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Keywords:
hyperbolic systems; periodic-Dirichlet problems; anisotropic Sobolev spaces; a priori estimates
hyperbolic systems; periodic-Dirichlet problems; anisotropic Sobolev spaces; a priori estimates
Summary:
We study one-dimensional linear hyperbolic systems with $L^{\infty}$-coeffici\-ents subjected to periodic conditions in time and reflection boundary conditions in space. We derive a priori estimates and give an operator representation of solutions in the whole scale of Sobolev-type spaces of periodic functions. These spaces give an optimal regularity trade-off for our problem.
References:
[1] Bandelow U., Recke L., Sandstede B.: Frequency regions for forced locking of self-pulsating multi-section DFB lasers. Optics Comm. 147 (1998), 212-218.
[2] Bourbaki N.: Integration, Chapters 1/4. Actualités Scientifiques et Industrielles, Hermann, Paris, 1966. Zbl 1116.28002
[3] Jochmann F., Recke L.: Well-posedness of an initial boundary value problem from laser dynamics. Math. Models Methods Appl. Sci. 12 (2002), 593-606. MR 1899843 | Zbl 1025.35011
[4] Garrett P.: Functions on circles. 2006; Eprint: www.math.umn.edu/$^\sim $garrett/m/mfms/notes/09\_sobolev.ps.
[5] Gorbachuk V.I., Gorbachuk M.L.: Boundary value problems for operator differential equations. Naukova Dumka, Kiev, 1984; English translation: Kluwer Academic Publishers, Dordrecht, 1991. MR 0776604 | Zbl 0845.34065
[6] Herrmann L.: Periodic solutions of abstract differential equations: the Fourier method. Czechoslovak Math. J. 30 (105) (1980), 177-206. MR 0566046 | Zbl 0445.35013
[7] Kmit I., Recke L.: Fredholm alternative for periodic-Dirichlet problems for linear hyperbolic systems. J. Math. Anal. Appl. 335 (2007), 355-370. MR 2340326 | Zbl 1160.35046
[8] Recke L., Schneider K.R., Strygin V.V.: Spectral properties of coupled wave equations. Z. Angew. Math. Phys. 50 (1999), 6 925-933. MR 1735638
[9] Rehberg J., Wünsche H.-J., Bandelow U., Wenzel H.: Spectral properties of a system describing fast pulsating DFB lasers. Z. Angew. Math. Mech. 77 (1997), 75-77. MR 1433576
[10] Robinson J.C.: Infinite-Dimensional Dynamical Systems. Cambridge Texts in Appl. Math., Cambridge University Press, Cambridge, 2001. MR 1881888 | Zbl 1084.37063
[11] Sieber J.: Numerical bifurcation analysis for multi-section semiconductor lasers. SIAM J. Appl. Dyn. Syst. 1 (2002), 248-270. MR 1968370
[12] Sieber J., Recke L., Schneider K.: Dynamics of multisection semiconductor lasers. J. Math. Sci. (New York) 124 (2004), 5 5298-5309. MR 2129136
[13] Tromborg B., Lassen H.E., Olesen H.: Traveling wave analysis of semiconductor lasers. IEEE J. of Quant. El. 30 (1994), 5 939-956.
[14] Vejvoda O. et al.: Partial Differential Equations: Time-Periodic Solutions. Sijthoff Noordhoff, 1981. Zbl 0501.35001
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