American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. Google Scholar

[10]

M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum, 51 (1995), 343-346. Google Scholar

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X. Yu, Differentiability of the age-dependent population system with time delay in the birth process, J. Math. Anal. Appl., 303 (2005), 576-584. Google Scholar

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X. Yu and K. Liu, Eventual differentiability of functional differential equations in Banach spaces, J. Math. Anal. Appl., 327 (2007), 792-800. Google Scholar

[13]

L. Zhang, Differentiability of the population semigroup, (Chinese) J. Systems Sci. Math. Sci., 8 (1988), 181-187. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

G. Di Blasio, K. Kunisch and E. Sinestrari, Stability for abstract linear functional differential equations, Israel J. Math., 50 (1985), 231-263.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. Google Scholar

[10]

M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum, 51 (1995), 343-346. Google Scholar

[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. Google Scholar

[12]

X. Yu and K. Liu, Eventual differentiability of functional differential equations in Banach spaces, J. Math. Anal. Appl., 327 (2007), 792-800. Google Scholar

[13]

L. Zhang, Differentiability of the population semigroup, (Chinese) J. Systems Sci. Math. Sci., 8 (1988), 181-187. Google Scholar

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