# 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 and Related Fields, 2012, 2 (1) : 17-28. doi: 10.3934/mcrf.2012.2.17
##### References:
 [1] C. J. K. Batty, Differentiability and growth bounds of solutions of delay equations, J. Math. Anal. Appl., 299 (2004), 133-146. doi: 10.1016/j.jmaa.2004.04.063. [2] C. J. K. Batty, Differentiability of perturbed semigroups and delay semigroups, in "Perspectives in Operator Theory," Banach Center Publ., 75, Polish Acad. Sci., Warsaw, (2007), 39-53. [3] G. Chen and J. Zhou, "Vibration and Damping in Distributed Systems. Vol. I. Analysis, Estimation, Attenuation, and Design," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. [4] G. Di Blasio, Differentiability of the solution semigroup for delay differential equations, in "Evolution Equations," Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, (2003), 147-158. [5] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263. [6] B. D. Doytchinov, W. J. Hrusa and S. J. Watson, On perturbations of differentiable semigroups, Semigroup Forum, 54 (1997), 100-111. doi: 10.1007/BF02676591. [7] R. H. Fabiano and K. Ito, Semigroup theory in linear viscoelasticity: Weakly and strongly singular kernels, in "Control and Estimation of Distributed Parameter Systems" (Vorau, 1988), Internat. Ser. Numer. Math., 91, Birkhäuser, Basel, (1989), 109-121. [8] K. Liu and Z. Liu, Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces, J. Differential Equations, 141 (1997), 340-355. [9] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [10] M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum, 51 (1995), 343-346. [11] X. Yu, Differentiability of the age-dependent population system with time delay in the birth process, J. Math. Anal. Appl., 303 (2005), 576-584. [12] X. Yu and K. Liu, Eventual differentiability of functional differential equations in Banach spaces, J. Math. Anal. Appl., 327 (2007), 792-800. [13] L. Zhang, Differentiability of the population semigroup, (Chinese) J. Systems Sci. Math. Sci., 8 (1988), 181-187.

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##### References:
 [1] C. J. K. Batty, Differentiability and growth bounds of solutions of delay equations, J. Math. Anal. Appl., 299 (2004), 133-146. doi: 10.1016/j.jmaa.2004.04.063. [2] C. J. K. Batty, Differentiability of perturbed semigroups and delay semigroups, in "Perspectives in Operator Theory," Banach Center Publ., 75, Polish Acad. Sci., Warsaw, (2007), 39-53. [3] G. Chen and J. Zhou, "Vibration and Damping in Distributed Systems. Vol. I. Analysis, Estimation, Attenuation, and Design," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. [4] G. Di Blasio, Differentiability of the solution semigroup for delay differential equations, in "Evolution Equations," Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, (2003), 147-158. [5] G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263. [6] B. D. Doytchinov, W. J. Hrusa and S. J. Watson, On perturbations of differentiable semigroups, Semigroup Forum, 54 (1997), 100-111. doi: 10.1007/BF02676591. [7] R. H. Fabiano and K. Ito, Semigroup theory in linear viscoelasticity: Weakly and strongly singular kernels, in "Control and Estimation of Distributed Parameter Systems" (Vorau, 1988), Internat. Ser. Numer. Math., 91, Birkhäuser, Basel, (1989), 109-121. [8] K. Liu and Z. Liu, Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces, J. Differential Equations, 141 (1997), 340-355. [9] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [10] M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum, 51 (1995), 343-346. [11] X. Yu, Differentiability of the age-dependent population system with time delay in the birth process, J. Math. Anal. Appl., 303 (2005), 576-584. [12] X. Yu and K. Liu, Eventual differentiability of functional differential equations in Banach spaces, J. Math. Anal. Appl., 327 (2007), 792-800. [13] L. Zhang, Differentiability of the population semigroup, (Chinese) J. Systems Sci. Math. Sci., 8 (1988), 181-187.
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