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Journal of Evolution Equations

, Volume 16, Issue 2, pp 441–482 | Cite as

Kirchhoff equations with strong damping

  • Marina Ghisi
  • Massimo GobbinoEmail author
Article

Abstract

We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the “elastic” operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space, assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space, we prove local existence and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces.

Keywords

Quasi-linear hyperbolic equation Degenerate hyperbolic equation Kirchhoff equation Global existence Strong damping Fractional damping Interpolation spaces 

Mathematics Subject Classification

35L70 35L80 35L90 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di PisaPisaItaly

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