Nonlinear wave equation

A general nonlinear wave equation can be expressed as

(NWE)\;\;\left\{\begin{array}[]{ll}u_{tt}-u_{xx}=f(x,t,u)&\,\mbox{ in }(0,\pi)% \times\mathbb{R}\\ u(0,t)=u(\pi,t)=0&\,\forall t\in\mathbb{R}\end{array}\right.

where $f(x,t,s)$ is a continuous, monotonic increasing or decreasing (in $s$ ) real function on $[0,\pi]\times{\mathbb{R}}^{2}$ and $2\pi$ -periodic in $t$ . Nontrivial periodic solutions are of interest in many applications. Existence of multiple or even infinitely many periodic solutions to some special cases of (NWE), e.g., $f(x,t,s)$ is superlinear in $s$ and/or independent of the time $t$ , have been proved by many authors such as P. Rabinowitz, V. Benci, M. Berti, etc.

Title	Nonlinear wave equation
Canonical name	NonlinearWaveEquation1
Date of creation	2013-03-11 19:28:45
Last modified on	2013-03-11 19:28:45
Owner	linor (11198)
Last modified by	(0)
Numerical id	1
Author	linor (0)
Entry type	Definition