Global existence and blow-up results for an equation of Kirchhoff type on $\mathbb R^N$

Abstract

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type $$ u_{tt}-\phi (x)\Vert\nabla u(t)\Vert^{2}\Delta u+\delta u_{t} = |u|^{a}u,\quad x\in {\mathbb R}^N,\ t\geq 0,$$ with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3$, $\delta \geq 0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(\mathbb R^N)\cap L^{\infty}(\mathbb R^N )$. When the initial energy $ E(u_{0},u_{1})$, which corresponds to the problem, is non-negative and small, there exists a unique global solution in time. When the initial energy $E(u_{0},u_{1})$ is negative, the solution blows-up in finite time. A combination of the modified potential well method and the concavity method is widely used.