Solution to the semilinear wave equation with a pyramid-shaped blow-up surface

Merle, Frank; Zaag, Hatem

Séminaire Laurent Schwartz — EDP et applications (2016-2017), Exposé no. 6, 13 p.

We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with an isolated characteristic blow-up point at the origin, and a blow-up surface which is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in ${ℝ}^2$. The blow-up surface is differentiable outside the bisectrices. As for the asymptotic behavior in similarity variables, the solution converges to the classical one-dimensional soliton outside the bisectrices. On the bisectrices outside the origin, it converges (up to a subsequence) to a genuinely two-dimensional stationary solution, whose existence is a by-product of the proof. At the origin, it behaves like the sum of 4 solitons localized on the two axes, with opposite signs for neighbors.

This is the first example of a blow-up solution with a characteristic point in higher dimensions, showing a really two-dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the first example of non-characteristic points where the blow-up surface is non-differentiable.

This note gives only the main ideas. For details, see [52.

[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. | DOI | Zbl

[2] 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. | DOI | Zbl

[3] P. Bizoń. Threshold behavior for nonlinear wave equations. J. Nonlinear Math. Phys., 8(suppl.):35–41, 2001. Nonlinear evolution equations and dynamical systems (Kolimbary, 1999). | DOI | MR

[4] P. Bizoń, T. Chmaj, and N. Szpak. Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation. J. Math. Phys., 52(10):103703, 11, 2011. | DOI | MR | Zbl

[5] P. Bizoń, T. Chmaj, and Z. Tabor. On blowup for semilinear wave equations with a focusing nonlinearity. Nonlinearity, 17(6):2187–2201, 2004. | DOI | MR | Zbl

[6] P. Bizoń and A. Zenginoğlu. Universality of global dynamics for the cubic wave equation. Nonlinearity, 22(10):2473–2485, 2009. | DOI | MR | Zbl

[7] A. Bressan. On the asymptotic shape of blow-up. Indiana Univ. Math. J., 39(4):947–960, 1990. | DOI | Zbl

[8] A. Bressan. Stable blow-up patterns. J. Differential Equations, 98(1):57–75, 1992. | DOI | MR | Zbl

[9] J. Bricmont and A. Kupiainen. Universality in blow-up for nonlinear heat equations. Nonlinearity, 7(2):539–575, 1994. | DOI | MR | Zbl

[10] 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. | DOI | MR | Zbl

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

[12] R. Côte. Construction of solutions to the subcritical gKdV equations with a given asymptotical behavior. J. Funct. Anal., 241(1):143–211, 2006. | DOI | MR | Zbl

[13] R. Côte. Construction of solutions to the $L^2$-critical KdV equation with a given asymptotic behaviour. Duke Math. J., 138(3):487–531, 2007. | DOI | MR | Zbl

[14] R. Côte, Y. Martel, and F. Merle. Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations. Rev. Mat. Iberoamericana, 27(1):273–302, 2011. | DOI | MR | Zbl

[15] R. Côte and H. Zaag. Construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension. Comm. Pure Appl. Math., 66(10):1541–1581, 2013. | DOI | MR | Zbl

[16] R. Donninger, W. Schlag, and A. Soffer. On pointwise decay of linear waves on a Schwarzschild black hole background. Comm. Math. Phys., 309(1):51–86, 2012. | DOI | MR | Zbl

[17] R. Donninger and B. Schörkhuber. Stable self-similar blow up for energy subcritical wave equations. Dyn. Partial Differ. Equ., 9(1):63–87, 2012. | DOI | MR | Zbl

[18] R. Donninger and B. Schörkhuber. Stable blow up dynamics for energy supercritical wave equations. Trans. Amer. Math. Soc., 366(4):2167–2189, 2014. | DOI | MR | Zbl

[19] Thomas Duyckaerts, Carlos Kenig, and Frank Merle. Classification of radial solutions of the focusing, energy-critical wave equation. Camb. J. Math., 1(1):75–144, 2013. | DOI | MR | Zbl

[20] M. A. Ebde and H. Zaag. Construction and stability of a blow up solution for a nonlinear heat equation with a gradient term. S$\overrightarrow{\mathrm{e}}$MA J., (55):5–21, 2011. | DOI | MR | Zbl

[21] T. Ghoul and N. Masmoudi. Stability of infinite time aggregation for the critical Patlak-Keller-Segel model in 2 dimension. Submitted, 2016.

