具有记忆项的对数 Boussinesq型方程解的长时间行为研究
Study on the Longtime Behavior of the Solution of Logarithmic Boussinesq Type Equations with Memory
DOI: 10.12677/PM.2023.1311343, PDF, 下载: 84  浏览: 160 
作者: 王爽, 闫龙：东北电力大学理学院，吉林 吉林
关键词: 对数梁方程；记忆项；整体存在性；指数增长；能量衰减；Logarithmic Beam Equations； Memory； Global Existence； Exponential Growth； Energy Decay
摘要: 本文考虑一类具有记忆项的对数梁方程的初边值问题。利用 Galerkin 方法结合对数 Sobolev 不等式及对数 Gronwall 不等式，我们证明了解的全局存在性。在此基础上，我们借助位势井思想进一步得到了系统在适当初值条件下的指数烹减及指数增长。
Abstract: This paper is concerned with the initial value problem of a logarithmic beam equations with memory. Using Galerkin method, logarithmic Sobolev inequality and the Gronwall inequality, we obtain the global existence of the solutions. Moreover, we prove the exponential decay and exponential growth of the system by using potential well theory.
文章引用：王爽, 闫龙. 具有记忆项的对数 Boussinesq型方程解的长时间行为研究[J]. 理论数学, 2023, 13(11): 3295-3315. https://doi.org/10.12677/PM.2023.1311343

参考文献

[1] Boussinesq, J. (1872) Théorie des ondes et des remous qui se propagent le long d’un canalrectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitessessensiblement pareilles de la surface au fond. Journal de Mathématiques Pures et Appliquées, 17, 55-108.
[2] Zakharov, V. (1973) On Stochastization of One-Dimensional Chains of Nonlinear Oscillators. Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 65, 219-225.
[3] Liu, Y. (1995) Instability and Blow-Up of Solutions to a Generalized Boussinesq Equation. SIAM Journal on Mathematical Analysis, 26, 1527-1546.
https://doi.org/10.1137/S0036141093258094
[4] Wang, Y. and Mu, C. (2007) Global Existence and Blow-Up of the Solutions for the Multidimensional Generalized Boussinesq Equation. Mathematical Methods in the Applied Sciences, 30, 1403-1417.
https://doi.org/10.1002/mma.846
[5] Górka, P. (2009) Logarithmic Klein-Gordon Equation. Acta Physica Polonica B, 40, 59-66.
[6] Wazwaz, A. (2015) Gaussian Solitary Waves for the Logarithmic Boussinesq Equation and the Logarithmic Regularized Boussinesq Equation. Ocean Engineering, 94, 111-115.
https://doi.org/10.1016/j.oceaneng.2014.11.024
[7] Hu, Q., Zhang, H. and Liu, G. (2016) Global Existence and Exponential Growth of Solution for the Logarithmic Boussinesq-Type Equation. Journal of Mathematical Analysis and Applications, 436, 990-1001.
https://doi.org/10.1016/j.jmaa.2015.11.082
[8] Hu, Q. and Zhang, H. (2017) Initial Boundary Value Problem for Generalized Logarithmic Improved Boussinesq Equation. Mathematical Methods in the Applied Sciences, 40, 3687-3697.
https://doi.org/10.1002/mma.4255
[9] Renardy, M., Hrusa, W.J. and Nohel, J.A. (1987) Mathematical Problems in Viscoelasticity. American Mathematical Society, New York.
[10] Cavalcanti, M.M., et al. (2017) Exponential Stability for the Wave Equation with Degenerate Nonlocal Weak Damping. Israel Journal of Mathematics, 219, 189-213.
https://doi.org/10.1007/s11856-017-1478-y
[11] Park, J. and Kim, J. (2004) Existence and Uniform Decay for Euler-Bernoulli Beam Equation with Memory Term. Mathematical Methods in the Applied Sciences, 27, 1629-1640.
https://doi.org/10.1002/mma.512
[12] Narciso, V. (2015) Long-Time Behavior of a Nonlinear Viscoelastic Beam Equation with Past History. Mathematical Methods in the Applied Sciences, 38, 775-784.
https://doi.org/10.1002/mma.3109
[13] Peyravi, A. (2020) General Stability and Exponential Growth for a Class of Semi-Linear Wave Equations with Logarithmic Source and Memory Terms. Applied Mathematics Optimization, 81, 545-561.
https://doi.org/10.1007/s00245-018-9508-7
[14] 吴晓霞,马巧珍.带有线性记忆的波方程在 Rn上的时间依赖吸引子 [J].应用数学,2021,34(1)73-85.
[15] 代辉亚，张宏伟.一类具记忆项的非线性强阻尼双曲方程解的爆破性 [J].数学的实践与认识2014,44(18): 266-270.
[16] Evans, L. (2003) Entropy and Partial Differential Equations. Library of Congress Catalogingin- Publication.

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