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:
[1]

C. J. K. Batty, Differentiability and growth bounds of solutions of delay equations,, J. Math. Anal. Appl., 299 (2004), 133. doi: 10.1016/j.jmaa.2004.04.063. Google Scholar

[2]

C. J. K. Batty, Differentiability of perturbed semigroups and delay semigroups,, in, 75 (2007), 39. 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, (1993). Google Scholar

[4]

G. Di Blasio, Differentiability of the solution semigroup for delay differential equations,, in, 234 (2003), 147. Google Scholar

[5]

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

[6]

B. D. Doytchinov, W. J. Hrusa and S. J. Watson, On perturbations of differentiable semigroups,, Semigroup Forum, 54 (1997), 100. doi: 10.1007/BF02676591. Google Scholar

[7]

R. H. Fabiano and K. Ito, Semigroup theory in linear viscoelasticity: Weakly and strongly singular kernels,, in, 91 (1989), 109. 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. Google Scholar

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar

[10]

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

[12]

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

[13]

L. Zhang, Differentiability of the population semigroup,, (Chinese) J. Systems Sci. Math. Sci., 8 (1988), 181. 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. doi: 10.1016/j.jmaa.2004.04.063. Google Scholar

[2]

C. J. K. Batty, Differentiability of perturbed semigroups and delay semigroups,, in, 75 (2007), 39. 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, (1993). Google Scholar

[4]

G. Di Blasio, Differentiability of the solution semigroup for delay differential equations,, in, 234 (2003), 147. Google Scholar

[5]

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

[6]

B. D. Doytchinov, W. J. Hrusa and S. J. Watson, On perturbations of differentiable semigroups,, Semigroup Forum, 54 (1997), 100. doi: 10.1007/BF02676591. Google Scholar

[7]

R. H. Fabiano and K. Ito, Semigroup theory in linear viscoelasticity: Weakly and strongly singular kernels,, in, 91 (1989), 109. 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. Google Scholar

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar

[10]

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

[12]

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

[13]

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

[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

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]