带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破

数学物理学报 ›› 2024, Vol. 44 ›› Issue (6): 1550-1562.

带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破

石金诚¹(),刘炎^2,^*()

¹广州华商学院应用数学系广州 511300
²广东金融学院应用数学系广州 510521

收稿日期:2023-08-31 修回日期:2024-04-29 出版日期:2024-12-26 发布日期:2024-11-22
通讯作者: ^*刘炎, Email: ly801221@163.com
作者简介:石金诚, Email: hning0818@163.com
基金资助:
国家自然科学基金(11223344);广东省自然基金(2023A1515012044);广州华商学院科研团队(2021HSKT01);广州华商学院(2024HSTS09)

Global Existence and Blow-Up for Semilinear Third Order Evolution Equation with Different Power Nonlinearities

Shi Jincheng¹(),Liu Yan^2,^*()

¹Department of Apllied Mathematics, Guangzhou Huashang College, Guangzhou 511300
²Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521

Received:2023-08-31 Revised:2024-04-29 Online:2024-12-26 Published:2024-11-22
Supported by:
NSFC(11223344);Guangdong Natural Science foundation(2023A1515012044);Scientific Research Team Funding of Guangzhou Huashang College(2021HSKT01);Science foundation of Guangzhou Huashang College (Qualitative Study of Thermoelastic Equations)(2024HSTS09)

摘要/Abstract

摘要：

该文研究了一类带不同幂次型非线性项的半线性三阶发展方程的 Cauchy 问题, 其线性化模型来自于考虑 Fourier 法则的经典热弹性板方程组. 首先, 通过适当的远离渐近线的 $L^r$--$L^q$ 估计并结合 Banach 不动点原理, 得到了在小初值条件下整体解的存在性; 其次, 对于满足特定条件的非线性项, 利用检验函数法证明了解的爆破; 最后, 基于这些研究结果, 得出该半线性三阶模型的一些临界指标.

关键词: 整体解的存在性, 半线性三阶发展方程, 爆破, 热弹性板方程组

Abstract:

This paper studies the Cauchy problem of a class of semilinear third-order evolution equations with different power-type nonlinear terms. Its linearized model is derived from the classical thermoelastic plate equations considering Fourier's law. Firstly, by using the appropriate $L^r\!-\!L^q$ estimation away from the asymptote and combining with the Banach fixed point theorem, the existence of the global solution under small initial conditions is obtained. Secondly, for the nonlinear terms that satisfy specific conditions, the explosion of the solution is proved by the test function method. Finally, based on these research results, some critical indicators of the semilinear third-order model are obtained.

Key words: Global existence of solution, Semilinear third order evolution equation, Blow-up, Thermoelastic plate equations

中图分类号:

0175.2

石金诚, 刘炎. 带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破[J]. 数学物理学报, 2024, 44(6): 1550-1562.

Shi Jincheng, Liu Yan. Global Existence and Blow-Up for Semilinear Third Order Evolution Equation with Different Power Nonlinearities[J]. Acta mathematica scientia,Series A, 2024, 44(6): 1550-1562.

