Convergence to nonlinear diffusion waves for solutions of p-system with time-dependent damping

In this paper, we study the Cauchy problem for the hyperbolic p-system with time-gradually-degenerate damping term −1(1+t)λu for 0⩽λ<1, and show that the damped p-system has a couple of global solutions uniquely, and such solutions tend time-asymptotically to the shifted nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation governed by the Darcy's law. We further derive the convergence rates when the initial perturbations are in L2. The approach adopted is the technical time-weighted energy method.