On the Parabolic Regime of a Hyperbolic Equation with Weak Dissipation: The Coercive Case

We consider a family of Kirchhoff equations with a small parameter in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 at the infinity. 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 at the infinity. 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.