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Effect of additional regularity for the initial data on semi-linear $σ$-evolution equations with different damping types

Dinh Van Duong, Tuan Anh Dao

TL;DR

This work analyzes semi-linear $σ$-evolution equations with multiple damping terms under added $L^m$ regularity of the initial data. It identifies the critical exponent $p_{crit}(m,σ,σ_1)=1+\dfrac{2mσ}{n-2mσ_1}$ and shows how regularity shifts the global existence vs. blow-up regime, with $m=1$ yielding blow-up at criticality and $m\in(1,2]$ enabling global solutions at the critical value in some low dimensions. A fixed-point framework combined with frequency decomposition and $L^m$–$L^2$ estimates for the linear problem underpins local and global existence results, while a test-function method establishes blow-up and yields lifespan estimates. The results illuminate the interplay between high-order damping, dispersion, and initial regularity in determining long-time dynamics and decay properties of solutions to high-order evolution equations.

Abstract

In this paper, we would like to study the critical exponent for semi-linear $σ$-evolution equations with different damping types under the influence of additional regularity for the initial data. On the one hand, we establish the existence of global (in time) solutions for small initial data and the blow-up in finite time solutions in the supercritical case and the subcritical case, respectively. The very interesting phenomenon is that the critical case belonging to the global solution range or the blow-up solution range depends heavily on the assumption of additional regularity for the initial data. Furthermore, we are going to provide lifespan estimates for solutions when the blow-up phenomenon occurs.

Effect of additional regularity for the initial data on semi-linear $σ$-evolution equations with different damping types

TL;DR

This work analyzes semi-linear -evolution equations with multiple damping terms under added regularity of the initial data. It identifies the critical exponent and shows how regularity shifts the global existence vs. blow-up regime, with yielding blow-up at criticality and enabling global solutions at the critical value in some low dimensions. A fixed-point framework combined with frequency decomposition and – estimates for the linear problem underpins local and global existence results, while a test-function method establishes blow-up and yields lifespan estimates. The results illuminate the interplay between high-order damping, dispersion, and initial regularity in determining long-time dynamics and decay properties of solutions to high-order evolution equations.

Abstract

In this paper, we would like to study the critical exponent for semi-linear -evolution equations with different damping types under the influence of additional regularity for the initial data. On the one hand, we establish the existence of global (in time) solutions for small initial data and the blow-up in finite time solutions in the supercritical case and the subcritical case, respectively. The very interesting phenomenon is that the critical case belonging to the global solution range or the blow-up solution range depends heavily on the assumption of additional regularity for the initial data. Furthermore, we are going to provide lifespan estimates for solutions when the blow-up phenomenon occurs.

Paper Structure

This paper contains 9 sections, 11 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Proofs of local and global (in time) existence
  3. Philosophy of our approach
  4. Proof of Theorem \ref{['Local-existence']}
  5. Proof of Theorem \ref{['Global-Existance']}
  6. Blow-up results and Lifespan estimates
  7. Preliminaries
  8. Proof of Theorem \ref{['Blow-up results']}
  9. Estimates for lifespan

Key Result

Theorem 1.1

Let $n \geq 1$ and $0 \leq \sigma_1 < \sigma/2 < \sigma_2 \leq \sigma$. We assume that the parameter $m$ satisfies and the exponent $p$ satisfies the following conditions: Moreover, if the initial data $(u_0, u_1) \in \mathcal{D}_{\sigma}^m$, then there exists $T > 0$ and a constant $\varepsilon_0 > 0$ such that for any $\varepsilon \in (0, \varepsilon_0]$, the Cauchy problem Main.Eq.1 admits a

Theorems & Definitions (21)

  • Theorem 1.1: Local existence
  • Theorem 1.2: Global existence
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3: Blow-up
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2: Fractional Gagliardo-Nirenberg inequality, see Ozawa
  • Proposition 2.1
  • ...and 11 more