ON THE EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR SOME NONLINEAR WAVE EQUATIONS OF KIRCHHOFF TYPE

Abstract

In this work, we consider the following nonlinear problem \begin{gather*} u_{tt} - M(\| \nabla u(t) \|^{2}_2) \Delta u - \frac{\partial}{\partial t} \Delta u = f(u), \\ u = 0 \text{ \ in } \Gamma_{0} \times (0,T), \\ M(\| \nabla u(t) \|^{2}_2) \frac{\partial u}{\partial \nu} + \frac{\partial}{\partial t} (\frac{\partial u}{\partial \nu}) = -u_{t} \text{ \ in } \Gamma_{1} \times (0,T), \\ u(x,0) = u_{0}(x), \text{ \ } u_{t}(x,0) = u_{1}(x), \end{gather*} in a bounded domain $\Omega$. The existence, asymptotic behavior and nonexistence of solutions are discussed under some conditions.