On smooth global solutions of a Kirchhoff type equation on unbounded domains. (English) Zbl 0822.35087

Summary: We consider the quasilinear hyperbolic equation \[ u_{tt}- M\biggl( \int_ \Omega |\text{grad } u|^ 2 dx\biggr) \Delta u= 0, \tag{1} \] where \(x\in \Omega= \mathbb{R}^ n\), \(t\) denotes time and \(M(s)\) is a smooth function satisfying \(M(s) >0\) for all \(s\geq 0\). 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.


35L15 Initial value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations