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Large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation

Ikki Fukuda, Masahiro Ikeda

TL;DR

This work analyzes the large-time behavior of solutions to the Cauchy problem for the BBM--Burgers equation, focusing on how the decay rate $\alpha$ of the initial data $u_0$ influences asymptotics. The authors first prove global existence and time-decay for small data, establishing convergence to the nonlinear diffusion wave $\chi(x,t)$. They then construct second asymptotic profiles, $Z$ for $1<\alpha<2$ and $V$ (with an accompanying $Z$ term when necessary) for $\alpha\ge 2$, and prove refined convergence rates to $\chi$, $\chi+Z$, and $\chi+V$ (and $\chi+Z+V$ in the borderline $\alpha=2$ case). The results yield optimal rates to the diffusion wave and extend prior work (notably Hayashi–Kaikina–Naumkin) to slowly decaying data, providing a detailed asymptotic expansion and highlighting the role of initial decay in the long-time dynamics of dissipative BBM-type equations.

Abstract

We consider the large time behavior of the solutions to the Cauchy problem for the BBM-Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers equation called the nonlinear diffusion wave. Moreover, we construct the appropriate second asymptotic profiles of the solutions depending on the initial data. Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. Especially, the important point of this study is that the second asymptotic profiles of the solutions with slowly decaying data, whose case has not been studied, are obtained.

Large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation

TL;DR

This work analyzes the large-time behavior of solutions to the Cauchy problem for the BBM--Burgers equation, focusing on how the decay rate of the initial data influences asymptotics. The authors first prove global existence and time-decay for small data, establishing convergence to the nonlinear diffusion wave . They then construct second asymptotic profiles, for and (with an accompanying term when necessary) for , and prove refined convergence rates to , , and (and in the borderline case). The results yield optimal rates to the diffusion wave and extend prior work (notably Hayashi–Kaikina–Naumkin) to slowly decaying data, providing a detailed asymptotic expansion and highlighting the role of initial decay in the long-time dynamics of dissipative BBM-type equations.

Abstract

We consider the large time behavior of the solutions to the Cauchy problem for the BBM-Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers equation called the nonlinear diffusion wave. Moreover, we construct the appropriate second asymptotic profiles of the solutions depending on the initial data. Based on that discussion, we investigate the effect of the initial data on the large time behavior of the solution, and derive the optimal asymptotic rate to the nonlinear diffusion wave. Especially, the important point of this study is that the second asymptotic profiles of the solutions with slowly decaying data, whose case has not been studied, are obtained.
Paper Structure (7 sections, 19 theorems, 183 equations)

This paper contains 7 sections, 19 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Global Existence and Time Decay Estimates
  3. First Asymptotic Profile
  4. Second Asymptotic Profiles
  5. Auxiliary Problem
  6. Proof of Theorem \ref{['main.thm-2nd.AP']} for $\bm{1<\alpha<2}$
  7. Proof of Theorem \ref{['main.thm-2nd.AP']} for $\bm{\alpha\ge2}$

Key Result

Theorem 1.1

Let $s\ge1$ be an integer. Assume that the initial data $u_{0}(x)$ satisfies the condition data, $u_{0}\in H^{s}(\mathbb{R})$ and $\|u_{0}\|_{H^{s}}+\|u_{0}\|_{L^{1}}$ is sufficiently small. Then, BBMB has a unique global mild solution $u\in C^{0}([0, \infty); H^{s}(\mathbb{R}))$. Moreover, for any holds for any integer $l$ satisfying $0\le l\le s$. Furthermore, holds for any integer $l$ satisfy

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 23 more