On the energy critical focusing non-linear wave equation

Kenig, Carlos E.; Merle, Frank

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2006-2007), Exposé no. 5, 12 p.

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

[2] N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417–453. | MR | Zbl

[3] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant le courbure scalaire, J. Math. Pures Appl. (9), 55, 1976, 3, 269–296. | MR | Zbl

[4] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math 121 (1999), 131–175. | MR | Zbl

[5] H. Brézis and M. Marcus, Hardy’s inequalities revisited, Ann. Scuola Norm. Piza 25 (1997), 217–237. | Numdam | Zbl

[6] M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equation, J. Funct. Anal. 100 (1991), 87–109. | MR | Zbl

[7] P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233.

[8] Y. Giga and R. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42 (1989), 223–241. | MR | Zbl

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

[10] J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133 (1995), 50–68. | MR | Zbl

[11] M. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. 132 (1990), 485–509. | MR | Zbl

[12] M. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (1992), 749–774. | MR | Zbl

[13] L. Hörmander, “The analysis of linear partial differential operators III”, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. | Zbl

[14] D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. 121 (1985), 463–494. | MR | Zbl

[15] L. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1 (1994), no. 2, 211–223. | MR | Zbl

[16] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, non-linear Schrödinger equation, Preprint.

[17] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, non-linear Schrödinger equation in the radical case, Invent. Math. 166 (2006), no. 3, 645–675. | MR | Zbl

[18] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527–620. | MR | Zbl

[19] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), 353–392. | MR | Zbl

[20] J. Krieger and W. Schlag, On the focusing critical semi-linear wave equation, to appear, Amer. J. of Math. | MR

[21] H. 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 (1974), 1–21. | MR | Zbl

[22] H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357–426. | MR | Zbl

[23] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), 555–578. | MR | Zbl

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

[25] F. Merle and H. Zaag, A Liouville theorem for vector-valued nonlinear heat equations and applications, Math. Ann. 316 (2000), no. 1, 103–137. | MR | Zbl

[26] L.E. Payne and D.H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22, (1975), 273–303. | MR | Zbl

[27] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), 261–270. | MR | Zbl

[28] D.H. Sattinger, On global solutions of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30, (1968), 148–172. | MR | Zbl

[29] J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. 138 (1993), 503–518. | MR | Zbl

[30] J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices 7 (1994), 303–309. | MR | Zbl

[31] J. Shatah and M. Struwe, “Geometric wave equations,” Courant Lecture Notes in Mathematics, 2 (1998). | Zbl

[32] C. Sogge, “Lectures on nonlinear wave equations,” Monographs in Analysis II, International Press, 1995. | Zbl

[33] G. Staffilani, On the generalized Korteweg-de Vries-type equations, Differential Integral Equations 10 (1997), 777–796. | MR | Zbl

[34] W. Strauss, “Nonlinear wave equations,” CBMS Regional Conference Series in Mathematics, 73, American Math. Soc., Providence, RI, 1989. | Zbl

[35] M. Struwe, Globally regular solutions to the $u^5$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1988), 495–513. | Numdam | MR | Zbl

[36] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372. | MR | Zbl

[37] T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, preprint, . | arXiv | MR

[38] T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 118 (2005), 28 pp. (electronic). | MR | Zbl

[39] M. Taylor, “Tools for PDE. Pseudodifferential operators, paradifferential operators and layer potentials,” Math. Surveys and Monographs 81, AMS, Providence RI 2000. | Zbl

[40] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 (1968), 265–274. | Numdam | MR | Zbl

[41] T. Wolff, Recent work on sharp estimates in second-order elliptic unique continuation problems, J. Geom. Anal. 3 (1993), 621–650. | MR | Zbl