)(x), in Omega, where Omega subset of R-N (N >= 1) is a bounded domain with smooth boundary partial derivative Omega, p, s : (Omega) over bar -> R+ are continuous functions that satisfy some technical conditions and -Delta(p(x)) is the p(x)-Laplacian, which generalizes thep-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.