A lower bound on the blow-up rate for the Davey–Stewartson system on the torus
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, p. 691-703
We consider the hyperbolic–elliptic version of the Davey–Stewartson system with cubic nonlinearity posed on the two-dimensional torus. A natural setting for studying blow-up solutions for this equation takes place in H s , 1/2<s<1. In this paper, we prove a lower bound on the blow-up rate for these regularities.
@article{AIHPC_2013__30_4_691_0,
     author = {Godet, Nicolas},
     title = {A lower bound on the blow-up rate for the Davey--Stewartson system on the torus},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {30},
     number = {4},
     year = {2013},
     pages = {691-703},
     doi = {10.1016/j.anihpc.2012.12.001},
     zbl = {1288.35113},
     mrnumber = {3082480},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2013__30_4_691_0}
}
Godet, Nicolas. A lower bound on the blow-up rate for the Davey–Stewartson system on the torus. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, pp. 691-703. doi : 10.1016/j.anihpc.2012.12.001. http://www.numdam.org/item/AIHPC_2013__30_4_691_0/

[1] V. Barros, The Davey Stewartson system in weak L p spaces, arXiv:1104.1454 (2011) | MR 2985685

[2] N. Burq, P. Gérard, N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math. 159 no. 1 (2005), 187-223 | MR 2142336 | Zbl 1092.35099

[3] N. Burq, P. Gérard, N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. Éc. Norm. Supér. 38 (2005), 255-301 | Numdam | MR 2144988 | Zbl 1116.35109

[4] H. Chihara, The initial value problem for the elliptic–hyperbolic Davey–Stewartson equation, J. Math. Kyoto Univ. 39 no. 1 (1999), 41-66 | MR 1684172 | Zbl 0947.35036

[5] J.-M. Ghidaglia, J.-C. Saut, On the initial value problem for the Davey–Stewartson systems, Nonlinearity 3 no. 2 (1990), 475-506 | MR 1054584 | Zbl 0727.35111

[6] J.-M. Ghidaglia, J.-C. Saut, Nonexistence of travelling wave solutions to nonelliptic nonlinear Schrödinger equations, J. Nonlinear Sci. 6 no. 2 (1996), 139-145 | MR 1381400 | Zbl 0848.35123

[7] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables dʼespace, Séminaire Bourbaki 796 (1995), 163-187 | Numdam | MR 1423623

[8] N. Godet, N. Tzvetkov, Strichartz estimates for the periodic non elliptic Schrödinger equation, C. R. Acad. Sci. Paris Sér. I 350 (2012), 955-958 | MR 2996773 | Zbl 1260.35161

[9] Z. Hani, Global well-posedness of the cubic nonlinear Schrödinger equation on closed manifolds, Comm. Partial Differential Equations 37 no. 7 (2012), 1186-1236 | MR 2942981 | Zbl 1256.35136

[10] N. Hayashi, Local existence in time of solutions to the elliptic–hyperbolic Davey–Stewartson system without smallness condition on the data, J. Anal. Math. 73 (1997), 133-164 | MR 1616477 | Zbl 0907.35120

[11] C. Klein, B. Muite, K. Roidot, Numerical study of blowup in the Davey–Stewartson system, arXiv:1112.4043 (2011) | MR 3038758

[12] Y. Li, B. Guo, M. Jiang, Existence and blow-up of solutions to degenerate Davey–Stewartson equations, J. Math. Phys. 41 no. 5 (2000), 2943-2956 | MR 1755480 | Zbl 0974.35115

[13] F. Linares, G. Ponce, On the Davey–Stewartson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 no. 5 (1993), 523-548 | MR 1249105 | Zbl 0807.35136

[14] T. Ozawa, Exact blow-up solutions to the Cauchy problem for the Davey–Stewartson systems, Proc. R. Soc. Lond. Ser. A 436 no. 1897 (1992), 345-349 | MR 1177134 | Zbl 0754.35114

[15] G. Richards, Mass concentration for the Davey–Stewartson system, Differential Integral Equations 24 no. 3–4 (2011), 261-280 | MR 2757460 | Zbl 1240.35520

[16] Y. Wang, Periodic cubic hyperbolic Schrödinger equation on T 2 , arXiv:1205.5205 (2012) | MR 3056710

[17] J. Zhang, S. Zhu, Sharp blow-up criteria for the Davey–Stewartson system in R 3 , Dyn. Partial Differ. Equ. 8 no. 3 (2011), 239-260 | MR 2901605 | Zbl 1256.35144