一类带有加权非线性项的波动方程解的破裂

Blow-up of solutions for a class of wave equations with weighted nonlinear terms

摘要

非线性波动方程在声波传播、弹性力学等领域具有广泛应用。解的破裂现象反映物理系统的能量聚焦机制,为相关物理过程建模等提供理论基础。本文在一维空间中研究一类带加权导数型非线性项波动方程的小初值问题,其中非线性项包含未知函数的空间偏导数或时间偏导数。针对加权参数导致经典能量估计失效的困难,通过引入适当的泛函并且应用迭代方法,证明问题的解会在有限时间内破裂,并且得出解的生命跨度的上界估计。本研究将已有文献的结果推广至不同形式加权非线性项情形,阐明加权函数中参数对解的生命跨度估计的影响。

Abstract

Nonlinear wave equations have wide applications in fields such as acoustic wave propagation and elastic mechanics. The blow-up of solutions reflects energy focusing mechanisms in physical systems, providing theoretical foundations for modeling wave propagation processes. This paper, investigates the initial value problem for wave equations with weighted derivative nonlinear terms in one dimensional space, where the nonlinear terms contain spatial or temporal partial derivatives of unknown functions. To address the challenge of classical energy estimates fail when dealing with weighted parameters, this paper introduces appropriate functionals and employs the iterative method to prove that solutions of the problems will blow up in finite time. In addition, upper bound estimates for lifespan of solutions are obtained. Furthermore, this research extends existing results to cases involving weighted nonlinearities of diverse forms. And it also clarifies the influence of parameter in weighting function on lifespan estimates of solutions.

关键词

波动方程 / 加权非线性项 / 破裂 / 生命跨度估计 / 迭代方法

Key words

wave equation / weighted nonlinear terms / blow-up / lifespan estimates / iterative method

[1] TAKAMURA H, WAKASA K. Almost global solutions of semilinear wave equations with the critical exponent in high dimensions[J]. Nonlinear Anal Theory Meth Appl, 2014, 109: 187-229.
[2] SASAKI T, TAKAMATSU S, TAKAMURA H. The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative[J]. AIMS Math, 2023, 8(11): 25477-25486.
[3] KITAMURA S, MORISAWA K, TAKAMURA H. Semilinear wave equations of derivative type with spatial weights in one space dimension[J]. Nonlinear Anal Theory Meth Appl, 2023, 72: 103764.
[4] JOHN F. Blow-up for quasi-linear wave equations in three space dimensions[J]. Communications on Pure and Applied Mathematics, 1981, 34(1): 29-51.
[5] KUBO H, OSAKA A, YAZICI M. Global existence and blow-up for wave equations with weighted nonlinear terms in one space dimension[J]. Interdisciplinary Information Sciences, 2013, 19(2): 143-148.
[6] HARUYAMA Y, TAKAMURA H. Blow-up of classical solutions of quasilinear wave equations in one space dimension[J]. Nonlinear Anal Theory Meth Appl, 2025, 81: 104212.
[7] WAKASA K. The lifespan of solutions to wave equations with weighted nonlinear terms in onespace dimension[J]. Hokkaido Mathematical Journal, 2017, 46(2): 257-276.
[8] KIDO R, SASAKI T, TAKAMATSU S, et al. The generalized combined effect for one dimensional wave equations with semilinear terms including product type[J]. Journal of Differential Equations, 2024, 403: 576-618.
[9] PALMIERI A, TAKAMURA H. A blow-up result for a Nakao-type weakly coupled system with nonlinearities of derivative type[J]. Journal of Mathematical Analysis and Applications, 2023, 387(1): 111-132.
[10] LAI N A, LIU M Y, WAKASA K, et al. Lifespan estimates for 2-dimensional semilinear wave equations in asymptotically Euclidean exterior domains[J]. Journal of Functional Analysis, 2021, 281(12): 109253.
[11] MING S, DU J Y, XIE J. Blow-up of solutions to the wave equations with memory terms in Schwarzschild spacetime[J]. Journal of Mathematical Analysis and Applications, 2024, 540(2):128637.
[12] KITAMURA S, TAKAMURA H, WAKASA K. The lifespan estimates of classical solutions of one dimensional semilinear wave equations with characteristic weights[J]. Journal of Mathematical Analysis and Applications, 2023, 528(2):127516.
[13] IKEDA M, TANAKA T, WAKASA K. Critical exponent for the wave equation with a time dependent scale invariant damping and a cubic convolution[J]. Journal of Differential Equations, 2021, 270:916-946.
[14] LAI N G, ZHOU Y. Blow-up and lifespan estimate to a nonlinear wave equation in Schwarzschild spacetime[J]. Journal de Mathematiques Pures et Appliquees, 2023, 173:172-194.
[15] IKEDA M, SOBAJIMA M, WAKASA K. Blow-up phenomena of semilinear wave equations and their weakly coupled systems[J]. Journal of Differential Equations, 2019, 267 (9):5165-5201.
[16] LIU C M, SHENG H, LAI N A. Global smooth solutions for hyperbolic systems with time-dependent damping[J]. Nonlinear Analysis, 2024, 247:113608.
[17] HAO J H, ZHANG T. General energy decay estimates for a variable coefficient viscoelastic semilinear wave equation with logarithmic source term and acoustic boundary conditions[J]. Mathematical Methods in the Applied Sciences, 2024, 47(12):9897-9917.
[18] HAN J B, WANG K Y, XU R Z, et al. Global quantitative stability of wave equations with strong and weak dampings[J]. Journal of Differential Equations, 2024, 390:228-344.
[19] SHAO K R, TAKAMURA H, WANG C B. Blow-up of solutions to semilinear wave equations with spatial derivatives[J]. Discrete and Continuous Dynamical Systems, 2025, 45(2):410-424.
[20] LIU M Y, WANG C B. The blow-up of solutions to semilinear wave equations on asymptotically Euclidean manifolds[J]. Discrete and Continuous Dynamical Systems, 2023, 43(11):3987-4009.

基金资助

山西省基础研究计划项目(20210302123045)