Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping
Abstract
We study the maximal existence time T*(ϱ) of the solution of the semilinear wave equation utt−Δu=u|u|p−1−ut|ut|q−1 in a bounded domain, with Dirichlet boundary condition and initial data (ϱ f,ϱ g), where 1<q<p and the amplitude ϱ is large. For nontrivial f and sufficiently large ϱ, the concavity method gives T*(ϱ)≤ Cϱ−(p−1)/2 for q≤2p/(p+1) and T*(ϱ)≤ Cϱ−(p−q)/q for q>2p/(p+1), whereas the energy method gives a lower bound of order ϱ1−p only. We prove lower bounds with the same exponents as the upper ones, so that T*(ϱ)asympϱ−ϑ(p,q) with ϑ(p,q)=min{(p−1)/2, (p−q)/q}. The proof rests on a hyperbolic rescaling which converts the large amplitude into a dilation of the domain and a coefficient ϱ(q(p+1)−2p)/2 in front of the damping term, on a local existence theory in uniformly local energy norms whose existence time does not depend on that coefficient, and on a quantitative use of the dissipation when the coefficient is large. The threshold 2p/(p+1) is the value of q at which the damping term is invariant under the rescaling. No attempt is made to optimize the constants: sharpness is meant throughout at the level of the exponent.
Related
- New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data
- Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source
- The critical exponent for the two-dimensional semilinear wave equation with strong damping
- Stable determination of damping and potential coefficients in a semilinear wave equation
- Lifespan estimate for one dimensional wave equation with semilinear terms of spatial derivative
BibTeX
@misc{kaabi2026sharp,
title = {{Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping}},
author = {Firas Kaabi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10819},
eprint = {2609.10819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10819},
}
RIS
TY - UNPB
TI - Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping
AU - Firas Kaabi
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10819
UR - https://arxiv.org/abs/2609.10819
AB - We study the maximal existence time T^*(ϱ) of the solution of the semilinear wave equation u_tt−Δu=u|u|^p−1−u_t|u_t|^q−1 in a bounded domain, with Dirichlet boundary condition and initial data (ϱ f,ϱ g), where 1<q<p and the amplitude ϱ is large. For nontrivial f and sufficiently large ϱ, the concavity method gives T^*(ϱ)≤ Cϱ^−(p−1)/2 for q≤2p/(p+1) and T^*(ϱ)≤ Cϱ^−(p−q)/q for q>2p/(p+1), whereas the energy method gives a lower bound of order ϱ^1−p only. We prove lower bounds with the same exponents as the upper ones, so that T^*(ϱ)asympϱ^−ϑ(p,q) with ϑ(p,q)=min{(p−1)/2, (p−q)/q}. The proof rests on a hyperbolic rescaling which converts the large amplitude into a dilation of the domain and a coefficient ϱ^(q(p+1)−2p)/2 in front of the damping term, on a local existence theory in uniformly local energy norms whose existence time does not depend on that coefficient, and on a quantitative use of the dissipation when the coefficient is large. The threshold 2p/(p+1) is the value of q at which the damping term is invariant under the rescaling. No attempt is made to optimize the constants: sharpness is meant throughout at the level of the exponent.
ER -