Quadratic forms for the Liouville equation $w_{tt}+{\lambda }^{2}a(t)w=0$ with applications to Kirchhoff equation

Abstract

We introduce a set of quadratic forms for the solutions of the Liouville equation $w_{tt}+{\lambda }^{2}a(t)w=0$. From these forms we derive estimates for the wave equation $u_{tt}-a(t)\Delta u=0$ and then prove the global solvability for the Kirchhoff equation in suitable classes of not necessarily smooth or small initial data.