Nonlinear wave equation

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 NonlinearWaveEquation1 2013-03-11 19:28:45 2013-03-11 19:28:45 linor (11198) (0) 1 linor (0) Definition