热弹性方程的快速增长或衰减估计

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 1042-1052.

热弹性方程的快速增长或衰减估计

李远飞*(),石金诚,朱慧珊,黄诗淇

^{广东财经大学华商学院应用数学系广州 511300}

收稿日期:2020-10-20 出版日期:2021-08-26 发布日期:2021-08-09
通讯作者: 李远飞 E-mail:liqfd@163.com
基金资助:
广东省普通高校重点项目（自然科学）(2019KZDXM042);广州华商学院科研团队项目(2021HSKT01)

Fast Growth or Decay Estimates of Thermoelastic Equations in an External Domain

Yuanfei Li*(),Jincheng Shi,Huishan Zhu,Shiqi Huang

^{Department of Apllied Mathematics, Huashang College Guangdong University of Finance & Economics, Guangzhou 511300}

Received:2020-10-20 Online:2021-08-26 Published:2021-08-09
Contact: Yuanfei Li E-mail:liqfd@163.com
Supported by:
Key Projects of Colleges and Universities in Guangdong Province(Natural Science)(2019KZDXM042);the Science Research Team Project in Guangzhou Huashang College(2021HSKT01)

摘要/Abstract

摘要：

考虑了定义在三维球体外部区域上的Ⅲ型热弹性方程，其中假设方程的解在球体边界上满足一定的边界条件.通过对边界条件做出一定约束之后，在能量表达式中设置一个任意的大于零的参数，利用能量分析法和微分不等式技术，获得了解随距原点的距离的快速增长率或衰减率.在衰减的情况下，证明了解对系数的连续依赖性.

关键词: Ⅲ型热弹性方程, 衰减率, 增长率, 连续依赖性

Abstract:

The Thermoelastic equation of type Ⅲ defined on the outer region of a three-dimensional sphere is considered. It is assumed that the solution of the equation satisfies certain boundary conditions at the boundary of the sphere. By setting an arbitrary parameter greater than zero in the energy expression after making certain constraints on the boundary conditions, the fast growth rate or decay rate of the solution with the distance from the origin is obtained by using energy analysis method and differential inequality technology. In the case of decay, the continuous dependence of solutions on coefficients is proved.

Key words: Type Ⅲ thermoelastic equation, Decay rate, Growth rate, Continuous dependence

中图分类号:

O175.29

李远飞,石金诚,朱慧珊,黄诗淇. 热弹性方程的快速增长或衰减估计[J]. 数学物理学报, 2021, 41(4): 1042-1052.

Yuanfei Li,Jincheng Shi,Huishan Zhu,Shiqi Huang. Fast Growth or Decay Estimates of Thermoelastic Equations in an External Domain[J]. Acta mathematica scientia,Series A, 2021, 41(4): 1042-1052.

