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On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term

Yuequn Li, Alessandro Palmieri

TL;DR

This work analyzes finite-time blow-up for a weakly coupled semilinear wave–damped wave system with a time-dependent damping term in the second equation. The authors develop an iterative framework based on space-averaged functionals $U(t)$ and $V(t)$ and establish a general ODI comparison principle that yields explicit blow-up conditions and lifespan bounds. For scale-invariant damping $b(t)=\frac{\mu}{1+t}$, the blow-up region shifts with the damping strength, effectively modifying the critical curve to $\Gamma(n,p,q,\mu)>0$ and producing lifespans $T(\varepsilon)\le C\varepsilon^{-1/\Gamma(n,p,q,\mu)}$; for scattering-producing damping $b\in L^1([0,\infty))$, the blow-up region coincides with the Nakao/Chen–Reissig regime at $\mu=0$, and the lifespan bound remains of the same type. These results clarify how time-dependent damping shapes blow-up thresholds in Nakao-type problems and connect to established results for classical damped-wave and wave–heat systems, while highlighting open questions on global small-data existence and possible regime changes for large damping.

Abstract

In this paper, we study a semilinear weakly coupled system of wave equations with power nonlinearities. More precisely, we couple (through the nonlinear terms) a wave equation and a damped wave equation with a time-dependent coefficient for the damping term. For the coefficient of the damping term we consider two cases: the scale-invariant case and the scattering producing case. By applying an iteration argument, we get a blow-up result and upper bound estimates for the lifespan of the solutions. In the scale-invariant case, we obtain a shift of the space dimension in the blow-up region for the same weakly coupled system with a classical damping (i.e. with a constant coefficient), while for the scattering producing case we find the same blow-up region as for the classical Nakao problem.

On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term

TL;DR

This work analyzes finite-time blow-up for a weakly coupled semilinear wave–damped wave system with a time-dependent damping term in the second equation. The authors develop an iterative framework based on space-averaged functionals and and establish a general ODI comparison principle that yields explicit blow-up conditions and lifespan bounds. For scale-invariant damping , the blow-up region shifts with the damping strength, effectively modifying the critical curve to and producing lifespans ; for scattering-producing damping , the blow-up region coincides with the Nakao/Chen–Reissig regime at , and the lifespan bound remains of the same type. These results clarify how time-dependent damping shapes blow-up thresholds in Nakao-type problems and connect to established results for classical damped-wave and wave–heat systems, while highlighting open questions on global small-data existence and possible regime changes for large damping.

Abstract

In this paper, we study a semilinear weakly coupled system of wave equations with power nonlinearities. More precisely, we couple (through the nonlinear terms) a wave equation and a damped wave equation with a time-dependent coefficient for the damping term. For the coefficient of the damping term we consider two cases: the scale-invariant case and the scattering producing case. By applying an iteration argument, we get a blow-up result and upper bound estimates for the lifespan of the solutions. In the scale-invariant case, we obtain a shift of the space dimension in the blow-up region for the same weakly coupled system with a classical damping (i.e. with a constant coefficient), while for the scattering producing case we find the same blow-up region as for the classical Nakao problem.
Paper Structure (14 sections, 8 theorems, 142 equations)

This paper contains 14 sections, 8 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Iteration frames and first lower bound estimates
  4. Scale-invariant damping
  5. The first kind of lower bounds for the functionals
  6. The second kind of lower bounds for the functionals
  7. Scattering producing damping
  8. The first kind of lower bounds for the functionals
  9. The second kind of lower bounds for the functionals
  10. Comparison argument for a weakly coupled system of ODI
  11. Proof of the main results
  12. Proof of Theorem \ref{['theorem1']}
  13. Proof of Theorem \ref{['theorem2']}
  14. Final remarks and open problem

Key Result

Theorem 2.1

Let $b(t)=\frac{\mu}{1+t}$, with $\mu\geq0$ in eqs. Let $(u_0,u_1,v_0,v_1)\in (W^{1,1}_{\mathrm{loc}}(\mathbb{R}^n)\times L^1_{\mathrm{loc}}(\mathbb{R}^n))^2$ be nonnegative and nontrivial, and satisfy initialass1. Let $p,q>1$ such that There exists a $\varepsilon_0=\varepsilon_0(n,\mu,p,q,R,u_0,u_1,v_0,v_1)>0$ such that for any $\varepsilon\in(0,\varepsilon_0]$, the local (in time) weak solution

Theorems & Definitions (18)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 8 more