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On the Parabolic Regime of a Hyperbolic Equation with Weak Dissipation: The Coercive Case

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Progress in Partial Differential Equations

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 44))

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Abstract

We consider a family of Kirchhoff equations with a small parameter ε in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 as t→+∞.

It is well-known that, when the decay of the coefficient is slow enough, solutions behave as solutions of the corresponding parabolic equation, and in particular they decay to 0 as t→+∞.

In this paper we consider the nondegenerate and coercive case, and we prove optimal decay estimates for the hyperbolic problem, and optimal decay-error estimates for the difference between solutions of the hyperbolic and the parabolic problem. These estimates show a quite surprising fact: in the coercive case the analogy between parabolic equations and dissipative hyperbolic equations is weaker than in the noncoercive case.

This is actually a result for the corresponding linear equations with time-dependent coefficients. The nonlinear term comes into play only in the last step of the proof.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Marina Ghisi

  2. Dipartimento di Matematica Applicata, Università di Pisa, Pisa, Italy

    Massimo Gobbino

Authors
  1. Marina Ghisi
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  2. Massimo Gobbino
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Corresponding author

Correspondence to Marina Ghisi .

Editor information

Editors and Affiliations

  1. Institut of Applied Analysis, TU Bergakademie Freiberg, Prüferstr. 9, Freiberg, 09596, Sachsen, Germany

    Michael Reissig

  2. Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, United Kingdom

    Michael Ruzhansky

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Ghisi, M., Gobbino, M. (2013). On the Parabolic Regime of a Hyperbolic Equation with Weak Dissipation: The Coercive Case. In: Reissig, M., Ruzhansky, M. (eds) Progress in Partial Differential Equations. Springer Proceedings in Mathematics & Statistics, vol 44. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00125-8_5

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