JournalspmVol. 65 / No. 4DOI 10.4171/pm/1821

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.