Eventual regularity of a wave equation with boundary dissipation

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

Eventual regularity of a wave equation with boundary dissipation

Kangsheng Liu ^1, , Xu Liu ^2, and Bopeng Rao ^3,

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

Abstract
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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.

Keywords: eventual differentiability, Eventual regularity, $C_0$-semigroups..

Mathematics Subject Classification: Primary: 47D06; Secondary: 35L0.

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.

show all references

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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