## Differential and Integral Equations

### On smooth global solutions of a Kirchhoff type equation on unbounded domains

#### Abstract

We consider the quasilinear hyperbolic equation $$u_{tt}-M\Bigl( \int_\Omega|\text{ grad }u|^2\,dx\Bigr)\,\Delta u=0, \tag1$$ where $x\in\Omega=\Bbb R^n$, $t$ denotes time and $M(s)$ is a smooth function satisfying $M(s)>0$ for all $s\ge 0$. We prove that there are no non-trivial breathers" for equation (1). Here, a "breather" means a time periodic solution which is "small" as $|x|\to +\infty$. We also present a simpler proof of the so-called Pohozaev's second conservation law for (1) solving the global Cauchy problem for "non-physical" nonlinearity $M$ arising from this conservation law.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 6 (1995), 1571-1583.

Dates
First available in Project Euclid: 15 May 2013

https://projecteuclid.org/euclid.die/1368638182

Mathematical Reviews number (MathSciNet)
MR1329857

Zentralblatt MATH identifier
0822.35087

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations

#### Citation

Menzala, Gustavo Perla; Pereira, Jardel Morais. On smooth global solutions of a Kirchhoff type equation on unbounded domains. Differential Integral Equations 8 (1995), no. 6, 1571--1583. https://projecteuclid.org/euclid.die/1368638182