Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension
Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 11, 10 p.

We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution u(x,t), the graph xT(x) of its blow-up points and 𝒮 the set of all characteristic points and show that 𝒮 is locally finite. Finally, given x 0 𝒮, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that T(x) forms a corner of angle π 2.

@article{SEDP_2009-2010____A11_0,
     author = {Merle, Frank and Zaag, Hatem},
     title = {Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2009-2010},
     note = {talk:11},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2009-2010____A11_0}
}
Merle, Frank; Zaag, Hatem. Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension. Séminaire Équations aux dérivées partielles (Polytechnique) (2009-2010), Talk no. 11, 10 p. http://www.numdam.org/item/SEDP_2009-2010____A11_0/

[1] S. Alinhac. Blowup for nonlinear hyperbolic equations, volume 17 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc., Boston, MA, 1995. | MR 1339762 | Zbl 0820.35001

[2] S. Alinhac. A numerical study of blowup for wave equations with gradient terms. 2006. preprint.

[3] C. Antonini and F. Merle. Optimal bounds on positive blow-up solutions for a semilinear wave equation. Internat. Math. Res. Notices, (21):1141–1167, 2001. | MR 1861514 | Zbl 0989.35090

[4] L. A. Caffarelli and A. Friedman. Differentiability of the blow-up curve for one-dimensional nonlinear wave equations. Arch. Rational Mech. Anal., 91(1):83–98, 1985. | MR 802832 | Zbl 0593.35055

[5] L. A. Caffarelli and A. Friedman. The blow-up boundary for nonlinear wave equations. Trans. Amer. Math. Soc., 297(1):223–241, 1986. | MR 849476 | Zbl 0638.35053

[6] J. Ginibre, A. Soffer, and G. Velo. The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal., 110(1):96–130, 1992. | MR 1190421 | Zbl 0813.35054

[7] S. Kichenassamy and W. Littman. Blow-up surfaces for nonlinear wave equations. I. Comm. Partial Differential Equations, 18(3-4):431–452, 1993. | MR 1214867 | Zbl 0803.35093

[8] S. Kichenassamy and W. Littman. Blow-up surfaces for nonlinear wave equations. II. Comm. Partial Differential Equations, 18(11):1869–1899, 1993. | MR 1243529 | Zbl 0803.35094

[9] H. A. Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt =-Au+(u). Trans. Amer. Math. Soc., 192:1–21, 1974. | MR 344697 | Zbl 0288.35003

[10] Y. Martel and F. Merle. 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. | MR 1888800 | Zbl 1005.35081

[11] F. Merle and P. Raphael. On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. Math., 156(3):565–672, 2004. | MR 2061329 | Zbl 1067.35110

[12] F. Merle and H. Zaag. Optimal estimates for blowup rate and behavior for nonlinear heat equations. Comm. Pure Appl. Math., 51(2):139–196, 1998. | MR 1488298 | Zbl 0899.35044

[13] F. Merle and H. Zaag. A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Annalen, 316(1):103–137, 2000. | MR 1735081 | Zbl 0939.35086

[14] F. Merle and H. Zaag. Determination of the blow-up rate for the semilinear wave equation. Amer. J. Math., 125:1147–1164, 2003. | MR 2004432 | Zbl 1052.35043

[15] F. Merle and H. Zaag. Blow-up rate near the blow-up surface for semilinear wave equations. Internat. Math. Res. Notices, (19):1127–1156, 2005. | MR 2147056 | Zbl 1160.35478

[16] F. Merle and H. Zaag. Determination of the blow-up rate for a critical semilinear wave equation. Math. Annalen, 331(2):395–416, 2005. | MR 2115461 | Zbl 1136.35055

[17] F. Merle and H. Zaag. 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. | MR 2362418 | Zbl 1133.35070

[18] F. Merle and H. Zaag. Openness of the set of non characteristic points and regularity of the blow-up curve for the 1 d semilinear wave equation. Comm. Math. Phys., 282:55–86, 2008. | MR 2415473 | Zbl 1159.35046

[19] F. Merle and H. Zaag. Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension. Amer. J. Math., 2010. to appear.

[20] F. Merle and H. Zaag. Isolatedness of characteristic points for a semilinear wave equation in one space dimension. 2010. preprint.

[21] N Nouaili. A simplified proof of a Liouville theorem for nonnegative solution of a subcritical semilinear heat equations. J. Dynam. Differential Equations, 2008. to appear. | MR 2482011 | Zbl pre05550487

[22] H. Zaag. 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. | MR 2228461 | Zbl 1096.35062