Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian

Abstract

In this paper, we are concerned with a wave problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following hyperbolic problem involving the fractional Laplacian

$$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt} +[u]^{2 (\theta -1)}_{s}(-\Delta )^su=|u|^{p-1}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\quad u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N {\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array}\right. } \end{aligned}$$

where $[u]_{s}$ is the Gagliardo seminorm of u, $s\in (0,1)$, $\theta \in [1, 2_s^*/2)$, with $2_s^*=2N/(N-2s)$, $p\in (2\theta -1, 2_s^*-1]$, $\Omega \subset {\mathbb {R}}^N$ is a bounded domain with Lipschitz boundary $\partial \Omega$, $(-\Delta )^s$ is the fractional Laplacian. Under some appropriate assumptions, we obtain the global existence, vacuum isolating and blowup of solutions for the above problem by combining the Galerkin method with potential wells theory. Finally, we investigate the existence of global solutions for the above problem with the critical initial conditions. The significant feature and difficulty of the above problem are that the coefficient of $(-\Delta )^s$ can vanish at zero.