[22] M. A. Hamza and H. Zaag. A Lyapunov functional and blow-up results for a class of perturbations for semilinear wave equations in the critical case. J. Hyperbolic Differ. Equ., 9:195–221, 2012. | DOI | MR | Zbl

[23] M. A. Hamza and H. Zaag. A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations. Nonlinearity, 25(9):2759–2773, 2012. | DOI | MR | Zbl

[24] M. A. Hamza and H. Zaag. Blow-up behavior for the Klein–Gordon and other perturbed semilinear wave equations. Bull. Sci. Math., 137(8):1087–1109, 2013. | DOI | MR | Zbl

[25] M. A. Hamza and H. Zaag. Blow-up results for semilinear wave equations in the super-conformal case. Discrete Contin. Dyn. Syst. Ser. B, 18(9):2315–2329, 2013. | DOI | MR | Zbl

[26] R. Killip, B. Stovall, and M. Vişan. Blowup behaviour for the nonlinear Klein–Gordon equation. Math. Ann., 358(1-2):289–350, 2014. | DOI | MR | Zbl

[27] R. Killip and M. Vişan. Smooth solutions to the nonlinear wave equation can blow up on Cantor sets. 2011. | arXiv

[28] 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. | DOI | MR

[29] F. Mahmoudi, N. Nouaili, and H. Zaag. Construction of a stable periodic solution to a semilinear heat equation with a prescribed profile. Nonlinear Anal., 131:300–324, 2016. | DOI | MR | Zbl

[30] Y. Martel. Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math., 127(5):1103–1140, 2005. | DOI | MR | Zbl

[31] Y. Martel. Multi-solitons and large time dynamics of some nonlinear dispersive equations. Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{\mathrm{e}}$MA, (33):79–111, 2005. | Zbl

[32] Y. Martel and F. Merle. Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23(6):849–864, 2006. | DOI | Numdam | Zbl

[33] Y. Martel, F. Merle, and T.P. Tsai. Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations. Comm. Math. Phys., 231(2):347–373, 2002. | DOI | Zbl

[34] Y. Martel, F. Merle, and T.P. Tsai. Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations. Duke Math. J., 133(3):405–466, 2006. | DOI | Zbl

[35] N. Masmoudi and H. Zaag. Blow-up profile for the complex Ginzburg-Landau equation. J. Funct. Anal., 255(7):1613–1666, 2008. | DOI | MR | Zbl

[36] F. Merle. Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys., 129(2):223–240, 1990. | DOI | Zbl

[37] F. Merle. Solution of a nonlinear heat equation with arbitrarily given blow-up points. Comm. Pure Appl. Math., 45(3):263–300, 1992. | DOI | MR | Zbl

[38] F. Merle, P. Raphaël, and I. Rodnianski. Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem. Invent. Math., 193(2):249–365, 2013. | DOI | Zbl

[39] F. Merle and H. Zaag. Stabilité du profil à l’explosion pour les équations du type $u_t=\Delta u+{\left|u\right|}^{p-1}u$. C. R. Acad. Sci. Paris Sér. I Math., 322(4):345–350, 1996. | Zbl

[40] F. Merle and H. Zaag. Stability of the blow-up profile for equations of the type $u_t=\Delta u+{\left|u\right|}^{p-1}u$. Duke Math. J., 86(1):143–195, 1997. | DOI | MR | Zbl

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

[42] 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. | DOI | Zbl

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

[44] 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. | DOI | MR | Zbl

[45] 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. | DOI | MR | Zbl

[46] F. Merle and H. Zaag. On characteristic points at blow-up for a semilinear wave equation in one space dimension. In Singularities in Nonlinear Problems, Kyoto. 2009. | DOI | MR | Zbl

[47] F. Merle and H. Zaag. Isolatedness of characteristic points for a semilinear wave equation in one space dimension. In Séminaire sur les Équations aux Dérivées Partielles, 2009–2010. École Polytech., Palaiseau, 2010. Exp. No. 11, 10p.

