Asymptotic behavior of solutions to the Cauchy problem for 1-D p-system with space dependent damping

TL;DR

This paper investigates the long-term behavior of solutions to a one-dimensional p-system with space-dependent damping, modeling compressible flow in porous media, and demonstrates their convergence to a diffusion wave using energy methods.

Contribution

It reformulates the p-system into a second-order quasilinear hyperbolic equation and analyzes its asymptotic behavior, providing new insights into the diffusion wave phenomenon.

Findings

Solutions asymptotically behave like diffusion waves

Reformulation enables detailed energy analysis

Results support Darcy law-based diffusion approximation

Abstract

We consider the Cauchy problem for one-dimensional p-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangean coordinate. Our concern is an asymptotic behavior of solutions, which is expected to be the diffusion wave based on the Darcy law. To show this expectation, the problem is reformulated to the Cauchy problem for the second order quasilinear hyperbolic equation with space dependent damping, which is analyzed by the energy method.