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

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 at the infinity. 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 goes to infinity and as the parameter goes to 0.