Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

TL;DR

This paper studies the long-term behavior of solutions to a nonlinear damped wave equation with slowly decaying initial data, revealing how decay rates influence asymptotic profiles and convergence to diffusion waves.

Contribution

It extends previous results by analyzing solutions with slower initial decay, deriving new asymptotic profiles, and examining the impact of decay rate changes on asymptotic behavior.

Findings

Solutions converge to nonlinear diffusion waves under certain decay conditions

Slower initial decay alters the asymptotic profile and convergence rate

The decay rate of initial data significantly affects the large time behavior

Abstract

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profiles. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.