参考文献

[1]	Aslan H S, Chen W. Global existence and blow-up for weakly coupled systems of semilinear thermoelastic plate equations. Preprint. arXiv:1904.00778, 2019
[2]	Avalos G, Lasiecka I. Exponential stability of a thermoelastic system without mechanical dissipation. Rend Istit Mat Univ Trieste Suppl, 1997, 28: 1-28
[3]	Bongarti M, Sutthirut C, Lasiecka I. Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics. J Evol Equ, 2021, 21(3): 3553-3584
[4]	Chen W. Cauchy problem for thermoelastic plate equations with different damping mechanisms. Commun Math Sci, 2020, 18(2): 429-457
[5]	Chen W, Ikehata R. The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case. J Differential Equations, 2021, 292: 176-219
[6]	Chen W, Ikehata R, Palmieri A. Asymptotic behaviors for Blackstock's model of thermoviscous flow. Indiana Univ Math J, 2023, 72(2): 489-552
[7]	Chen W, Palmieri A. Nonexistence of global solutions for the semilinear Moore-Gibson-Thompson equation in the conservative case. Discrete Contin Dyn Syst, 2020, 40(9): 5513-5540
[8]	Chen W, Palmieri A. A blow-up result for the semilinear Moore-Gibson-Thompson equation with nonlinearity of derivative type in the conservative case. Evol Equ Control Theory, 2021, 10(4): 673-687
[9]	D'Abbicco M, Ebert M R. A new phenomenon in the critical exponent for structurally dampedsemi-linear evolution equations. Nonlinear Anal, 2017, 149: 1-40
[10]	D'Abbicco M, Reissig M. Semilinear structural damped waves. Math Methods Appl Sci, 2014, 37(11): 1570-1592
[11]	Dell'Oro F, Lasiecka I, Pata V. The Moore-Gibson-Thompson equation with memory in the critical case. J Differential Equations, 2016, 261(7): 4188-4222
[12]	Denk R, Racke R. $L^p$-resolvent estimates and time decay for generalized thermoelastic plate equations. Electron J Differential Equations, 2006, 48: Article 16
[13]	Denk R, Racke R, Shibata Y. $L_p$ theory for the linear thermoelastic plate equations in bounded and exterior domains. Adv Differential Equations, 2009, 14(7/8): 685-715
[14]	Denk R, Racke R, Shibata Y. Local energy decay estimate of solutions to the thermoelastic plate equations in two,- and three-dimensional exterior domains. Z Anal Anwend, 2010, 29(1): 21-62
[15]	Fischer L. Globale L?sungen zu Cauchy problemen bei nichtlinearen thermoelastischen Plattengleichungen. Konstanz, Germany: Master Thesis, University of Konstanz, 2016
[16]	Hidano K, Yokoyama K. Lifespan of small solutions to a system of wave equations. Nonlinear Anal, 2016, 139: 106-130
[17]	Ikeda M, Sobajima M, Wakasa K. Blow-up phenomena of semilinear wave equations and their weakly coupledsystems. J Differential Equations, 2019, 267(9): 5165-5201
[18]	Ikehata R, Tanizawa K. Global existence of solutions for semilinear damped wave equations in ${\rm R}^N$ with noncompactly supported initial data. Nonlinear Anal, 2005, 61(7): 1189-1208
[19]	Kaltenbacher B, Lasiecka I, Pospieszalska M K. Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound. Math Models Methods Appl Sci, 2012, 22(11): 1250035
[20]	Kim J U. On the energy decay of a linear thermoelastic bar and plate. SIAM J Math Anal, 1992, 23(4): 889-899
[21]	Lasiecka I, Triggiani R. Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations. Adv Differential Equations, 1998, 3(3): 387-416
[22]	Lasiecka I, Triggiani R. Analyticity, and lack thereof, of thermo-elastic semigroups. ESAIM Proc EDP Sciences, 1998, 4: 199-222
[23]	Lasiecka I, Triggiani R. Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C. Abstr Appl Anal, 1998, 3(1/2): 153-169
[24]	Lasiecka I, Triggiani R. Analyticity of thermo-elastic semigroups with free boundary conditions. Ann Scuola Norm Sup Pisa CI Sci, 1998, 4(3/4): 457-482
[25]	Liu Y, Chen W. Asymptotic profiles of solutions for regularity-loss-type generalized thermoelastic plate equations and their applications. Z Angew Math Phys, 2020, 71: Article 55
[26]	Liu K, Liu Z. Exponential stability and analyticity of abstract linear thermoelastic systems. Z Angew Math Phys, 1997, 48(6): 885-904
[27]	Mu?oz Rivera J E, Racke R. Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type. SIAM J Math Anal, 1995, 26(6): 1547-1563
[28]	Mu?oz Rivera J E, Racke R. Large solutions and smoothing properties for nonlinear thermoelastic systems. J Differential Equations, 1996, 127(2): 454-483
[29]	Narazaki T. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Discrete Contin Dyn Syst, Dynamical systems, differential equations and applications 7th AIMS Conference. 2009, 2009: 592-601. doi: 10.3934/proc.2009.2009.592
[30]	Nishihara K. Asymptotic behavior of solutions for a system of semilinear heat equations and the corresponding damped wave system. Osaka J Math, 2012, 49(2): 331-348
[31]	Nishihara K, Wakasugi Y. Critical exponent for the Cauchy problem to the weakly coupled damped wave system. Nonlinear Anal, 2014, 108: 249-259
[32]	Nishihara K, Wakasugi Y. Critical exponents for the Cauchy problem to the system of wave equations with time or space dependent damping. Bull Inst Math Acad Sin (NS), 2015, 10(3): 283-309
[33]	Ogawa T, Takeda H. Global existence of solutions for a system of nonlinear damped wave equations. Differential Integral Equations, 2010, 23(7/8): 635-657
[34]	Pham D T, Mohamed Kainane M, Reissig M. Global existence for semi-linear structurally damped $\sigma$-evolution models. J Math Anal Appl, 2015, 431: 569-596
[35]	Racke R, Said-Houari B. Decay rates and global existence for semilinear dissipative Timoshenko systems. Quart Appl Math, 2013, 71(2): 229-266
[36]	Racke R, Ueda Y. Dissipative structures for thermoelastic plate equation in $ R^n$. Adv Differential Equations, 2016, 21(7/8): 610-630
[37]	Racke R, Ueda Y. Nonlinear thermoelastic plate equations-global existence and decay rates for the Cauchy problem. J Differential Equations, 2017, 263(12): 8138-8177
[38]	Racke R, Wang W, Xue R. Optimal decay rates and global existence for a semilinear Timoshenko system with two damping effects. Math Methods Appl Sci, 2017, 40(1): 210-222
[39]	Said-Houari B. Decay properties of linear thermoelastic plates: Cattaneo versus Fourier law. Appl Anal, 2013, 92(2): 424-440
[40]	Sobolev S L. On a theorem of functional analysis. Mat Sbornik, 1938, 4(4): 471-497
[41]	Sun F, Wang M. Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping. Nonlinear Anal, 2007, 66(12): 2889-2910
[42]	Todorova G, Yordanov B. Critical exponent for a nonlinear wave equation with damping. J Differential Equations, 2001, 174(2): 464-489
[43]	Zhang Q S. A blow-up result for a nonlinear wave equation with damping: the critical case. C R Acad Sci Paris Sér I Math, 2001, 333(2): 109-114