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

This paper deals with a homogeneous Dirichlet initial-boundary value problem of the Kirchhoff equation of pseudo-parabolic type with logarithmic nonlinearity, \(\begin{aligned} u_{t}-\Delta u_t-M(\Vert \nabla u\Vert _2^{2})\Delta u=|u|^{q-2}u \,{\log }|u|, \ (x,t) \in \Omega \times (0,T), \end{aligned}\) $$\begin{array}{c}\hfill u_t-\mathrm{\Delta }{u}_t-{M(\Vert \mathrm{\nabla }u\Vert }_2^2{)\mathrm{\Delta }u=|u|}^{q-2}ulog|u|,(x,t)\in \mathrm{\Omega }\times (0,T),\end{array}$$ where \(M(s):=a+bs\) $M(s):=a+bs$ , \(a,b>0\) $a,b>0$ , \(q>2\) $q>2$ ; \(\Omega \subset \mathbb {R}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ 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 \(q>4\) $q>4$ , 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 \(q+\mu \le 4\) $q+\mu \le 4$ , we give the results on global solutions and large time estimate, where \(\mu \) $\mu$ is a positive constant.