Solutions en grand temps d'équations d'ondes non linéaires

Zuily, Claude 

Séminaire Bourbaki : volume 1993/94, exposés 775-789, Astérisque, no. 227 (1995), Exposé no. 779, 38 p.

Bibliographie

[A1] S. Alinhac - Temps de vie et comportement explosif des solutions d'équations d'ondes quasi linéaires en dimension deux I, Ann. Sc. ENS (1993) (à paraître). | Numdam | EuDML | Zbl

[A2] S. Alinhac - Temps de vie et comportement explosif des solutions d'équations d'ondes quasi linéaires en dimension deux II, Duke Math. Journal (1993) (à paraître). | Zbl

[A3] S. Alinhac - Approximation près du temps d'explosion des solutions d'équations d'ondes quasi-linéaires en dimension deux, Siam J. Math. An. (1993) (à paraître). | Zbl

[As] F. Asakura - Existence of a global solution to a semi-linear wave equation with slowly decreasing initial data in three space dimension, Comm. in P.D.E., 11 (13) (1986),1459-1487. | Zbl | MR | DOI

[BS] M. Beals - W. Strauss - Lp estimates for the wave equation with a potential, Comm. in P.D.E., Vol. 18 (n° 7 and 8), (1993),1365-1397. | Zbl | MR | DOI

[Br1] P. Brenner - On Lp - Lq estimates for the wave equation, Math. Z., 145 (1975), 251-254. | Zbl | MR | EuDML | DOI

[Br2] P. Brenner - On the existence of global smooth solutions of certain hyperbolic equations, Math. Z., 167 (1979), 99-135. | Zbl | MR | EuDML | DOI

[BrW] P. Brenner , V. Von Wahl - Global classical solutions of non linear wave equation Math. Z., 176 (1981), 87-121. | Zbl | MR | EuDML | DOI

[CF1] L. Caffarelli , A. Friedman - The blow up boundary for non linear wave equations, Trans. Amer. Math. Soc., 297 (1) (1986), 233-241. | Zbl | MR | DOI

[CF2] L. Caffarelli , A. Friedman - Differentiability of the blow up curve for one dimensional non linear wave equations, Arch. Rational Mech. Anal., 91, 1, (1985), 83-98. | Zbl | MR | DOI

[CaC] F. Cagnac, Y. Choquet-Bruhat - Solution globale d'une équation d'ondes non linéaire sur une variété hyperbolique, J. Math. Pures Appl., 63 (1984), 377-390. | Zbl | MR

[Ch] J. Chadam - Asymptotics for $\square u=m^{2}u+G(t,x,u,u_x,u_t),$ I, Ann. Scuola Norm. Sup. Pisa, 26 (1972), 33-65. | Numdam | Zbl | EuDML

[Cho1] Y. Choquet-Bruhat - Théorèmes de non existence de solutions d'une équation d'ondes non linéaire, C.R. Acad. Sci. Paris, 305 (1987), 263-266. | MR

[Cho2] Y. Choquet-Bruhat - Global existence for solutions of $\square u={A\left|u\right|}^p$, Journ. Diff. Equ., 82 (1989), 98-108. | Zbl | DOI

[Chr] D. Christodolou - Global solutions of non linear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. | Zbl | MR | DOI

[ChrK] D. Christodolou, S. Klainerman - Asymptotic properties of linear field equations in Minkowski space, Comm. Pure Appl. Math., 43 (1990), 137-199. | Zbl | MR | DOI

[F1] F. G. Friedlander - On the radiation field of pulse solutions of the wave equation I, II, Proc. Roy. Soc. A., 269 (1962), 53-65 et 279 (1964), 386-394. | Zbl | MR | DOI

[F2] F. G. Friedlander - The wave equation on a curved space time, Cambridge University Press 1975. | Zbl | MR

[GV1] J. Ginibre, G. Velo - The global Cauchy problem for the non linear Klein-Gordon equation, Math. Z., 189 (1985), 487-505. | Zbl | MR | EuDML | DOI

[GV2] J. Ginibre, G. Velo - Time decay of finite energy solutions of the non linear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 399-442. | Numdam | Zbl | MR | EuDML

[GV3] J. Ginibre, G. Velo - Conformal invariance and time decay for non linear wave equations, Ann. Inst. H. Poincaré Phy. Théor., 47 (1987), 221-276. | Numdam | Zbl | MR | EuDML

[GV4] J. Ginibre, G. Velo - The global Cauchy problem for the non linear Klein-Gordon equation II, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 6 (1989), 15-35. | Numdam | Zbl | MR | EuDML | DOI

