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Results for blow-up and sharp lifespan estimates to a weakly coupled system of structurally damped wave equations with critical nonlinearities

Trung Loc Tang, Tuan Anh Dao, The Anh Cung

Abstract

In this paper, we would like to study a weakly coupled system of semilinear structurally damped wave equations with moduli of continuity in nonlinear terms whose powers belong to the critical curve in the $p-q$ plane. Our aim is to find sharp conditions for these moduli of continuity by investigating the global (in time) existence of small data solutions and a finite blow-up of small data solutions. Furthermore, when the finite blow-up phenomenon occurs, we are going to achieve the sharp lifespan estimate for the local (in time) solution.

Results for blow-up and sharp lifespan estimates to a weakly coupled system of structurally damped wave equations with critical nonlinearities

Abstract

In this paper, we would like to study a weakly coupled system of semilinear structurally damped wave equations with moduli of continuity in nonlinear terms whose powers belong to the critical curve in the plane. Our aim is to find sharp conditions for these moduli of continuity by investigating the global (in time) existence of small data solutions and a finite blow-up of small data solutions. Furthermore, when the finite blow-up phenomenon occurs, we are going to achieve the sharp lifespan estimate for the local (in time) solution.
Paper Structure (9 sections, 11 theorems, 211 equations)

This paper contains 9 sections, 11 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Blow-up in finite time
  3. Preliminaries
  4. Construction of test functions
  5. Proof of Theorem \ref{['thr3.1']}
  6. Sharp lifespan estimates
  7. Estimates for upper bound
  8. Estimates for lower bound
  9. Conclusion and Open problems

Key Result

Theorem 1.1

Let $\sigma\in \left[0,1/2\right]$, $n>2\sigma$. We assume that the initial data $u_0=v_0=0$ and $u_1,v_1\in L^1$ satisfy the relations: Moreover, we suppose the following assumptions of modulus of countinuity and where $c>0$ is a suitable small constant. Then, there is no global (in time) Sobolev solution to eqsys where $p^*,q^*$ belong to the curve critcurve2.

Theorems & Definitions (20)

  • Theorem 1.1: Blow-up
  • Remark 1.1
  • Theorem 1.2: Lifespan estimates
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: AnhFino, Lemma 2.3
  • ...and 10 more