Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic profiles of solutions for the generalized Fornberg-Whitham equation with dissipation

Ikki Fukuda

TL;DR

This paper analyzes the Cauchy problem for the generalized Fornberg–Whitham equation with dissipation and nonlocal dispersion, establishing that solutions converge toward a modified heat kernel $G_0$ as $t\to\infty$ and identifying a second-order correction that depends on the nonlinearity degree $p$. The authors develop a Fourier-based framework to approximate the nonlocal term by a linearized dispersive-diffusive kernel $G_0$ and derive precise Duhamel-term asymptotics, yielding explicit second-order profiles $W_p$ for $2<p<3$, a logarithmic correction for $p=3$, and combined $(m+M)\partial_x G_0$ and $t\partial_x^3 G_0$-type terms for $p>3$. The results reveal how dissipation dominates the first-order asymptotics, while dispersion and nonlinear effects shape higher-order profiles, with sharp decay rates in $L^q$ spaces. The work thus clarifies the interplay between nonlocal dispersion, dissipation, and nonlinearity in the long-time behavior of solutions to dissipative dispersive equations.

Abstract

We consider the Cauchy problem for the generalized Fornberg-Whitham equation with dissipation. This is one of the nonlinear, nonlocal and dispersive-dissipative equations. The main topic of this paper is an asymptotic analysis for the solutions to this problem. We prove that the solution to this problem converges to the modified heat kernel. Moreover, we construct the second term of asymptotics for the solutions depending on the degree of the nonlinearity. In view of those second asymptotic profiles, we investigate the effects of the dispersion, dissipation and nonlinear terms on the asymptotic behavior of the solutions.

Asymptotic profiles of solutions for the generalized Fornberg-Whitham equation with dissipation

TL;DR

This paper analyzes the Cauchy problem for the generalized Fornberg–Whitham equation with dissipation and nonlocal dispersion, establishing that solutions converge toward a modified heat kernel as and identifying a second-order correction that depends on the nonlinearity degree . The authors develop a Fourier-based framework to approximate the nonlocal term by a linearized dispersive-diffusive kernel and derive precise Duhamel-term asymptotics, yielding explicit second-order profiles for , a logarithmic correction for , and combined and -type terms for . The results reveal how dissipation dominates the first-order asymptotics, while dispersion and nonlinear effects shape higher-order profiles, with sharp decay rates in spaces. The work thus clarifies the interplay between nonlocal dispersion, dissipation, and nonlinearity in the long-time behavior of solutions to dissipative dispersive equations.

Abstract

We consider the Cauchy problem for the generalized Fornberg-Whitham equation with dissipation. This is one of the nonlinear, nonlocal and dispersive-dissipative equations. The main topic of this paper is an asymptotic analysis for the solutions to this problem. We prove that the solution to this problem converges to the modified heat kernel. Moreover, we construct the second term of asymptotics for the solutions depending on the degree of the nonlinearity. In view of those second asymptotic profiles, we investigate the effects of the dispersion, dissipation and nonlinear terms on the asymptotic behavior of the solutions.
Paper Structure (7 sections, 17 theorems, 140 equations)

This paper contains 7 sections, 17 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Global Existence and Decay Estimates
  3. Auxiliary Lemmas and Propositions
  4. Proof of the Main Result
  5. Proof of Theorem \ref{['thm.main']} for $\bm{2<p<3}$
  6. Proof of Theorem \ref{['thm.main']} for $\bm{p=3}$
  7. Proof of Theorem \ref{['thm.main']} for $\bm{p>3}$

Key Result

Theorem 1.1

Let $p>2$. Assume that $u_{0} \in H^{1}(\mathbb{R}) \cap L^{1}(\mathbb{R})$ and $E_{0}:=\|u_{0}\|_{H^{1}}+\|u_{0}\|_{L^{1}}$ is sufficiently small. Then, VFW has a unique global mild solution $u \in C([0, \infty); H^{1}(\mathbb{R}))$ satisfying Moreover, the solution $u(x, t)$ satisfies the following estimate: Furthermore, if $xu_{0} \in L^{1}(\mathbb{R})$, then the solution $u(x, t)$ satisfies

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • ...and 24 more