Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension

Merle, Frank; Zaag, Hatem

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé 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 $x\mapsto T\left(x\right)$ of its blow-up points and $𝒮\subset ℝ$ the set of all characteristic points and show that $𝒮$ is locally finite. Finally, given $x_0\in 𝒮$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that $T\left(x\right)$ forms a corner of angle $\frac{\pi }{2}$.

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[9] H. A. Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form $P{u}_{tt}=-Au+ℱ\left(u\right)$. Trans. Amer. Math. Soc., 192:1–21, 1974. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl