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(+) not equal 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 <=lambda< 3/5, lambda= 3/5 and 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.