# Nonlinear wave equation

Nonlinear wave equation

A general nonlinear wave equation can be expressed as

$$(NWE)\{\begin{array}{cc}{u}_{tt}-{u}_{xx}=f(x,t,u)\hfill & \text{in}(0,\pi )\times \mathbb{R}\hfill \\ u(0,t)=u(\pi ,t)=0\hfill & \forall t\in \mathbb{R}\hfill \end{array}$$ |

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 |