[GV5] J. Ginibre, G. Velo - Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. | Zbl | MR | DOI

[GV6] J. Ginibre, G. Velo - Regularity of solutions of critical and subcritical non linear wave equation, Non Linear Analysis | Zbl

[GSV] J. Ginibre, A. Soffer, G. Velo - The global Cauchy problem for the critical non linear wave equation, Journ. of Funct. Anal., 110 (1) (1992), 96-130. | Zbl | MR | DOI

[Gl1] R. Glassey - On the asymptotic behavior of non linear wave equations, Trans. Amer. Math. Soc., 182 (1973), 187-200. | Zbl | MR | DOI

[G12] R. Glassey - Blow-up theorems for non linear wave equations, Math. Z., 132 (1973), 183-203. | Zbl | MR | EuDML | DOI

[G13] R. Glassey - Finite time blow up for solutions of non linear wave equations, Math. Z., 177 (1981), 323-340. | Zbl | MR | EuDML | DOI

[G14] R. Glassey - Existence in the large for $\square u=F\left(u\right)$ in two space dimensions, Math. Z., 178 (1981), 233-261. | Zbl | EuDML | DOI

[Go] P. Godin - Lifespan of solutions of semi linear wave equations in two space dimensions, Comm. in P.D.E., 18 (5 and 6) (1993), 895-916. | Zbl | MR | DOI

[Gr1] M. Grillakis - Regularity and asymptotic behavior of the wave equation with a critical non linearity, Annals of Math., 132 (1990), 485-509. | Zbl | MR | DOI

[Gr2] M. Grillakis - Regularity for the wave equation with a critical non linearity, Comm. Pure Appl. Math., XLV (1992), 749-774. | Zbl | MR | DOI

[HJ] B. Hanouzet, J. L. Joly - Explosion pour des problèmes hyperboliques semi-linéaires avec second membres non compatibles, C.R. Acad. Sc. Paris, 301 (1985), 581-584. | Zbl | MR

[Ho1] L. Hörmander - On global existence of solutions of non linear hyperbolic equations in R1+3, Institut Mittag-Leffler, Report n° 9, (1985).

[Ho2] L. Hörmander - On Sobolev spaces associated with some Lie algebra, Current topics in P.D.E. Kinokuniya Company Ltd Tokyo, (1985), 261-287. | Zbl | MR

[Ho3] L. Hörmander - The lifespan of classical solutions of non linear hyperbolic equations, Springer Lectures Notes in Math., 1256 (1986), 214-280. | Zbl | MR

[Ho4] L. Hörmander - Non linear hyperbolic differential equations, Lectures at University of Lund 1986-87.

[Ho5] L. Hörmander - L1 - L∞ estimates for the wave operator, Mittag-Leffler report 1 (1986), 1-29, and Analyse Math. et Appl. Gauthier-Villars Paris (1988), 211-234. | Zbl

[Ho6] L. Hörmander - The fully non-linear Cauchy problem with small data, Bol. Soc. Bras. Mat. Vol 20 , 1, 1-27, (1989). | Zbl | DOI

[Jo1] F. John - Delayed singularity formation in solutions of non linear wave equations in higher dimensions, Comm. Pure Appl. Math., 29 (1976), 649-682. | Zbl | MR | DOI

[Jo2] F. John - Blow-up of solutions of non linear wave equations in three space dimensions for small initial data, Manuscripta Math., 28 (1979), 235-268. | Zbl | MR | EuDML | DOI

[Jo3] F. John - Blow up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51. | Zbl | MR | DOI

[Jo4] F. John - Lower bounds for the lifespan of solutions to non linear wave equations in three dimensions, Comm. Pure Appl. Math., 36 (1983), 1-35. | Zbl | MR | DOI

[Jo5] F. John - Blow-up of radial solutions of utt = c2(ut)Δu in three space dimension, Mat. Apl. Comput. V (1985), 3-18. | Zbl

[Jo6] F. John - Existence for large times of strict solutions of non linear wave equations in three space dimensions, Comm. Pure Appl. Math., 40 (1987), 79-109. | Zbl | MR | DOI

[Jo7] F. John - Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math., 41 (1988), 615-666. | Zbl | MR | DOI

[Jo8] F. John - Solutions of quasi-linear wave equations with small initial data. The third phase, Lecture Notes in Math., 1402 (1989), 155-173, (Springer Verlag). | Zbl | MR

[Jo9] F. John - Non linear wave equations. Formation of singularities, Lehigh University, University Lecture Series, Amer. Math. Soc. Providence, (1990). | Zbl

