Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms

Abstract

In this paper, we consider the following Kirchhoff-type wave problems, with nonlinear damping and source terms involving the fractional Laplacian, $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} u_{tt} +[u]^{2\gamma -2}_{s}(-\Delta )^su+|u_t|^{a-2}u_t+u=|u|^{b-2}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\ \ \ \ u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N{\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array} \right. \end{aligned}$$ where $$(-\Delta )^s$$ is the fractional Laplacian, $$[u]_{s}$$ is the Gagliardo semi-norm of u, $$s\in (0,1)$$ , $$2<a<2\gamma<b<2_s^*=2N/(N-2s)$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary $$\partial \Omega $$ . Under some natural assumptions, we obtain the global existence, vacuum isolating, asymptotic behavior and blowup of solutions for the problem above by combining the Galerkin method with potential wells theory. The significant feature and difficulty of the problem are that the coefficient of $$(-\Delta )^s$$ can vanish at zero.