Openness of the set of non-characteristic points and regularity of the blow-up curve for the 1 D semilinear wave equation

Communications in Mathematical Physics

Volumn 282, Issue 1, 2008, Pages 55-86

  Merle, Frank ^a   Zaag, Hatem ^b  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b UNIVERSITÉ PARIS 13   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 46749129675     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-008-0532-3     Document Type: Article

References (25)

1. Alinhac, S.: Blowup for nonlinear hyperbolic equations. Volume 17 of Progress in Nonlinear Differential Equations and their Applications. Boston, MA: Birkhäuser Boston Inc. 1995
2. Alinhac, S.: A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations. In: Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), Nantes:Univ. Nantes, 2002 pp. Exp. No. I, 33
3. Antonini C., Merle F.: Optimal bounds on positive blow-up solutions for a semilinear wave equation. Internat. Math. Res. Notices. 21, 1141-1167 (2001)
4. Caffarelli L.A., Friedman A.: Differentiability of the blow-up curve for one-dimensional nonlinear wave equations. Arch. Rati. Mech. Anal. 91(1), 83-98 (1985)
5. Caffarelli L.A., Friedman. A.: The blow-up boundary for nonlinear wave equations. Trans. Amer. Math. Soc. 297(1), 223-241 (1986)
6. Ginibre J., Soffer A., Velo G.: The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal. 110(1), 96-130 (1992)
7. Kichenassamy S., Littman W.: Blow-up surfaces for nonlinear wave equations. I. Comm. Partial Differe. Eq. 18(3-4), 431-452 (1993)
8. Kichenassamy S., Littman W.: Blow-up surfaces for nonlinear wave equations. II. Comm. Partial Differe. Eq. 18(11), 1869-1899 (1993)
9. Levine H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu ⊂ tt=-Au F}u). Trans. Amer.Math. Soc. 192, 1-21 (1974)
10. Martel Y., Merle F.: A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. (9). 79(4), 339-425 (2000)
11. Martel Y., Merle F.: Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation. Ann. of Math. (2). 155(1), 235-280 (2002)
12. 2 critical nonlinear Schrödinger equation. Invent. Math. 156(3), 565-672 (2004)
13. Merle F., Raphael P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Annals of Math (2). 161((1), 157-222 (2005)
14. Merle F., Zaag H.: Optimal estimates for blowup rate and behavior for nonlinear heat equations. Comm. Pure Appl. Math. 51(2), 139-196 (1998)
15. Merle F., Zaag H.: A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Ann. 316(1), 103-137 (2000)
16. Merle F., Zaag H.: Determination of the blow-up rate for the semilinear wave equation. Amer. J. Math. 125, 1147-1164 (2003)
17. Merle F., Zaag H.: Blow-up rate near the blow-up surface for semilinear wave equations. Internat. Math. Res. Notices. 19, 1127-1156 (2005)
18. Merle F., Zaag H.: Determination of the blow-up rate for a critical semilinear wave equation. Math. Annalen. 331(2), 395-416 (2005)
19. Merle F., Zaag H.: Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension. J. Funct. Anal. 253(1), 43-121 (2007)
20. Merle, F., Zaag, H.: Existence and characterization of characteristic points for a semilinear wave equation in one space dimension. In preparation (2008)
21. 1, α regularity of the blow-up curve at non-characteristic points for the one dimensional semilinear wave equation. Comm. Partial Diff. Eq. (2008) (to appear)
22. Zaag H.: On the regularity of the blow-up set for semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 19(5), 505-542 (2002)
23. Zaag H.: One dimensional behavior of singular N dimensional solutions of semilinear heat equations. Comm. Math. Phys. 225(3), 523-549 (2002)
24. Zaag, H.: Regularity of the blow-up set and singular behavior for semilinear heat equations. In: Mathematics & mathematics education (Bethlehem, 2000), River Edge, NJ: World Sci. Publishing, (2002) pp 337-347
25. Zaag H.: Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation. Duke Math. J. 133(3), 499-525 (2006)