[JoK] F. John, S. Klainerman - Almost global existence to non linear wave equations in three space dimension, Comm. Pure Appl. Math., 37 (1984), 443-455. | Zbl | MR | DOI

[Jor] K. Jörgens - Das Ang faugswertproblem in Grossen für ein Klasse nichtlinearer Wellengleichungen, Math. Z., 77 (1961), 295-307. | Zbl | MR | EuDML | DOI

[K1] L. V. Kapitanskii - Some generalizations of the Strichartz-Brenner inequality, Leningrad Math. Journ., 1 (1990), 693-726. | Zbl | MR

[K2] L. V. Kapitanskii - The Cauchy problem for a semi linear wave equation, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI), 181 (1990), 24- 64; 182 (190), 38-85.

[Ka1] T. Kato - Linear and quasi linear equations of evolution of hyperbolic type, CIME II Ciclo, (1976). | Zbl

[Ka2] T. Kato - Blow up of solutions of some non linear hyperbolic equations, Comm. Pure Appl. Math., 32 (1980), 501-505. | Zbl | MR | DOI

[Ke] Keller - On solutions of non linear wave equations, Comm. Pure Appl. Math., 10 (1957), 523-530. | Zbl | MR | DOI

[KiL] S. Kichenassamy, W. Littman - Blow up surfaces for non linear wave equations I, Comm. in P.D.E., 18 (3 and 4), (1993), 431-452. | Zbl | MR | DOI

[Kl1] S. Klainerman - Global existence of non linear wave equations, Comm. Pure Appl. Math., 33 (1980), 43-101. | Zbl | MR | DOI

[Kl2] S. Klainerman - Long time behavior of solutions to non linear evolution equations, Arch. Rat. Mech. Anal., 78 (1982),73-98. | Zbl | MR | DOI

[K13] S. Klainerman - On almost global existence to non linear wave equations in three space dimensions, Comm. Pure Appl. Math., 36 (1983), 325-344. | Zbl | MR | DOI

[Kl4] S. Klainerman - Long time behavior of solutions to non linear wave equations, Proc. Int. Congr. Math. Warszawa, (1983), 1209-1215. | Zbl | MR

[Kl5] S. Klainerman - Weighted L∞ and L1 estimates for solutions to the classical wave equation in three space dimensions, Comm. Pure Appl. Math., 37 (1984), 269-284. | Zbl | DOI

[Kl6] S. Klainerman - Uniform decay estimates and the Lorentz invariance of the wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332. | Zbl | MR | DOI

[Kl7] S. Klainerman - Global existence of small amplitude solutions to non linear Klein-Gordon equations in four space time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641. | Zbl | MR | DOI

[Kl8] S. Klainerman - The null condition and global existence to non linear wave equations, Lectures in Applied Mathematics, 23 (1986), 293-326. | Zbl | MR

[Kl9] S. Klainerman - Remarks on the global Sobolev inequalities in the Minkowski space Rn+1, Comm. Pure Appl. Math., 40 (1987), 111-117. | Zbl | MR | DOI

[Kl10] S. Klainerman - Remark on the asymptotic behavior of the Klein-Gordon equation in Rn+1, Comm. Pure Appl. Math., 46 (1993), 137-144. | Zbl | MR | DOI

[KIM] S. Klainerman, M. Machedon - Space time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), 1221- 1268. | Zbl | MR | DOI

[KlMa] S. Klainerman, A. Majda - Formation of singularities for wave equations including the non linear vibrating string, Comm. Pure Appl. Math., 33 (1980), 241-263. | Zbl | MR | DOI

[KlP] S. Klainerman, G. Ponce - Global small amplitude solutions to non linear evolution equations, Comm. Pure Appl. Math., 36 (1983), 133-141. | Zbl | MR | DOI

[Le1] H. Levine - Instability and non existence of global solutions to non linear wave equations of the form Putt = -Au + F(u), Trans. Amer. Math. Soc., 192 (1974), 1-21. | Zbl | MR

[Le2] H. Levine - The role of critical exponents in blow-up theorems, SIAM Review, 32 (1990), 262-288. | Zbl | MR | DOI

[LiY1] Li T. T., Yu Xin - Lifespan of classical solutions to fully non linear wave equations. | Zbl

[LiY2] Li T. T., Yu Xin - Initial value problems for non linear wave equations, Comm. in P.D.E., 13 (4) (1988), 383-422. | Zbl | DOI

[LiY3] Li T. T., Yu Xin - Non linear stability for two space dimensional wave equations with higher order perturbations, UMPA Prépublication 49, E.N.S. Lyon, (1991). | Zbl

