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

Ikki Fukuda

TL;DR

The paper studies the large-time behavior of solutions to a nonlinear damped wave equation derived from the Jin-Xin hyperbolic relaxation model with slowly decaying initial data. It extends previous diffusion-wave results by introducing second-order asymptotic profiles to account for tail decay: a nonlinear diffusion wave $\chi(x,t)$ solving $\chi_t + (a\chi + \frac{b}{2}\chi^2)_x = \mu \chi_{xx}$, together with a first correction $Z(x,t)$ (and a further correction $V(x,t)$ in the borderline case where $\min\{\alpha,\beta\}=2$) to capture slower decays. For $1<\min\{\alpha,\beta\}<2$, the solution satisfies $u(x,t) = \chi(x,t) + Z(x,t) + o(1)$ with the remainder decaying at rate $t^{-\min\{\alpha,\beta\}/2}$ in $L^\infty$, and the paper shows the optimality of these rates via the second asymptotic profile. In the borderline case $\min\{\alpha,\beta\}=2$, the authors prove $u(x,t) = \chi(x,t) + Z(x,t) + V(x,t) + o(1)$ with a logarithmic correction, and establish sharp lower bounds for the combined profile $Z+V$. The results illuminate how the decay rate of initial data governs the precise large-time asymptotics of nonlinear hyperbolic-relaxation systems and provide a comprehensive framework for the optimal diffusion-wave-type behavior under slowly decaying tails.

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.

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

TL;DR

The paper studies the large-time behavior of solutions to a nonlinear damped wave equation derived from the Jin-Xin hyperbolic relaxation model with slowly decaying initial data. It extends previous diffusion-wave results by introducing second-order asymptotic profiles to account for tail decay: a nonlinear diffusion wave solving , together with a first correction (and a further correction in the borderline case where ) to capture slower decays. For , the solution satisfies with the remainder decaying at rate in , and the paper shows the optimality of these rates via the second asymptotic profile. In the borderline case , the authors prove with a logarithmic correction, and establish sharp lower bounds for the combined profile . The results illuminate how the decay rate of initial data governs the precise large-time asymptotics of nonlinear hyperbolic-relaxation systems and provide a comprehensive framework for the optimal diffusion-wave-type behavior under slowly decaying tails.

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.

Paper Structure

This paper contains 5 sections, 17 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main1']}
  4. Proof of Theorem 1.2 for $1<\min\{\alpha, \beta\}< 2$
  5. Proof of Theorem \ref{['main2']} for $\min\{\alpha, \beta\}=2$

Key Result

Theorem 1.1

Assume the condition 1-3 holds with $1<\min\{\alpha, \beta\} \le 2$. Let $s$ be a positive integer and $1\le p\le \infty$. Suppose that $u_{0}\in W^{s, p}(\R)$, $u_{1}\in W^{s-1,p}(\R)$ and $\|u_{0}\|_{W^{s, p}}+\|u_{0}\|_{L^{1}}+\|u_{1}\|_{W^{s-1, p}}+\|u_{1}\|_{L^{1}}$ is sufficiently small. Then where $\sigma=\min\{2, s\}$. Moreover, for any ${\varepsilon}>0$, the estimate holds for any $q$ w

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 17 more