2011/08/18 by Marina Ghisi, Massimo Gobbino, Ghisi, Marina +1 · 1 citation
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1108.3758
We consider degenerate Kirchhoff equations with a small parameter epsilon in\nfront of the second-order time-derivative. It is well known that these\nequations admit global solutions when epsilon is small enough, and that these\nsolutions decay as t -> +infinity with the same rate of solutions of the limit\nproblem (of parabolic type).\n In this paper we prove decay-error estimates for the difference between a\nsolution of the hyperbolic problem and the solution of the corresponding\nparabolic problem. These estimates show in the same time that the difference\ntends to zero both as epsilon -> 0, and as t -> +infinity. Concerning the decay\nrates, it turns out that the difference decays faster than the two terms\nseparately (as t -> +infinity).\n Proofs involve a nonlinear step where we separate Fourier components with\nrespect to the lowest frequency, followed by a linear step where we exploit\nweighted versions of classical energies.\n