Decay rates for nonlinear diffusion waves of p-systems with nonlinear and time-dependent damping

In this article, we study the Cauchy problem for a \(p\)-system with both nonlinear and time-dependent damping. By combining the a priori estimates with energy methods, we prove that under sufficiently small initial data, the Cauchy problem admits a unique global solution. Furthermore, we show that the solution converges time- asymptotically to the corresponding nonlinear diffusion wave governed by the classical Darcy's law. In particular, \(L^{\infty}\) convergence rates are established via weighted energy estimates. This result fills a significant gap in the theoretical framework by explicitly accounting for the combined effects of nonlinear and time-dependent damping, an aspect that has not been adequately addressed in previous studies. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/88/abstr.html