Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 327-358.

We prove that the Cauchy problem for a class of hyperbolic equations with non-Lipschitz coefficients is well-posed in 𝒞 ∞ and in Gevrey spaces. Some counter examples are given showing the sharpness of these results.

Classification : 35L15, 35L10, 35A05 
@article{ASNSP_2002_5_1_2_327_0,
     author = {Colombini, Ferruccio and del Santo, Daniele and Kinoshita, Tamotu},
     title = {Well-posedness of the {Cauchy} problem for a hyperbolic equation with {non-Lipschitz} coefficients},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {327--358},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 1},
     number = {2},
     year = {2002},
     mrnumber = {1991143},
     zbl = {1098.35094},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2002_5_1_2_327_0/}
}
TY  - JOUR
AU  - Colombini, Ferruccio
AU  - del Santo, Daniele
AU  - Kinoshita, Tamotu
TI  - Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2002
SP  - 327
EP  - 358
VL  - 1
IS  - 2
PB  - Scuola normale superiore
UR  - http://archive.numdam.org/item/ASNSP_2002_5_1_2_327_0/
LA  - en
ID  - ASNSP_2002_5_1_2_327_0
ER  - 
%0 Journal Article
%A Colombini, Ferruccio
%A del Santo, Daniele
%A Kinoshita, Tamotu
%T Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2002
%P 327-358
%V 1
%N 2
%I Scuola normale superiore
%U http://archive.numdam.org/item/ASNSP_2002_5_1_2_327_0/
%G en
%F ASNSP_2002_5_1_2_327_0
Colombini, Ferruccio; del Santo, Daniele; Kinoshita, Tamotu. Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 2, pp. 327-358. http://archive.numdam.org/item/ASNSP_2002_5_1_2_327_0/

[1] F. Colombini - E. De Giorgi - S. Spagnolo, Sur les équations hyperboliques avec des coefficients qui ne dépendent que du temp, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 6 (1979), 511-559. | Numdam | MR | Zbl

[2] F. Colombini - D. Del Santo - T. Kinoshita, On the Cauchy problem for hyperbolic operators with non-regular coefficients, to appear in Proceedings of the Conference “À la mémoire de Jean Leray” Karlskrona 2000, M. de Gosson - J. Vaillant (eds.), Kluwer, New York. | MR | Zbl

[3] F. Colombini - N. Lerner, Hyperbolic operators with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657-698. | MR | Zbl

[4] F. Colombini - S. Spagnolo, Some examples of hyperbolic equations without local solvability, Ann. Sci. École Norm. Sup. (4) 22 (1989), 109-125. | Numdam | MR | Zbl

[5] L. Hörmander, “Linear Partial Differential Operators”, Springer-Verlag, Berlin, 1963. | Zbl

[6] E. Jannelli, Regularly hyperbolic systems and Gevrey classes, Ann. Mat. Pura Appl. 140 (1985), 133-145. | MR | Zbl

[7] T. Nishitani, Sur les équations hyperboliques à coefficients höldériens en t et de classe de Gevrey en x, Bull. Sci. Math. 107 (1983), 113-138. | MR | Zbl