# American Institute of Mathematical Sciences

February  2020, 19(2): 747-769. doi: 10.3934/cpaa.2020035

## Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

 1 Universidad Privada del Norte, Campus Breña, Av. Tingo María 1122, Lima, Peru 2 Institute of Mathematics, Federal University of Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

* Corresponding author

Received  October 2018 Revised  July 2019 Published  October 2019

In this paper we are concerned with a Boussinesq system for small-amplitude long waves arising in nonlinear dispersive media. Considerations will be given for the global well-posedness and the time decay rates of solutions when the model is posed on a periodic domain and a general class of damping operator acts in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, the result is extended for the full system.

Citation: George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747-769. doi: 10.3934/cpaa.2020035
##### References:

show all references

##### References:
 [1] Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353 [2] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [3] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [4] Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 [5] Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 [6] Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 [7] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [8] Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345 [9] Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346 [10] Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 [11] Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171 [12] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [13] Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291 [14] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [15] Aihua Fan, Jörg Schmeling, Weixiao Shen. $L^\infty$-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 [16] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [17] Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345 [18] Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A socp relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104 [19] Yen-Luan Chen, Chin-Chih Chang, Zhe George Zhang, Xiaofeng Chen. Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models. Journal of Industrial & Management Optimization, 2021, 17 (1) : 501-516. doi: 10.3934/jimo.2020149 [20] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

2019 Impact Factor: 1.105