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.
CITATION STYLE
Ghisi, M., & Gobbino, M. (2013). On the parabolic regime of a hyperbolic equation with weak dissipation: The coercive case. In Springer Proceedings in Mathematics and Statistics (Vol. 44, pp. 93–123). Springer New York LLC. https://doi.org/10.1007/978-3-319-00125-8_5
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