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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

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 & 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, 2007 (): 160.   Google Scholar

[2]

M. Chipot, On some anisotropic singular perturbation problems, Asymptot. Ana., 55 (2007), 125-144.  Google Scholar

[3]

M. Chipot and S. Guesmia, On the asymptotic behavior of elliptic, anisotropic singular perturbations problems, Commun. Pure Appl. Anal., 8 (2009), 179-193.  Google Scholar

[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.  Google Scholar

[5]

M. Chipot and S. Guesmia, On some anisotropic, nonlocal, parabolic singular perturbations problems,, Applicable Analysis, ().   Google Scholar

[6]

M. Chipot and S. Guesmia, Correctors for some asymptotic problems, Proc. Steklov Inst. Math., 270 (2010), 263-277. doi: 10.1134/S0081543810030211.  Google Scholar

[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.  Google Scholar

[8]

M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci. Paris, 346 (2008), 21-26.  Google Scholar

[9]

S. Guesmia, "Etude du Comportement Asymptotique de certaines Équations aux Dérivées Partielles dans des Domaines Cylindriques," Thèse Université de Haute Alsace, 2006. Google Scholar

[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.  Google Scholar

[11]

S. Guesmia, Asymptotic behaviour of elliptic boundary-value problems with some small coefficients, Electron. J. Differential Equations, 59 (2008), 1-13.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.  Google Scholar

[15]

W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551.  Google Scholar

show all references

References:
[1]

B. Brighi and S. Guesmia, Asymptotic behaviour of solutions of hyperbolic problems on a cylindrical domain,, in, 2007 (): 160.   Google Scholar

[2]

M. Chipot, On some anisotropic singular perturbation problems, Asymptot. Ana., 55 (2007), 125-144.  Google Scholar

[3]

M. Chipot and S. Guesmia, On the asymptotic behavior of elliptic, anisotropic singular perturbations problems, Commun. Pure Appl. Anal., 8 (2009), 179-193.  Google Scholar

[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.  Google Scholar

[5]

M. Chipot and S. Guesmia, On some anisotropic, nonlocal, parabolic singular perturbations problems,, Applicable Analysis, ().   Google Scholar

[6]

M. Chipot and S. Guesmia, Correctors for some asymptotic problems, Proc. Steklov Inst. Math., 270 (2010), 263-277. doi: 10.1134/S0081543810030211.  Google Scholar

[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.  Google Scholar

[8]

M. Chipot and K. Yeressian, Exponential rates of convergence by an iteration technique, C. R. Math. Acad. Sci. Paris, 346 (2008), 21-26.  Google Scholar

[9]

S. Guesmia, "Etude du Comportement Asymptotique de certaines Équations aux Dérivées Partielles dans des Domaines Cylindriques," Thèse Université de Haute Alsace, 2006. Google Scholar

[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.  Google Scholar

[11]

S. Guesmia, Asymptotic behaviour of elliptic boundary-value problems with some small coefficients, Electron. J. Differential Equations, 59 (2008), 1-13.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.  Google Scholar

[15]

W. A. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math., 19 (1966), 543-551.  Google Scholar

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