Decay of solutions for a Benjamin-Bona-Mahony-Burgers equation on a periodic domain
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In this paper we are concerned with a Benjamin-Bona-Mahony-Burgers (BBMB) equation, when the model is posed on a periodic domain. We extend the results obtained in Bautista, 2023, for the linear equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linear equation decay uniformly to zero. In the case of uniform decay, the result is extended for the nonlinear equation.
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References
Arora, S., Jain, R., & Kukreja, V. K. (2020). Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines. Applied Numerical Mathematics, 154, 1 - 16. https://doi.org/10.1016/j.apnum.2020.03.015.
Bautista, G. (2023). Buena colocación para la ecuación de Benjamin-Bona-Mahony-Burgers lineal con condiciones de frontera periódicas. Hatun Yachay Wasi, 3(1), 68 - 76. https://doi.org/10.57107/hyw.v3i1.58.
Benjamin, T., Bona, J., & Mahony, J. (1972). Model equations for long waves in nonlinear dispersive systems. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272(1220), 47 - 78. https://doi.org/10.1098/rsta.1972.0032.
Bona, J., Chen, M., & Saut, J. (2004). Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media II: Nonlinear theory. Nonlinearity, 17(3), 925 - 952. https://doi.org/10.1088/0951-7715/17/3/010.
Burgers, J. (1948). A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics, 1, 171 - 199. https://doi.org/10.1016/S0065-2156(08)70100-5.
Cazenave, T., & Haraux, A. (1998). An introduction to semilinear evolution equations. Oxford Lecture Series in Mathematics and its Applications, (13). The Clarendon Press, Oxford University Press. https://books.google.com.pe/books/about/An_Introduction_to_Semilinear_Evolution.html?id=pcBo9WZBHyMC&redir_esc=y.
Korteweg, D., & de Vries, G. (1985). On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39(240), 422 - 443. https://doi.org/10.1080/14786449508620739.
Pazy, A. (1983). Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, (44). SpringerVerlag. https://doi.org/10.1007/978-1-4612-5561-1.
Zhang, Q., & Liu, L. (2021). Convergence and stability in maximum norms of linearized fourth-order conservative compact scheme for Benjamin–Bona–Mahony–Burgers’ equation. Journal of Scientific Computing, 87(59). https://doi.org/10.1007/s10915-021-01474-3.