# American Institute of Mathematical Sciences

March  2012, 2(1): 17-28. doi: 10.3934/mcrf.2012.2.17

## Eventual regularity of a wave equation with boundary dissipation

 1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China 2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China 3 Institut de Recherche Mathématique Avancée, Université Louis Pasteur de Strasbourg, 7 rue René-Descartes, 67084 Strasbourg

Received  March 2011 Revised  October 2011 Published  January 2012

This paper addresses a study of the eventual regularity of a wave equation with boundary dissipation and distributed damping. The equation under consideration is rewritten as a system of first order and analyzed by semigroup methods. By a certain asymptotic expansion theorem, we prove that the associated solution semigroup is eventually differentiable. This implies the eventual regularity of the solution of the wave equation.
Citation: Kangsheng Liu, Xu Liu, Bopeng Rao. Eventual regularity of a wave equation with boundary dissipation. Mathematical Control & Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17
##### References:

show all references

##### References:
 [1] Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577 [2] José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195 [3] Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711 [4] Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361 [5] Jiří Neustupa. On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body. Conference Publications, 2007, 2007 (Special) : 758-767. doi: 10.3934/proc.2007.2007.758 [6] Benjamin Webb. Dynamics of functions with an eventual negative Schwarzian derivative. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1393-1408. doi: 10.3934/dcds.2009.24.1393 [7] Alberto Ferrero, Filippo Gazzola, Hans-Christoph Grunau. Decay and local eventual positivity for biharmonic parabolic equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1129-1157. doi: 10.3934/dcds.2008.21.1129 [8] Filippo Gazzola, Hans-Christoph Grunau. Eventual local positivity for a biharmonic heat equation in RN. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 83-87. doi: 10.3934/dcdss.2008.1.83 [9] Chi Hin Chan, Magdalena Czubak, Luis Silvestre. Eventual regularization of the slightly supercritical fractional Burgers equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 847-861. doi: 10.3934/dcds.2010.27.847 [10] Jinlong Bai, Xuewei Ju, Desheng Li, Xiulian Wang. On the eventual stability of asymptotically autonomous systems with constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4457-4473. doi: 10.3934/dcdsb.2019127 [11] Jerry Bona, Jiahong Wu. Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1141-1168. doi: 10.3934/dcds.2009.23.1141 [12] P. Magal, H. R. Thieme. Eventual compactness for semiflows generated by nonlinear age-structured models. Communications on Pure & Applied Analysis, 2004, 3 (4) : 695-727. doi: 10.3934/cpaa.2004.3.695 [13] Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199 [14] Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963 [15] Tetsuya Ishiwata. Motion of polygonal curved fronts by crystalline motion: v-shaped solutions and eventual monotonicity. Conference Publications, 2011, 2011 (Special) : 717-726. doi: 10.3934/proc.2011.2011.717 [16] Piotr Kościelniak, Marcin Mazur. On $C^0$ genericity of various shadowing properties. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 523-530. doi: 10.3934/dcds.2005.12.523 [17] Kingshook Biswas. Maximal abelian torsion subgroups of Diff( C,0). Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 839-844. doi: 10.3934/dcds.2011.29.839 [18] Lavinia Roncoroni. Exact lumping of feller semigroups: A $C^{\star}$-algebras approach. Conference Publications, 2015, 2015 (special) : 965-973. doi: 10.3934/proc.2015.0965 [19] Wael Bahsoun, Benoît Saussol. Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6657-6668. doi: 10.3934/dcds.2016089 [20] Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385

2018 Impact Factor: 1.292