Hyperbolic–parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates

By Marina Ghisi, Massimo Gobbino. 2008

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We consider degenerate Kirchhoff equations with a small parameter e in front of the second-order time-derivative. It is well known that these equations admit global solutions when e is small enough, and that these solutions decay as t→+∞ with the same rate of solutions of the limit problem (of parabolic type). In this paper we prove decay-error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem. These estimates show in the same time that the difference tends to zero both as e→0+, and as t→+∞. Concerning the decay rates, it turns out that the difference decays faster than the two terms separately (as t→+∞). Proofs involve a nonlinear step where we separate Fourier components with respect to the lowest frequency, followed by a linear step where we exploit weighted versions of classical energies.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4d9d599edf3d4b2243b27378723c1156997f8367