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.