# Quadratic forms for the Liouville equation <var>w<sub>tt</sub></var> + <var>λ</var><sup>2</sup><var>a</var>(<var>t</var>)<var>w</var> = 0 with applications to Kirchhoff equation

### Renato Manfrin

Università IUAV di Venezia, Italy

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## Abstract

We introduce a set of quadratic forms for the solutions of the Liouville equation wtt + λ2a(t)w = 0. From these forms we derive estimates for the wave equation utt − a(t)Δu = 0 and then prove the global solvability for the Kirchhoff equation in suitable classes of not necessarily smooth or small initial data.

## Cite this article

Renato Manfrin, Quadratic forms for the Liouville equation <var>w<sub>tt</sub></var> + <var>λ</var><sup>2</sup><var>a</var>(<var>t</var>)<var>w</var> = 0 with applications to Kirchhoff equation. Port. Math. 65 (2008), no. 4, pp. 447–484

DOI 10.4171/PM/1821