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Spectral gap global solutions for degenerate Kirchhoff equations

Ghisi, Marina and Gobbino, Massimo (2009) Spectral gap global solutions for degenerate Kirchhoff equations. Nonlinear Analysis: Theory, Methods & Applications, 71/200 (9). pp. 4115-4124. ISSN 0362-546X

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    SUMMARY: We consider the second order Cauchy problem $$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that $u_{0}$ and $u_{1}$ are regular enough, depending on the continuity modulus of $m$, and on the strict/weak hyperbolicity of the equation. We prove that for such initial data $(u_{0},u_{1})$ there exist two pairs of initial data $(\overline{u}_{0},\overline{u}_{1})$, $(\widehat{u}_{0},\widehat{u}_{1})$ for which the solution is global, and such that $u_{0}=\overline{u}_{0}+\widehat{u}_{0}$, $u_{1}=\overline{u}_{1}+\widehat{u}_{1}$. This is a byproduct of a global existence result for initial data with a suitable spectral gap, which extends previous results obtained in the strictly hyperbolic case with a smooth nonlinearity $m$.

    Item Type: Article
    Uncontrolled Keywords: uniqueness, integro-differential hyperbolic equation, degenerate hyperbolic equation, continuity modulus, Kirchhoff equations, Gevrey spaces.
    Subjects: Area01 - Scienze matematiche e informatiche > MAT/05 - Analisi matematica
    Divisions: Dipartimenti (until 2012) > DIPARTIMENTO DI MATEMATICA "L. TONELLI"
    Depositing User: dott. Marina Ghisi
    Date Deposited: 22 Feb 2010
    Last Modified: 20 Dec 2010 11:47
    URI: http://eprints.adm.unipi.it/id/eprint/673

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