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

  • Marina GhisiEmail author
  • Massimo Gobbino
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 44)

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.

Keywords

Hyperbolic-parabolic singular perturbation Nondegenerate hyperbolic equations Quasilinear hyperbolic equations Kirchhoff equations Decay-error estimates Linear equations with time-dependent coefficients 

Mathematics Subject Classification (2000)

35B25 35L72 35B40 

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Dipartimento di Matematica ApplicataUniversità di PisaPisaItaly

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