[Lin1] H. Linblad - Blow up of solutions of $\square u={\left|u\right|}^p$ with small initial data, Comm. P.D.E., 15 (6) (1990), 757-821. | Zbl | DOI

[Lin2] H. Linblad - On the lifespan of solutions of non linear wave equations with small initial data, Comm. Pure Appl. Math., 43 (1990), 445-472. | Zbl | MR | DOI

[Lin3] H. Linblad - Global solutions of non linear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096. | Zbl | DOI

[Lio] J. L. Lions - Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier Villars, Paris (1965). | Zbl | MR

[Lit] W. Littman - $L^p$ - $L^q$ estimates for singular integral operators arising from hyperbolic equations…, Part. Diff. Equ. Proceedings of Symposia in Pure Mathematics, Providence, R.I., A.M.S., 23 (1973), 479-481. | Zbl | MR | DOI

[MSW] B. Marshall, W. Strauss, S. Wainger - LP - Lq estimates for the Klein-Gordon equation, Journ. Math. Pure et Appl., (9) 59 (1980), 417-440. | Zbl | MR

[Ma] K. Masuda - Blow-up of solutions for quasi linear wave equations in two space dimensions, Lecture Notes in Num. Appl. Anal., 6 (1983), 87-91, North-Holland. | Zbl | MR

[Mat] A. Matsumura - Global existence and asymptotics of the solutions ..., Publ. RIMS Kyoto Univ., 13 (1977), 349-379. | Zbl | MR

[Mo] C. Morawetz - Time decay for the non linear Klein-Gordon equation, Proc. Roy. Soc., London Ser. A, 306 (1968), 291-296. | Zbl | MR | DOI

[MoS] C. Morawetz, W. Strauss - Decay and scattering of solutions of a non linear relativistic wave equation, Comm. Pure Appl. Math., 25 (1972), 1-31 and 26 (1973), 47-54. | Zbl | DOI

[P1] H. Pecher - $L^p$-Abschätzungen und Klassiche Lösungen für nichtlinear ... I, Math. Z., 150 (1976), 159-183, II Manuscripta Math., 20 (1977), 227-244. | Zbl | EuDML | DOI

[P2] H. Pecher - Ein nichtlinear Interpolationsatz und seine Anwadung ..., Math. Z., 161 (1978), 9-40. | Zbl | MR | EuDML | DOI

[P3] H. Pecher - Global smooth solutions to a class of semi linear wave equations with strong non linearity, Manuscripta Math., 69 (1990), 71-92. | Zbl | MR | EuDML | DOI

[Pe] I. Peral-Alonso - Some remarks on semi linear wave equations in ${ℝ}^n$, Contr. to Non linear PDE Pitman, (1986). | Zbl

[Po1] G. Ponce - Global existence of small solutions to a class of non linear evolution equations, Non Linear Analysis, T.M.A. 9 (1985), 399-418. | Zbl | MR | DOI

[Po2] G. Ponce - Regularity of solutions to non linear dispersive equations, Journ. Diff. Equ., 78 (1989), 122-135. | Zbl | MR | DOI

[PoS] G. Ponce, T. Sideris - Local regularity of non linear wave equations in three space dimensions, Comm. in P.D.E., 18 (1 and 2) (1993), 169-177. | Zbl | MR | DOI

[R] R. Racke - Lectures on non linear evolution equations, Aspects of Math., Vol E 19, Braunschweig Wiesbaden, Vieweg. | Zbl

[Ra] J. Rauch - The $u^5$-Klein-Gordon equation, Non linear PDE's and Applications, Pitman Research Notes in Math., 53, 335-364. | Zbl | MR

[Re] M. Reed - Abstract non linear wave equations, Lectures Notes in Math., Vol. 507, (1976). | Zbl | MR

[S1] J. Schaeffer - The equation $\square u={\left|u\right|}^p$ for the critical value of $p$, Proc. Roy. Soc. Edinburgh, 101 A (1985), 31-44. | Zbl | DOI

[S2] J. Schaeffer - Finite time blow up for $u_{tt}-\Delta u=H(u_r,u_t)$ in two space dimensions, Comm. P.D.E., 11 (5) (1986), 513-543. | Zbl | DOI

[Se1] I. E. Segal - The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129-135. | Numdam | Zbl | MR | EuDML | DOI

[Se2] I. E. Segal - Quantization and dispersion for non linear relativistic equation, Proc. Conf. Math. Th. El. Particles, MIT Press (1966), 79-108. | Zbl | Numdam

