Global Existence and Blow-Up for a Parabolic Problem of Kirchhoff Type with Logarithmic Nonlinearity

Abstract

In this paper, we study the following parabolic problem of Kirchhoff type with logarithmic nonlinearity:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {u_t} +{M([u]^2_s){\mathcal {L}}_Ku}={|u|^{p-2}u\log |u|},\ \ \ &{}\hbox { in } \Omega \times (0,+\infty ),\\ \displaystyle u(x,t)=0,&{}\hbox { in }({\mathbb {R}}^N\setminus \Omega )\times (0,+\infty ),\\ \displaystyle u(x,0)=u_0(x),&{}\hbox { in }\Omega , \end{array}\right. \end{aligned}$$

where $[u]_s$ is the Gagliardo seminorm of u, $\Omega \subset {\mathbb {R}}^N$ is a bounded domain with Lipschitz boundary, $0<s<1$, ${\mathcal {L}}_K$ is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator $(-\Delta )^s$, $u_0$ is the initial function, and $M:[0,+\infty )\rightarrow [0,+\infty )$ is continuous. Let $J(u_0)$ be the initial energy (see (2.1) for the definition of J), $d>0$ be the mountain-pass level given in (2.4), and ${\widetilde{M}}\in (0,d]$ be the constant defined in (2.6). Firstly, we get the conditions on global existence and finite time blow-up for $J(u_0)\le d$. Then we study the lower and upper bounds of blow-up time to blow-up solutions under some appropriate conditions. Secondly, for $J(u_0)\le {\widetilde{M}}$, the growth rate of the solution is got. Moreover, we give some blow-up conditions independent of d and study the upper bound of the blow-up time. Thirdly, the behavior of the energy functional as $t\rightarrow T$ is also discussed, where T is the blow-up time. In addition, for $J(u_0)\le d$, we give some equivalent conditions for the solutions blowing up in finite time or existing globally. Finally, we consider the existence of ground state solutions and the asymptotical behavior of the general global solution.