Some singular perturbations results for semilinear hyperbolic problems

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June 2012, 5(3): 567-580. doi: 10.3934/dcdss.2012.5.567

Some singular perturbations results for semilinear hyperbolic problems

Senoussi Guesmia ^1, and Abdelmouhcene Sengouga ^2,

1.	University of Zürich, Institute of Mathematics, Winterthurerstrasse 190, CH-8057 Zurich
2.	Department of Mathematics, University of M’sila, BP:166, 28000, M’sila, Algeria

Received August 2010 Revised December 2010 Published October 2011

Abstract
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This paper is concerned with the asymptotic behaviour of the solutions of some semilinear hyperbolic problems. Using the monotonicity hypothesis, convergence results are shown in different spaces depending on the derivative directions of an arbitrary domain $\Omega .$ Some improvements are established when $\Omega $ is a cylinder.

Keywords: Semilinear hyperbolic problems, monotone operators., anisotropic singular perturbations, asymptotic behaviour.

Mathematics Subject Classification: 35B25, 35B40, 35L15, 35L20, 35L7.

Citation: Senoussi Guesmia, Abdelmouhcene Sengouga. Some singular perturbations results for semilinear hyperbolic problems. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 567-580. doi: 10.3934/dcdss.2012.5.567

References:

[1]	B. Brighi and S. Guesmia, Asymptotic behaviour of solutions of hyperbolic problems on a cylindrical domain, in "Discrete Contin. Dyn. Syst.," 2007, Dynamical Systems and Differential Equations, Proceedings of the 6^th AIMS International Conference, suppl., 160-169.
[2]	M. Chipot, On some anisotropic singular perturbation problems, Asymptot. Ana., 55 (2007), 125-144.
[3]	M. Chipot and S. Guesmia, On the asymptotic behavior of elliptic, anisotropic singular perturbations problems, Commun. Pure Appl. Anal., 8 (2009), 179-193.
[4]	M. Chipot and S. Guesmia, On a class of integro-differential problems, Commun. Pure Appl. Anal., 9 (2010), 1249-1262. doi: 10.3934/cpaa.2010.9.1249.
[5]	M. Chipot and S. Guesmia, On some anisotropic, nonlocal, parabolic singular perturbations problems, Applicable Analysis, to appear.
[6]	M. Chipot and S. Guesmia, Correctors for some asymptotic problems, Proc. Steklov Inst. Math., 270 (2010), 263-277. doi: 10.1134/S0081543810030211.
[7]	M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 319-338. doi: 10.3934/dcdsb.2001.1.319.
[8]	M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci. Paris, 346 (2008), 21-26.
[9]	S. Guesmia, "Etude du Comportement Asymptotique de certaines Équations aux Dérivées Partielles dans des Domaines Cylindriques," Thèse Université de Haute Alsace, 2006.
[10]	S. Guesmia, Some results on the asymptotic behavior for hyperbolic problems in cylindrical domains becoming unbounded, J. Math. Anal. Appl., 341 (2008), 1190-1212. doi: 10.1016/j.jmaa.2007.11.001.
[11]	S. Guesmia, Asymptotic behaviour of elliptic boundary-value problems with some small coefficients, Electron. J. Differential Equations, 59 (2008), 1-13.
[12]	S. Guesmia and A. Sengouga, Anisotropic singular perturbation of hyperbolic problems, Appl. Math. Comput. 217 (2011), 8983-8996. doi: 10.1016/j.amc.2011.03.104.
[13]	J.-L. Lions, "Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal," Lecture Notes in Math., 323, Springer-Verlag, Berlin-New York, 1973.
[14]	J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
[15]	W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551.

show all references

References:

[1]	B. Brighi and S. Guesmia, Asymptotic behaviour of solutions of hyperbolic problems on a cylindrical domain, in "Discrete Contin. Dyn. Syst.," 2007, Dynamical Systems and Differential Equations, Proceedings of the 6^th AIMS International Conference, suppl., 160-169.
[2]	M. Chipot, On some anisotropic singular perturbation problems, Asymptot. Ana., 55 (2007), 125-144.
[3]	M. Chipot and S. Guesmia, On the asymptotic behavior of elliptic, anisotropic singular perturbations problems, Commun. Pure Appl. Anal., 8 (2009), 179-193.
[4]	M. Chipot and S. Guesmia, On a class of integro-differential problems, Commun. Pure Appl. Anal., 9 (2010), 1249-1262. doi: 10.3934/cpaa.2010.9.1249.
[5]	M. Chipot and S. Guesmia, On some anisotropic, nonlocal, parabolic singular perturbations problems, Applicable Analysis, to appear.
[6]	M. Chipot and S. Guesmia, Correctors for some asymptotic problems, Proc. Steklov Inst. Math., 270 (2010), 263-277. doi: 10.1134/S0081543810030211.
[7]	M. Chipot and A. Rougirel, On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 319-338. doi: 10.3934/dcdsb.2001.1.319.
[8]	M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci. Paris, 346 (2008), 21-26.
[9]	S. Guesmia, "Etude du Comportement Asymptotique de certaines Équations aux Dérivées Partielles dans des Domaines Cylindriques," Thèse Université de Haute Alsace, 2006.
[10]	S. Guesmia, Some results on the asymptotic behavior for hyperbolic problems in cylindrical domains becoming unbounded, J. Math. Anal. Appl., 341 (2008), 1190-1212. doi: 10.1016/j.jmaa.2007.11.001.
[11]	S. Guesmia, Asymptotic behaviour of elliptic boundary-value problems with some small coefficients, Electron. J. Differential Equations, 59 (2008), 1-13.
[12]	S. Guesmia and A. Sengouga, Anisotropic singular perturbation of hyperbolic problems, Appl. Math. Comput. 217 (2011), 8983-8996. doi: 10.1016/j.amc.2011.03.104.
[13]	J.-L. Lions, "Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal," Lecture Notes in Math., 323, Springer-Verlag, Berlin-New York, 1973.
[14]	J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
[15]	W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551.

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