[Se3] I. E. Segal - Space time decay for solutions of wave equations, Adv. Math., 22 (1976), 305-311. | Zbl | MR | DOI

[Sh1] J. Shatah - Global existence of small solutions to non linear evolution equations, Journ. Diff. Equ., 46 (1982), 409-425. | Zbl | MR | DOI

[Sh2] J. Shatah - Normal forms and quadratic non linear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696. | Zbl | MR | DOI

[Sh3] J. Shatah - Weak solutions and the development of singularities in the $SU\left(2\right)\sigma$-model, Comm. Pure Appl. Math., 41 (1988), 459-469. | Zbl | DOI

[ShS] J. Shatah, M. Struwe - Regularity results for non linear wave equation, Annals of Math., (1993), (to appear). | Zbl | MR

[ShT] J. Shatah, A. Tahvildar-Zadeh - Regularity of harmonic maps from Minkowski space into rotationally symmetric manifolds. | Zbl

[Si1] T. Sideris - Global behavior of solutions to non linear wave equations in three dimensions, Comm. in P.D.E., 8 (1983), 1291-1323. | Zbl | MR | DOI

[Si2] T. Sideris - Non existence of global solutions to semi linear wave equations in high dimensions, Journ. Diff. Eq., 52 (1984), 378-406. | Zbl | DOI

[Si3] T. Sideris - Formation of singularities in solutions to non linear hyperbolic equations, Arch. Rat. Mech. Anal., 86 (4) (1984), 369-381. | Zbl | MR | DOI

[Si4] T. Sideris - Decay estimates for the three dimensional inhomogeneous Klein-Gordon equation and applications, Comm. P.D.E., 14 (1989), 1421-1455. | Zbl | MR | DOI

[St1] W. Strauss - Decay and asymptotics for $\square u=F\left(u\right)$, Journal of Funct. Anal., 2 (1968), 409-457. | Zbl | DOI

[St2] W. Strauss - On weak solutions of semi linear hyperbolic equations, Anais Acad. Brasil Cienc., 42 (1970), 645-651. | Zbl | MR

[St3] W. Strauss - Non linear invariant wave equations, Lecture Notes in Phys. (Springer), Vol. 78, (1978), 197-249. | MR

[St4] W. Strauss - Non linear wave equations, Conf. Board of the Math. Sciences, 73 (1989), Amer. Math. Soc. | Zbl | MR

[StV] W. Strauss, L. Vasquez - Numerical solution of a non linear Klein-Gordon equation, J. Comput. Phys., 28 (1978), 271-278. | Zbl | MR | DOI

[Str1] R. Strichartz - A priori estimates for the wave equation and some applications, J. Funct. Analysis, 5 (1970), 218-235. | Zbl | MR | DOI

[Str2] R. Strichartz - Restrictions of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. Journ., 44 (1977), 705-714. | Zbl | MR | DOI

[Stru1] M. Struwe - Globally regular solutions to the $u^5$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., (4) 15 (1988), 495-513. | Numdam | Zbl | MR | EuDML

[Stru2] M. Struwe - Semi linear wave equations, Bull. Amer. Math. Soc., 26 (1992), 53-86. | Zbl | MR | DOI

[T] H. Triebel - Interpolation theory, function spaces, differential operators, North Holland Math. Lib. 18, (1978). | Zbl | MR

[Ts] K. Tsutaya - A global existence theorem for semi linear wave equations with data of non compact support in two space dimensions, Comm. in P.D.E., 17 (11 and 12), (1992), 1925-1954. | Zbl | MR

[W1] W. Von Wahl - $L^p$ decay rates for homogeneous wave equations, Math. Z., 120 (1971), 93-106. | Zbl | MR | EuDML | DOI

[W2] W. Von Wahl - Über nichtlineare Wellengleichungen mit zeitabhängigem elliptischen Hamptteil, Math. Z., 142 (1975), 105-120. | Zbl | MR | EuDML | DOI

[W3] W. Von Wahl - Regular solutions of initial boundary value problems for linear and non linear wave equations II, Math. Z., 142 (1975), 121-130. | Zbl | MR | EuDML | DOI

[We] M. Weinstein - Solitary waves of non linear dispersive evolution equations with critical power non linearity, J. Diff. Equ., 19 (1987), 192-203. | Zbl | MR | DOI

[Z] C. Zuily - Existence globale de solutions régulières pour l'équation des ondes non linéaire amortie sur le groupe de Heisenberg, Indiana Univ. Math. Journ., Vol. 42, n° 2, (1993), 323-360. | Zbl | MR | DOI