参考文献 28

1	Saint-Venant A . Mémoire sur la flexion des prismes. J Math Pures Appl, 1856, 1 (2): 89- 189
2	Ames K A , Payne L E , Schaefer P W . Spatial decay estimates in time-dependent Stokes flow. SIAM J Math Anal, 1993, 24, 1395- 1413 doi: 10.1137/0524081
3	Liu Y , Li Y F , Lin Y W , Yao Z . Spatial decay bounds for the channel flow of the Boussinesq equations. J Math Anal Appl, 2011, 381, 87- 109 doi: 10.1016/j.jmaa.2011.02.066
4	Li Y F , Lin C . Spatial decay for solutions to 2-D Boussinesq system with variable thermal diffusivity. Acta Applicandae Mathematicae, 2017, 154, 111- 130
5	Knops R J , Quintanilla R . Spatial behaviour in thermoelastostatic cylinders of indefinitely increasing cross-section. J Elast, 2015, 121, 89- 117 doi: 10.1007/s10659-015-9523-8
6	Knops R J , Quintanilla R . Spatial decay in transient heat conduction for general elongated regions. Q Appl Math, 2018, 76 (4): 611- 625
7	Horgan C O , Payne L E , Wheeler L T . Spatial decay estimates in transient heat equation. Quart Appl Math, 1984, 42, 119- 127 doi: 10.1090/qam/736512
8	Lin C , Payne L E . Phragmén-Lindelöf alternative for a class of quasilinear second order parabolic problems. Diff Integ Equa, 1995, 8, 539- 551
9	Liu Y , Lin C . Phragmén-Lindelöf type alternative results for the Stokes flow equation. Mathematical Inequalities & Applications, 2006, 9 (4): 671- 694
10	Leseduarte M C , Quintanilla R . Phragmén-Lindelöf alternative for the Laplace equation with dynamic boundary conditions. Journal of Applied Analysis and Computation, 2017, 7 (4): 1323- 1335 doi: 10.11948/2017081
11	李远飞. 在一个半无穷柱体上的非标准Stokes流体方程的二择一问题. 应用数学和力学, 2020, 41 (4): 406- 419
	Li Y F . Phragmén-Lindelöf type results for Non-Standard stokes flow equations around semi-infinite cylinder. Applied Mathematics and Mechanics, 2020, 41 (4): 406- 419
12	李远飞, 肖胜中, 郭连红, 曾鹏. 一类二阶拟线性瞬态方程组的Phragmén-Lindelöf型二择性结果. 吉林大学学报(理学版), 2020, 58 (5): 1047- 1054
	Li Y F , Xiao S Z , Guo L H , Zeng P . Phragmén-Lindelöf type alternative results for a class of second order quasilinear transient equations. Journal of Jilin University(Science Edition), 2020, 58 (5): 1047- 1054
13	李远飞, 李志青. 具有非线性边界条件的瞬态热传导方程的二择一结果. 数学物理学报, 2020, 40A (5): 1248- 1258 doi: 10.3969/j.issn.1003-3998.2020.05.012
	Li Y F , Li Z Q . Phragmén-Lindelöf type Results for transient heat conduction equation with nonlinear boundary conditions. Acta Mathematica Scientia, 2020, 40A (5): 1248- 1258 doi: 10.3969/j.issn.1003-3998.2020.05.012
14	Horgan C O , Payne L E . Phragmén-Lindelöf type results for harmonic functions with nonlinear boundary conditions. Archive for Rational Mechanics and Analysis, 1993, 122 (2): 123- 144 doi: 10.1007/BF00378164
15	Quintanilla R . Some remarks on the fast spatial growth/decay in exterior regions. Z Angew Math Phys, 2019, 70, 83 doi: 10.1007/s00033-019-1127-x
16	Hameed A A , Harfash A J . Continuous dependence of double diffusive convection in a porous medium with temperature-dependent density. Basrah Journal of Science, 2019, 37, 1- 15
17	李远飞. 海洋动力学中二维粘性原始方程组解对热源的收敛性. 应用数学和力学, 2020, 41 (3): 339- 352
	Li Y F . Convergence results on heat source for 2D viscous primitive equations of ocean dynamics. Applied Mathematics and Mechanics, 2020, 41 (3): 339- 352
18	李远飞, 郭连红. 具有边界反应Brinkman-Forchheimer型多孔介质的结构稳定性. 高校应用数学学报, 2019, 34 (3): 315- 324
	LI Y F , Guo L H . Structural stability on boundary reaction terms in a porous medium of Brinkman-Forchheimer type. Applied Mathematics A Journal of Chinese Universities, 2019, 34 (3): 315- 324
19	李远飞. 大尺度湿大气原始方程组对黏性系数的连续依赖性. 吉林大学学报(理学版), 2020, 58 (4): 744- 752
	Li Y F . Continuous dependence of primitive equations of large-scale moist atmosphere on viscosity coefficient. Journal of Jilin University(Science Edition), 2020, 58 (4): 744- 752
20	Liu Y . Continuous dependence for a thermal convection model with temperature-dependent solubitity. Appl Math Comput, 2017, 308, 18- 30
21	Liu Y , Xiao S Z , Lin Y W . Continuous dependence for the Brinkman-Forchheimer fluid interfacing with a Darcy fluid in a bounded domain. Mathematics and Computers in Simulation, 2018, 150, 66- 88 doi: 10.1016/j.matcom.2018.02.009
22	Green A E , Naghdi P M . A re-examination of the basic postulates of thermomechanics. Proceeding of Royal Society London A, 1991, 432, 171- 194
23	Green A E , Naghdi P M . On undamped heat waves in an elastic solid. Journal of Thermal Stresses, 1992, 15, 253- 264 doi: 10.1080/01495739208946136
24	Zhang X , Zuazua E . Decay of solutions of the system of thermoelasticity of type Ⅲ. Communications in Contempotrary Mathematics, 2003, 5 (1): 25- 83 doi: 10.1142/S0219199703000896
25	Messaoundi S A , Soufyane A . Boundary stabilization of memory type in thermoelasticity of type Ⅲ. Applicable Analysis, 2008, 87 (1): 13- 28 doi: 10.1080/00036810701714180
26	Hirsch M W , Smale S . Differential Equations, Dynamical Systems, and Linear Algebra. New York: Academic Press, 1974
27	Payne L E , Song J C . Phragmén-Lindelöf and continuous dependence type results in generalized heat conduction. Z Angew Math Phys, 1996, 47, 527- 538 doi: 10.1007/BF00914869
28	Song J C , Yoon D S . Phragmén-Lindelöf type and continuous dependence results in generalized dissipative heat conduction. J Korean Math Soc, 1998, 35 (4): 945- 960

[1]	朱旭生, 李芳娥. 带非线性阻尼项的三维可压缩欧拉方程组的整体解[J]. 数学物理学报, 2014, 34(5): 1111-1122.
[2]	钟吉玉, 李晓培. 关于迭代方程的集值解[J]. 数学物理学报, 2011, 31(4): 970-977.
[3]	刘颖, 李佳. 半空间MHD方程组弱解的L²衰减[J]. 数学物理学报, 2010, 30(4): 1166-1175.
[4]	董柏青; 李用声. 一类修正的Navier-Stokes方程的长时间性态[J]. 数学物理学报, 2006, 26(4): 498-505.
[5]	赵新泉. 奇异积分方程解的稳定性[J]. 数学物理学报, 1998, 18(1): 63-69.
[6]	郭百昌. 高阶KdV方程(组)Cauchy问题解的唯一性[J]. 数学物理学报, 1995, 15(1): 98-102.