JournalspmVol. 65, No. 4pp. 447–484

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
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 cover

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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