Global existence and blow-up results for some nonlinear wave equations on {$\mathbb R^N$}

Abstract

We discuss the asymptotic behaviour of solutions of the semilinear hyperbolic problem $$u_{tt} +\delta u_t -\phi (x)\Delta u = \lambda \, u|u|^{\beta -1}, \, \;\; x \in \mathbb R^N, \;\; t \geq 0,$$ 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 $(\phi (x))^{-1} =g (x)$, a positive function belonging to $L^{{N/2}}(\mathbb R^N)\cap L^{\infty}(\mathbb R^N )$. Under certain conditions we prove the global existence of solutions. Also we examine blow-up in finite time when the initial data are sufficiently large. The space setting of the problem is the energy space ${\mathcal X}_0={\mathcal D}^{1,2}(\mathbb R^N) \times L_{g}^{2}(\mathbb R^N)$, where $L_{g}^{2}$ is an appropriate weighted Hilbert space; see Section 2.