| Title:
|
Linear hyperbolic problems in the whole scale of Sobolev-type spaces of periodic functions (English) |
| Author:
|
Kmit, I. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
4 |
| Year:
|
2007 |
| Pages:
|
631-645 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
hyperbolic systems |
| Keyword:
|
periodic-Dirichlet problems |
| Keyword:
|
anisotropic Sobolev spaces |
| Keyword:
|
a priori estimates |
| MSC:
|
35A05 |
| MSC:
|
35B10 |
| MSC:
|
35B45 |
| MSC:
|
35L50 |
| idZBL:
|
Zbl 1199.35213 |
| idMR:
|
MR2375164 |
| . |
| Date available:
|
2009-05-05T17:05:16Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119686 |
| . |
| Reference:
|
[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. |
| Reference:
|
[2] Bourbaki N.: Integration, Chapters 1/4.Actualités Scientifiques et Industrielles, Hermann, Paris, 1966. Zbl 1116.28002 |
| Reference:
|
[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. Zbl 1025.35011, MR 1899843 |
| Reference:
|
[4] Garrett P.: Functions on circles.2006; Eprint: www.math.umn.edu/$^\sim $garrett/m/mfms/notes/09\_sobolev.ps. |
| Reference:
|
[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. Zbl 0845.34065, MR 0776604 |
| Reference:
|
[6] Herrmann L.: Periodic solutions of abstract differential equations: the Fourier method.Czechoslovak Math. J. 30 (105) (1980), 177-206. Zbl 0445.35013, MR 0566046 |
| Reference:
|
[7] Kmit I., Recke L.: Fredholm alternative for periodic-Dirichlet problems for linear hyperbolic systems.J. Math. Anal. Appl. 335 (2007), 355-370. Zbl 1160.35046, MR 2340326 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[10] Robinson J.C.: Infinite-Dimensional Dynamical Systems.Cambridge Texts in Appl. Math., Cambridge University Press, Cambridge, 2001. Zbl 1084.37063, MR 1881888 |
| Reference:
|
[11] Sieber J.: Numerical bifurcation analysis for multi-section semiconductor lasers.SIAM J. Appl. Dyn. Syst. 1 (2002), 248-270. MR 1968370 |
| Reference:
|
[12] Sieber J., Recke L., Schneider K.: Dynamics of multisection semiconductor lasers.J. Math. Sci. (New York) 124 (2004), 5 5298-5309. MR 2129136 |
| Reference:
|
[13] Tromborg B., Lassen H.E., Olesen H.: Traveling wave analysis of semiconductor lasers.IEEE J. of Quant. El. 30 (1994), 5 939-956. |
| Reference:
|
[14] Vejvoda O. et al.: Partial Differential Equations: Time-Periodic Solutions.Sijthoff Noordhoff, 1981. Zbl 0501.35001 |
| . |
