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Higher-order asymptotic profiles for solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity

Ikki Fukuda, Yota Irino

TL;DR

This work addresses refining the large-time asymptotics for a dispersive-dissipative equation with a cubic nonlinearity by deriving a second asymptotic profile for the Duhamel term. It employs an integral formulation with the linear kernel $S_{\alpha}(t)$ and analyzes both the linear and nonlinear contributions to extract a new self-similar term $\Psi$ and a logarithmic correction to $\partial_x G$, achieving an improved decay rate. The main contributions include the explicit construction of the second asymptotic profile, a complete higher-order expansion of the solution $u(x,t)$ that respects parabolic self-similarity, and a detailed dependence on the fractional exponent $\alpha$ through different expansion regimes. These results extend prior work on higher-order asymptotics and provide sharper long-time approximations for this class of dispersive-dissipative systems.

Abstract

We consider the asymptotic behavior of solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity. It is known that the leading term of the asymptotic profile for the solution to this problem is the Gaussian. Moreover, by analyzing the corresponding integral equation, the higher-order asymptotic expansion for the solution to the linear part and the first asymptotic profile for the Duhamel term have already been obtained. In this paper, we construct the second asymptotic profile for the Duhamel term and give the more detailed higher-order asymptotic expansion of the solutions, which generalizes the previous works. Furthermore, we emphasize that the newly obtained higher-order asymptotic profiles have a good structure in the sense of satisfying the parabolic self-similarity.

Higher-order asymptotic profiles for solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity

TL;DR

This work addresses refining the large-time asymptotics for a dispersive-dissipative equation with a cubic nonlinearity by deriving a second asymptotic profile for the Duhamel term. It employs an integral formulation with the linear kernel and analyzes both the linear and nonlinear contributions to extract a new self-similar term and a logarithmic correction to , achieving an improved decay rate. The main contributions include the explicit construction of the second asymptotic profile, a complete higher-order expansion of the solution that respects parabolic self-similarity, and a detailed dependence on the fractional exponent through different expansion regimes. These results extend prior work on higher-order asymptotics and provide sharper long-time approximations for this class of dispersive-dissipative systems.

Abstract

We consider the asymptotic behavior of solutions to the Cauchy problem for a dispersive-dissipative equation with a cubic nonlinearity. It is known that the leading term of the asymptotic profile for the solution to this problem is the Gaussian. Moreover, by analyzing the corresponding integral equation, the higher-order asymptotic expansion for the solution to the linear part and the first asymptotic profile for the Duhamel term have already been obtained. In this paper, we construct the second asymptotic profile for the Duhamel term and give the more detailed higher-order asymptotic expansion of the solutions, which generalizes the previous works. Furthermore, we emphasize that the newly obtained higher-order asymptotic profiles have a good structure in the sense of satisfying the parabolic self-similarity.
Paper Structure (4 sections, 14 theorems, 94 equations)

This paper contains 4 sections, 14 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Proof of the Main Results

Key Result

Proposition 1.1

Let $1<\alpha<3$, $q\ge2$ and $u_{0} \in H^{1}(\R)$. Then, there exist $T>0$ and a unique local mild solution $u \in C([0, T); H^{1}(\R))$ to gKdVB. Moreover, if $u_{0} \in H^{1}(\R)\cap L^{1}(\R)$ and $\| u_{0} \|_{H^{1}} + \| u_0 \|_{L^{1}}$ is sufficiently small, then, there exists a unique globa Furthermore, the solution $u(x, t)$ satisfies the following estimates: for any $2\le p\le \infty$.

Theorems & Definitions (26)

  • Proposition 1.1: K99-1
  • Theorem 1.2: K99-1
  • Theorem 1.3: K99-1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Remark 2.7
  • ...and 16 more