| Title:
|
Global well-posedness for the Klein-Gordon-Schrödinger system with higher order coupling (English) |
| Author:
|
Soenjaya, Agus Leonardi |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
147 |
| Issue:
|
4 |
| Year:
|
2022 |
| Pages:
|
461-470 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Global well-posedness for the Klein-Gordon-Schrödinger system with generalized higher order coupling, which is a system of PDEs in two variables arising from quantum physics, is proven. It is shown that the system is globally well-posed in $(u,n)\in L^2\times L^2$ under some conditions on the nonlinearity (the coupling term), by using the $L^2$ conservation law for $u$ and controlling the growth of $n$ via the estimates in the local theory. In particular, this extends the well-posedness results for such a system in Miao, Xu (2007) for some exponents to other dimensions and in lower regularity spaces. (English) |
| Keyword:
|
low regularity |
| Keyword:
|
global well-posedness |
| Keyword:
|
Klein-Gordon-Schrödinger equation |
| Keyword:
|
higher order coupling |
| MSC:
|
35G55 |
| MSC:
|
35Q40 |
| idZBL:
|
Zbl 07655820 |
| idMR:
|
MR4512167 |
| DOI:
|
10.21136/MB.2021.0172-20 |
| . |
| Date available:
|
2022-11-16T11:15:26Z |
| Last updated:
|
2023-04-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151092 |
| . |
| Reference:
|
[1] Bachelot, A.: Problème de Cauchy pour des systèmes hyperboliques semi-linéaires.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984), 453-478 French. Zbl 0566.35068, MR 0778979, 10.1016/S0294-1449(16)30414-0 |
| Reference:
|
[2] Bekiranov, D., Ogawa, T., Ponce, G.: Interaction equations for short and long dispersive waves.J. Funct. Anal. 158 (1998), 357-388. Zbl 0909.35123, MR 1648479, 10.1006/jfan.1998.3257 |
| Reference:
|
[3] Biler, P.: Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling.SIAM J. Math. Anal. 21 (1990), 1190-1212. Zbl 0725.35020, MR 1062399, 10.1137/0521065 |
| Reference:
|
[4] Cazenave, T.: Semilinear Schrödinger Equations.Courant Lecture Notes in Mathematics 10. AMS, Providence (2003). Zbl 1055.35003, MR 2002047, 10.1090/cln/010 |
| Reference:
|
[5] Colliander, J.: Wellposedness for Zakharov systems with generalized nonlinearity.J. Differ. Equations 148 (1998), 351-363. Zbl 0921.35162, MR 1643187, 10.1006/jdeq.1998.3445 |
| Reference:
|
[6] Colliander, J., Holmer, J., Tzirakis, N.: Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems.Trans. Am. Math. Soc. 360 (2008), 4619-4638. Zbl 1158.35085, MR 2403699, 10.1090/S0002-9947-08-04295-5 |
| Reference:
|
[7] Fukuda, I., Tsutsumi, M.: On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions.Proc. Japan Acad. 51 (1975), 402-405. Zbl 0313.35065, MR 0380160, 10.3792/pja/1195518563 |
| Reference:
|
[8] Fukuda, I., Tsutsumi, M.: On coupled Klein-Gordon-Schrödinger equations. III: Higher order interaction, decay and blow-up.Math. Jap. 24 (1979), 307-321. Zbl 0415.35071, MR 0550215 |
| Reference:
|
[9] Ginibre, J., Tsutsumi, Y., Velo, G.: On the Cauchy problem for the Zakharov system.J. Funct. Anal. 151 (1997), 384-436. Zbl 0894.35108, MR 1491547, 10.1006/jfan.1997.3148 |
| Reference:
|
[10] Miao, C., Xu, G.: Low regularity global well-posedness for the Klein-Gordon-Schrödinger system with the higher-order Yukawa coupling.Differ. Integral Equ. 20 (2007), 643-656. Zbl 1212.35454, MR 2319459 |
| Reference:
|
[11] Pecher, H.: Some new well-posedness results for the Klein-Gordon-Schrödinger system.Differ. Integral Equ. 25 (2012), 117-142. Zbl 1249.35309, MR 2906550 |
| Reference:
|
[12] Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis.CBMS Regional Conference Series in Mathematics 106. AMS, Providence (2006). Zbl 1106.35001, MR 2233925, 10.1090/cbms/106 |
| Reference:
|
[13] Tzirakis, N.: The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space.Commun. Partial Differ. Equations 30 (2005), 605-641. Zbl 1075.35082, MR 2153510, 10.1081/PDE-200059260 |
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