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Sharp higher-order $L^2$-asymptotic expansion of solutions to σ-evolution equations with different damping types

Dinh Van Duong, Tuan Anh Dao

TL;DR

The paper analyzes high-order $L^2$-asymptotic expansions for linear $\sigma$-evolution equations with double damping, identifying how the coexisting parabolic-like damping $(-\Delta)^{\sigma_1}u_t$ and $\sigma$-evolution-like damping $(-\Delta)^{\sigma_2}u_t$ shape long-time behavior. It constructs $k$-th order asymptotic profiles via explicit Fourier multipliers $\mathcal{A}_0^k,\mathcal{A}_1^k$ (and $\mathcal{B}_0^k,\mathcal{B}_1^k$ for $\sigma_1=0$) and proves sharp $L^2$-decay rates for the remainder with detailed low-/high-frequency analyses. The results reveal the roles of $\sigma$, $\sigma_1$, and $\sigma_2$ in determining the decay and diffusion-like features, including precise lower bounds when the data have nonzero mean $P_1$. The approach advances understanding of the interplay between parabolic and hyperbolic damping mechanisms in fractional-diffusion-type models and provides tools for sharp long-time approximations in the $L^2$ setting.

Abstract

In this paper, our main goal is to achieve the high-order asymptotic expansion of solutions to $σ$-evolution equations with different damping types in the $L^2$ framework. Throughout this, we observe the influence of parabolic like models corresponding to $σ_1 \in [0, σ/2)$ and $σ$-evolution like models corresponding to $σ_2 \in (σ/2, σ]$ on the asymptotic behavior of solutions.

Sharp higher-order $L^2$-asymptotic expansion of solutions to σ-evolution equations with different damping types

TL;DR

The paper analyzes high-order -asymptotic expansions for linear -evolution equations with double damping, identifying how the coexisting parabolic-like damping and -evolution-like damping shape long-time behavior. It constructs -th order asymptotic profiles via explicit Fourier multipliers (and for ) and proves sharp -decay rates for the remainder with detailed low-/high-frequency analyses. The results reveal the roles of , , and in determining the decay and diffusion-like features, including precise lower bounds when the data have nonzero mean . The approach advances understanding of the interplay between parabolic and hyperbolic damping mechanisms in fractional-diffusion-type models and provides tools for sharp long-time approximations in the setting.

Abstract

In this paper, our main goal is to achieve the high-order asymptotic expansion of solutions to -evolution equations with different damping types in the framework. Throughout this, we observe the influence of parabolic like models corresponding to and -evolution like models corresponding to on the asymptotic behavior of solutions.
Paper Structure (9 sections, 11 theorems, 103 equations)

This paper contains 9 sections, 11 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Notations and Main results
  3. Notations
  4. Main results
  5. Proofs of main results
  6. Preliminaries
  7. Estimates for low frequencies
  8. Estimates for high frequencies
  9. Proof of main results

Key Result

Theorem 2.1

Let $n > 4\sigma_1, \,s \geq 0$ and $k \in \mathbb{N}$. Assuming that the initial data $(u_0, u_1)$ satisfy Then, the solution to Main.Eq.1 satisfy for all $t \geq 1$. Moreover, if we assume an additional condition $P_1 \neq 0$, then there exists a positive constant $C_1:= C_1 \left( P_1, \|(u_0, u_1)\|_{\mathcal{D}_s}\right)$ such that the following estimate hold: for all $t \geq 1$.

Theorems & Definitions (22)

  • Theorem 2.1: Asymptotic profile with $\sigma_1 > 0$
  • Remark 2.1
  • Theorem 2.2: Asymptotic profile with $\sigma_1 = 0$
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1: Faà di Bruno’s formula
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 12 more