Lifespan estimate for one dimensional wave equation with semilinear terms of spatial derivative
Abstract
This paper studies the upper and lower bounds of the lifespan for the classical solutions to the initial value problems of one dimensional wave equations with non-autonomous semilinear terms including the space-derivative of the unknown function.This is a non-trivial business comparing to the analogous results with time-derivative type semilinear terms, especially for the proof to obtain the sharp upper bound of the lifespan as we have to deal with space dependent weights among iteration procedures of the weighted functional of the solution. Also it is surprising that a part of them reaches to the same ordinary differential inequality for classical semilinear damped wave equations introduced by Li and Zhou (Discrete Contin. Dynam. Systems, 1995, 1(4): 503-520), and we show a simple proof for blow up result from this ordinary differential inequality by iteration argument and slicing method in more general situation.
Related
- Characteristic localization, sharp lifespan asymptotics and global dynamics for a one-dimensional derivative wave equation
- Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping
- New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data
- The critical exponent for the two-dimensional semilinear wave equation with strong damping
- The estimate of lifespan for ideal incompressible MHD systems
BibTeX
@misc{lai2026lifespan,
title = {{Lifespan estimate for one dimensional wave equation with semilinear terms of spatial derivative}},
author = {Ning-An Lai and Cui Ren and Takiko Sasaki and Hiroyuki Takamura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.07310},
eprint = {2605.07310},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.07310},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lifespan estimate for one dimensional wave equation with semilinear terms of spatial derivative
AU - Ning-An Lai
AU - Cui Ren
AU - Takiko Sasaki
AU - Hiroyuki Takamura
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2605.07310
UR - https://arxiv.org/abs/2605.07310
AB - This paper studies the upper and lower bounds of the lifespan for the classical solutions to the initial value problems of one dimensional wave equations with non-autonomous semilinear terms including the space-derivative of the unknown function.This is a non-trivial business comparing to the analogous results with time-derivative type semilinear terms, especially for the proof to obtain the sharp upper bound of the lifespan as we have to deal with space dependent weights among iteration procedures of the weighted functional of the solution. Also it is surprising that a part of them reaches to the same ordinary differential inequality for classical semilinear damped wave equations introduced by Li and Zhou (Discrete Contin. Dynam. Systems, 1995, 1(4): 503-520), and we show a simple proof for blow up result from this ordinary differential inequality by iteration argument and slicing method in more general situation.
ER -