Asymptotic behavior of solutions to the Cauchy problem for 1D p-system with spatiotemporal damping: Case 1. v+ = v−

This paper investigates the Cauchy problem for the p-system with spatiotemporal damping, modeling one-dimensional compressible flow through porous media in Lagrangian coordinates. We focus on the large-time asymptotic behavior of the system's solutions when the state constants for the specific volume are the same: $v_{+}=v_{-}$, but the state constants for the velocity are different: $u_{+}\ne u_{-}$. We show the convergence of the solutions to their diffusion waves with the different algebraic time decay rates according to different exponent of time-damping: $0\le \lambda <\frac{3}{5}$, $\lambda =\frac{3}{5}$ and $\frac{3}{5}<\lambda <1$, respectively. Our analysis employs an energy method to establish a series of a priori estimates, offering new insights and theoretical support for understanding the long-time dynamics of compressible flows in porous media with spatially heterogeneous damping.