Existence and nonexistence of global solutions to the Cauchy problem of thenonlinear hyperbolic equation with damping term

Abstract

This paper concerns with the Cauchy problem for two classes of nonlinear hyperbolic equations with double damping terms. Firstly, by virtue of the Fourier transform method, we prove that the Cauchy problem of a class of high order nonlinear hyperbolic equation admits a global smooth solution $u(x, t)\in C^{\infty}((0, T]; H^{\infty}(\mathbb{R}))$$\bigcap C([0, T]; H^{3}(\mathbb{R}))$$\bigcap C^{1}([0, T]; H^{-1}(\mathbb{R}))$ as long as initial value $u_{0}\in W^{4, 1}(\mathbb{R})\bigcap H^{3}(\mathbb{R}), u_{1}\in L^{1}(\mathbb{R})\bigcap H^{-1}(\mathbb{R})$. Moreover, we give the sufficient conditions on the blow-up of the solution of a nonlinear damped hyperbolic equation with the initial value conditions in finite time and an example.