Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic behavior of solutions to the generalized KdV-Burgers equation with slowly decaying data

Ikki Fukuda

TL;DR

This work analyzes the long-time behavior of global solutions to the generalized KdV-Burgers equation with slowly decaying initial data. It establishes global existence for small data and proves that solutions converge to the nonlinear diffusion wave χ, with optimal rates that depend on the decay exponent α of the data. For 1<α<2, a second asymptotic profile Z is constructed so that u−χ−Z vanishes at rate t−α/2, and lower bounds show this rate is sharp. In the critical case α=2, an additional profile V is introduced, yielding decay of u−χ−Z−V at rate t−1 log t, with matching lower bounds confirming optimality. The analysis combines Hopf-Cole transforms, Green-function estimates, and refined asymptotic expansions via the U-operator to capture slow spatial decay effects.

Abstract

We consider the asymptotic behavior of the global solutions to the initial value problem for the generalized KdV-Burgers equation. It is known that the solution to this problem converges to a self-similar solution to the Burgers equation called a nonlinear diffusion wave. In this paper, we derive the optimal asymptotic rate to the nonlinear diffusion wave when the initial data decays slowly at spatial infinity. In particular, we investigate that how the change of the decay rate of the initial value affects the asymptotic rate to the nonlinear diffusion wave.

Asymptotic behavior of solutions to the generalized KdV-Burgers equation with slowly decaying data

TL;DR

This work analyzes the long-time behavior of global solutions to the generalized KdV-Burgers equation with slowly decaying initial data. It establishes global existence for small data and proves that solutions converge to the nonlinear diffusion wave χ, with optimal rates that depend on the decay exponent α of the data. For 1<α<2, a second asymptotic profile Z is constructed so that u−χ−Z vanishes at rate t−α/2, and lower bounds show this rate is sharp. In the critical case α=2, an additional profile V is introduced, yielding decay of u−χ−Z−V at rate t−1 log t, with matching lower bounds confirming optimality. The analysis combines Hopf-Cole transforms, Green-function estimates, and refined asymptotic expansions via the U-operator to capture slow spatial decay effects.

Abstract

We consider the asymptotic behavior of the global solutions to the initial value problem for the generalized KdV-Burgers equation. It is known that the solution to this problem converges to a self-similar solution to the Burgers equation called a nonlinear diffusion wave. In this paper, we derive the optimal asymptotic rate to the nonlinear diffusion wave when the initial data decays slowly at spatial infinity. In particular, we investigate that how the change of the decay rate of the initial value affects the asymptotic rate to the nonlinear diffusion wave.

Paper Structure

This paper contains 5 sections, 15 theorems, 166 equations.

Table of Contents

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

Key Result

Theorem 1.1

Assume the condition 1-2 with $1<\alpha \le2$. Let $u_{0}\in H^{3}(\mathbb{R})$ and $\|u_{0}\|_{L^{1}}+\|u_{0}\|_{H^{3}}$ is sufficiently small. Then 1-1 has a unique global solution $u(x, t) \in C^{0}([0, \infty); H^{3})$. Moreover, the solution satisfies the following estimate: where $\chi(x, t)$ is defined by 1-4.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • ...and 16 more