[48] F. Merle and H. Zaag. Blow-up behavior outside the origin for a semilinear wave equation in the radial case. Bull. Sci. Math., 135(4):353–373, 2011. | DOI | MR | Zbl

[49] 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., 134(3):581–648, 2012. | DOI | MR | Zbl

[50] F. Merle and H. Zaag. Isolatedness of characteristic points for a semilinear wave equation in one space dimension. Duke Math. J., 161(15):2837–2908, 2012. | DOI | MR | Zbl

[51] F. Merle and H. Zaag. On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations. Comm. Math. Phys., pages 1–34, 2015. | DOI | MR | Zbl

[52] F. Merle and H. Zaag. Blow-up solutions to the semilinear wave equation with a nearly pyramid-shaped blow-up surface. Submitted, 2016. | DOI

[53] F. Merle and H. Zaag. Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions. Trans. Amer. Math. Soc., 368(1):27–87, 2016. | DOI | MR | Zbl

[54] F. Merle and H. Zaag. Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions. Trans. Amer. Math. Soc., 368(1):27–87, 2016. | DOI | MR | Zbl

[55] M. Ming, F. Rousset, and N. Tzvetkov. Multi-solitons and related solutions for the water-waves system. SIAM J. Math. Anal., 47(1):897–954, 2015. | DOI | MR | Zbl

[56] V. T. Nguyen and H. Zaag. Construction of a stable blow-up solution for a class of strongly perturbed semilinear heat equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2016. to appear, . | arXiv | Zbl

[57] N. Nouaili and H. Zaag. Profile for a Simultaneously Blowing up Solution to a Complex Valued Semilinear Heat Equation. Comm. Partial Differential Equations, 40(7):1197–1217, 2015. | DOI | MR | Zbl

[58] P. Raphaël and R. Schweyer. On the stability of critical chemotactic aggregation. Math. Ann., 359(1-2):267–377, 2014. | DOI | MR | Zbl

[59] R. Schweyer. Type II blow-up for the four dimensional energy critical semi linear heat equation. J. Funct. Anal., 263(12):3922–3983, 2012. | DOI | MR | Zbl

[60] S. Tayachi and H. Zaag. Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term. Submitted, , 2015. | arXiv | DOI | MR | Zbl

[61] S. Tayachi and H. Zaag. Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term. In C. Dogbe, editor, Actes du colloque EDP-Normandie, pages 119–136, 2015. | DOI | MR

[62] Martel Y. and Raphaël P. Strongly interacting blow up bubbles for the mass critical nls. Preprint, , 2015. | arXiv | DOI

[63] H. Zaag. Blow-up results for vector-valued nonlinear heat equations with no gradient structure. Ann. Inst. H. Poincaré Anal. Non Linéaire, 15(5):581–622, 1998. | DOI | Numdam | MR | Zbl

[64] H. Zaag. On the regularity of the blow-up set for semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 19(5):505–542, 2002. | DOI | Numdam | MR | Zbl

[65] H. Zaag. One dimensional behavior of singular $N$ dimensional solutions of semilinear heat equations. Comm. Math. Phys., 225(3):523–549, 2002. | DOI | MR | Zbl

[66] H. Zaag. Regularity of the blow-up set and singular behavior for semilinear heat equations. In Mathematics & mathematics education (Bethlehem, 2000), pages 337–347. World Sci. Publishing, River Edge, NJ, 2002. | DOI | Zbl

[67] 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. | DOI | MR | Zbl