Best Asymptotic Profile for Hyperbolic p -System with Damping

Source abstract

For the $2\times2$ hyperbolic p-system with damping, its asymptotic profile has been traditionally regarded as the self-similar solution, the so-called diffusion wave, to the corresponding parabolic equation by Darcy's law, and the convergence of the original solution to the diffusion wave has been intensively studied by many people. However, by a deep observation and a heuristic analysis, we realize that the best asymptotic profile for the solution to the Cauchy problem of the p-system with damping is a particular solution to the corresponding nonlinear parabolic equation with a specified initial data, and we further show the convergence rates to this particular asymptotic profile. These new rates are much better than the existing convergence rates to the diffusion waves. Our results essentially improve and develop the previous studies. Finally, some numerical simulations are carried out, which also confirm our theoretical results.