A Note on a Mixed Pseudo-Parabolic Kirchhoff Equation with Logarithmic Damping

Abstract

This paper deals with a homogeneous Dirichlet initial-boundary value problem of the Kirchhoff equation of pseudo-parabolic type with logarithmic nonlinearity,

$$\begin{aligned} u_{t}-\Delta u_t-M(\Vert \nabla u\Vert _2^{2})\Delta u=|u|^{q-2}u \,{\log }|u|, \ (x,t) \in \Omega \times (0,T), \end{aligned}$$

where $M(s):=a+bs$, $a,b>0$, $q>2$; $\Omega \subset \mathbb {R}^N$ is a bounded domain with Lipschitz boundary. Firstly, we employ the extended Galerkin method to prove the local existence and uniqueness of weak solution. Secondly, for $q>4$, we show the criteria on the existence of blow-up solutions or global solutions, which depend on the choosing of the initial energy and Nehari energy. Thirdly, for $q+\mu \le 4$, we give the results on global solutions and large time estimate, where $\mu$ is a positive constant.