Degenerate Kirchhoff-type fractional diffusion problem with logarithmic nonlinearity

Abstract

In this paper, we consider the following Kirchhoff-type diffusion problem involving the fractional Laplacian and logarithmic nonlinearity at high initial energy level: $$\begin{array}{l}\hfill \left\{\begin{array}{cc}{\mathit{u}}_{\mathit{t}}+{[\mathit{u}]}_{\mathit{s}}^{2\mathrm{(}\mathit{\theta }-1\mathrm{)}}{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}^{\mathit{s}}\mathit{u}=|\mathit{u}{|}^{\mathit{q}-2}\mathit{u}ln|\mathit{u}|\hfill & \mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\in \mathrm{\Omega }\times {\mathbb{R}}^{+}\mathrm{,}\hfill \\ \mathit{u}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}=0\hfill & \mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\in \mathrm{(}{\mathbb{R}}^{\mathit{N}}\setminus \mathrm{\Omega }\mathrm{)}\times {\mathbb{R}}^{+}\mathrm{,}\hfill \\ \mathit{u}\mathrm{(}\mathit{x}\mathrm{,}0\mathrm{)}={\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\hfill & \mathit{x}\in \mathrm{\Omega }\mathrm{,}\hfill \end{array}\right.\end{array}$$ where ${\mathrm{(}-\mathrm{\Delta }\mathrm{)}}^{\mathit{s}}$ is the fractional Laplacian with $\mathit{s}\in \mathrm{(}0\mathrm{,}1\mathrm{)}\mathrm{,}\mathit{N}\mathrm{>}2\mathit{s}$ , ${[\mathit{u}]}_{\mathit{s}}$ is the Gagliardo seminorm of u, $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{N}}$ is a bounded domain with Lipschitz boundary, $1⩽\mathit{\theta }\mathrm{<}\mathit{N}\mathrm{/}\mathrm{(}\mathit{N}-2\mathit{s}\mathrm{)}$ , $2\mathit{\theta }\mathrm{<}\mathit{q}\mathrm{<}{2}_{\mathit{s}}^{\ast }$ . Based on the potential well theory, a sufficient condition is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time under some appropriate assumptions. In particular, the existence of ground state solutions for the above stationary problem is obtained by restricting the related discussion on Nehari manifold.