A Bifurcation Problem for a Nonlinear Hyperbolic Partial Differential Equation

Abstract

The stability of equilibrium solutions of the damped nonlinear wave equation $w_{tt} + 2\alpha w_t - w_{xx} - \lambda f(w) = 0$, $\alpha > 0$, $\lambda \geqq 0$, is investigated using Lyapunov stability techniques. Under appropriate conditions on f it is shown that for $\lambda _n < \lambda \leqq \lambda _{n + 1} $, ${{\lambda _n = n^2 } /{f'(0)}}$, $n = 0,1,2, \cdots $, there are exactly $2n + 1$ equilibrium solutions and all solutions exist globally and approach exactly one of them as t approaches $\infty $.