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Blow-up results for a Nakao-type problem with a time-dependent damping term and derivative-type nonlinearities

Yuequn Li, Alessandro Palmieri

TL;DR

This work analyzes blow-up phenomena and lifespan estimates for a weakly coupled damped wave system with derivative-type nonlinearities under two time-dependent damping regimes: scale-invariant and scattering damping. By constructing an iteration framework based on adjoint problems and weighted functionals, the authors derive lower bounds that propagate through a slicing scheme, establishing blow-up regions in the $(p,q)$-plane. The scale-invariant case yields a shifted threshold $\Theta(n+\mu,p,q)\ge 0$, reflecting a shift in the effective space dimension, while the scattering case leaves the Glassey-type threshold intact. Overall, the paper clarifies how time-dependent damping influences blow-up and lifespan in Nakao-type systems, offering precise bounds $T(\varepsilon)\lesssim \varepsilon^{-1/\Theta}$ and $T(\varepsilon)\lesssim \exp(c\varepsilon^{-(pq-1)})$ in subcritical and critical regimes.

Abstract

In this paper, we consider a semilinear system of damped wave equations coupled through power nonlinearities of derivative-type. In particular, we consider a classical damped wave equation, i.e., with constant coefficients, and a wave equation with a time-dependent coefficient for the damping term. For this time-dependent coefficient we analyze two cases: the scale-invariant case and the scattering producing case. We prove blow-up results and derive upper bound estimates for the lifespan of local solutions. Our approach is based on an iteration argument for a couple of functionals related to the components of a local solution.

Blow-up results for a Nakao-type problem with a time-dependent damping term and derivative-type nonlinearities

TL;DR

This work analyzes blow-up phenomena and lifespan estimates for a weakly coupled damped wave system with derivative-type nonlinearities under two time-dependent damping regimes: scale-invariant and scattering damping. By constructing an iteration framework based on adjoint problems and weighted functionals, the authors derive lower bounds that propagate through a slicing scheme, establishing blow-up regions in the -plane. The scale-invariant case yields a shifted threshold , reflecting a shift in the effective space dimension, while the scattering case leaves the Glassey-type threshold intact. Overall, the paper clarifies how time-dependent damping influences blow-up and lifespan in Nakao-type systems, offering precise bounds and in subcritical and critical regimes.

Abstract

In this paper, we consider a semilinear system of damped wave equations coupled through power nonlinearities of derivative-type. In particular, we consider a classical damped wave equation, i.e., with constant coefficients, and a wave equation with a time-dependent coefficient for the damping term. For this time-dependent coefficient we analyze two cases: the scale-invariant case and the scattering producing case. We prove blow-up results and derive upper bound estimates for the lifespan of local solutions. Our approach is based on an iteration argument for a couple of functionals related to the components of a local solution.
Paper Structure (8 sections, 13 theorems, 162 equations)

This paper contains 8 sections, 13 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Deduction of the iteration frame
  4. Scale-invariant damping
  5. Scattering producing damping
  6. Proof of the main results: Iteration argument
  7. Proof of Theorem \ref{['theorem1']}
  8. Proof of Theorem \ref{['theorem2']}

Key Result

Theorem 2.1

Let $b(t)=\frac{\mu}{1+t}$, for some $\mu>0$ in eqs. Let $u_0,v_0\in W^{1,1}_{\mathrm{loc}}(\mathbb{R}^n), u_1,v_1\in L^{1}_{\mathrm{loc}}(\mathbb{R}^n)$ be nonnegative and compactly supported functions satisfying support condition data for some $R>0$ such that $u_1\geq u_0$ and $v_0$ is nontrivial. Then, there exists $\varepsilon_0=\varepsilon_0(n,\mu,p,q,R, v_0,v_1)>0$ such that for any $\vareps

Theorems & Definitions (26)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 16 more