Hyperbolic--parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation

Marina Ghisi; Massimo Gobbino

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arXiv · 0901.0185

Hyperbolic--parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation

Marina Ghisi·Massimo Gobbino

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Abstract

We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t tends to +infinity. The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t tends to +infinity and as epsilon goes to 0. In this paper we show that these results cannot be true for p>1, but they remain true in the critical case p=1.

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Marina Ghisi, Massimo Gobbino. 2009-01-01. Hyperbolic--parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation. https://arxiv.org/abs/0901.0185

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