## Acta Mathematica

### Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

#### Abstract

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H1 than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.

#### Note

The first author was supported in part by NSF and the second one in part by CNRS and by ANR ONDENONLIN. Part of this research was carried out during visits of the second author to the University of Chicago and I.H.E.S. and of the first author to Paris XIII.

#### Article information

Source
Acta Math., Volume 201, Number 2 (2008), 147-212.

Dates
Received: 30 October 2006
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892355

Digital Object Identifier
doi:10.1007/s11511-008-0031-6

Mathematical Reviews number (MathSciNet)
MR2461508

Zentralblatt MATH identifier
1183.35202

Rights
2008 © Institut Mittag-Leffler

#### Citation

Kenig, Carlos E.; Merle, Frank. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201 (2008), no. 2, 147--212. doi:10.1007/s11511-008-0031-6. https://projecteuclid.org/euclid.acta/1485892355

#### References

• A ntonini, C. & M erle, F., Optimal bounds on positive blow-up solutions for a semilinear wave equation. Int. Math. Res. Notices, 2001 (2001), 1141–1167.
• A ronszajn, N., K rzywicki, A. & S zarski, J., A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat., 4 (1962), 417–453.
• A ubin, T., Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., 55 (1976), 269–296.
• B ahouri, H. & G érard, P., High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math., 121 (1999), 131–175.
• B ahouri, H. & S hatah, J., Decay estimates for the critical semilinear wave equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 783–789.
• B rezis, H. & C oron, J.-M., Convergence of solutions of H-systems or how to blow bubbles. Arch. Rational Mech. Anal., 89 (1985), 21–56.
• B rezis, H. & M arcus, M., Hardy’s inequalities revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 217–237 (1998).
• C hrist, F. M. & W einstein, M. I., Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation. J. Funct. Anal., 100 (1991), 87–109.
• G érard, P., Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var., 3 (1998), 213–233.
• G iga, Y. & K ohn, R. V., Nondegeneracy of blowup for semilinear heat equations. Comm. Pure Appl. Math., 42 (1989), 845–884.
• G inibre, J., S offer, A. & V elo, G., The global Cauchy problem for the critical nonlinear wave equation. J. Funct. Anal., 110 (1992), 96–130.
• G inibre, J. & V elo, G., Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 133 (1995), 50–68.
• G rillakis, M. G., Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. of Math., 132 (1990), 485–509.
• — Regularity for the wave equation with a critical nonlinearity. Comm. Pure Appl. Math., 45 (1992), 749–774.
• H örmander, L., The Analysis of Linear Partial Differential Operators. III. Classics in Mathematics. Springer, Berlin–Heidelberg, 2007.
• J erison, D. & K enig, C. E., Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. of Math., 121 (1985), 463–494.
• K apitanski, L., Global and unique weak solutions of nonlinear wave equations. Math. Res. Lett., 1 (1994), 211–223.
• K enig, C. E., Global well-posedness and scattering for the energy critical focusing non-linear Schrödinger and wave equations. Lecture notes for a mini-course given at “Analyse des équations aux derivées partielles”, Evian-les-Bains, 2007.
• K enig, C. E. & M erle, F., Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math., 166 (2006), 645–675.
• — Scattering for $\dot {H}^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions. To appear in Trans. Amer. Math. Soc.
• K enig, C. E., P once, G. & V ega, L., Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Comm. Pure Appl. Math., 46 (1993), 527–620.
• K eraani, S., On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differential Equations, 175 (2001), 353–392.
• K rieger, J. & S chlag, W., On the focusing critical semi-linear wave equation. Amer. J. Math., 129 (2007), 843–913.
• L evine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+{\cal F}(u)$. Trans. Amer. Math. Soc., 192 (1974), 1–21.
• L indblad, H. & S ogge, C. D., On existence and scattering with minimal regularity for semilinear wave equations. J. Funct. Anal., 130 (1995), 357–426.
• L ions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223–283.
• M erle, F., Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Amer. Math. Soc., 14 (2001), 555–578.
• M erle, F. & V ega, L., Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation in 2D. Internat. Math. Res. Notices, 8 (1998), 399–425.
• M erle, F. & Z aag, H., A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Ann., 316 (2000), 103–137.
• — Determination of the blow-up rate for the semilinear wave equation. Amer. J. Math., 125 (2003), 1147–1164.
• O gawa, T. & T sutsumi, Y., Blow-up of H1 solution for the nonlinear Schrödinger equation. J. Differential Equations, 92 (1991), 317–330.
• P ayne, L. E. & S attinger, D. H., Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math., 22 (1975), 273–303.
• P echer, H., Nonlinear small data scattering for the wave and Klein–Gordon equation. Math. Z., 185 (1984), 261–270.
• S attinger, D. H., On global solution of nonlinear hyperbolic equations. Arch. Rational Mech. Anal., 30 (1968), 148–172.
• S hatah, J. & S truwe, M., Regularity results for nonlinear wave equations. Ann. of Math., 138 (1993), 503–518.
• — Well-posedness in the energy space for semilinear wave equations with critical growth. Int. Math. Res. Notices, 1994 (1994), 303–309.
• Geometric Wave Equations. Courant Lecture Notes in Mathematics, 2. New York University Courant Institute of Mathematical Sciences, New York, 1998.
• S ogge, C. D., Oscillatory integrals and unique continuation for second order elliptic differential equations. J. Amer. Math. Soc., 2 (1989), 491–515.
• Lectures on Nonlinear Wave Equations. Monographs in Analysis, II. International Press, Boston, MA, 1995.
• S taffilani, G., On the generalized Korteweg–de Vries-type equations. Differential Integral Equations, 10 (1997), 777–796.
• S trauss, W. A., Nonlinear Wave Equations. CBMS Regional Conference Series in Mathematics, 73. Amer. Math. Soc., Providence, RI, 1989.
• S truwe, M., Globally regular solutions to the u5 Klein–Gordon equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 495–513 (1989).
• T alenti, G., Best constant in Sobolev inequality. Ann. Mat. Pura Appl., 110 (1976), 353–372.
• T ao, T., Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions. Dyn. Partial Differ. Equ., 3 (2006), 93–110.
• T ao, T. & V isan, M., Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differential Equations, (2005), No. 118, 28 pp.
• T aylor, M. E., Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. Mathematical Surveys and Monographs, 81. Amer. Math. Soc., Providence, RI, 2000.
• T rudinger, N. S., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1968), 265–274.
• W olff, T. H., Recent work on sharp estimates in second-order elliptic unique continuation problems. J. Geom. Anal., 3 (1993), 621–650.