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Semi-linear wave equations
Author(s):
Michael
Struwe
Journal:
Bull. Amer. Math. Soc.
26
(1992),
53-85.
MSC (2000):
Primary 35L70
MathSciNet review:
1093058
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Abstract:
We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the -Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.
References:
-
- [1]
- P. Brenner, On
estimates for the wave equation, Math. Z. 145(1975), 251-254. MR 0387819 (52:8658) - [2]
- P. Brenner and W. von Wahl, Global classical solutions of non-linear wave equations, Math. Z. 176 (1981), 87-121. MR 606174 (82m:35103)
- [3]
- F. E. Browder, On nonlinear wave equations, Math. Z. 80 (1962), 249-264. MR 0147769 (26:5283)
- [4]
- J. Ginibre and G. Velo, The global Cauchy problem for the non-linear Klein-Gordon equation, Math. Z. 189 (1985), 487-505. MR 786279 (86f:35149)
- [5]
- -, Scattering theory in the energy space for a large class of non-linear wave equations, Comm. Math. Phys. 123 (1989), 535-573. MR 1006294 (90i:35172)
- [6]
- M. G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2) (to appear). MR 1078267 (92c:35080)
- [7]
- F. John, Blow-up of solutions to nonlinear wave equations in three space dimensions, Manuscripta Math. 28 (1979), 235-268. MR 535704 (80i:35114)
- [8]
- K. Jörgens, Das Anfangswertproblem im Grossen für eine Klasse nicht-linearer Wellengleichungen, Math. Z. 77 (1961), 295-308. MR 0130462 (24:A323)
- [9]
- J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), 37-92. MR 888880 (88f:53001)
- [10]
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris 1969. MR 0259693 (41:4326)
- [11]
- H. Pecher, Ein nichtlinearer Interpolationssatz und seine Anwendung auf nichtlineare Wellengleichungen, Math. Z. 161 (1978), 9-40. MR 499812 (81i:46039)
- [12]
- J. Rauch, The
-Klein-Gordon equation, (Brezis and Lions, eds.), Pitman Research Notes in Math., no. 53, pp. 335-364. MR 631403 (83a:35066) - [13]
- L. I. Schiff, Nonlinear meson theory of nuclear forces. I, Phys. Rev. 84 (1951), 1-9.
- [14]
- I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France 91 (1963), 129-135. MR 0153967 (27:3928)
- [15]
- J. Shatah, Weak solutions and the development of singularities in the
-model, Comm. Pure Appl. Math. 41 (1988), 459-469. MR 933231 (89f:58044) - [16]
- W. Strauss, Nonlinear wave equations, CBMS Lecture Notes, no. 73, Amer. Math. Soc., Providence, RI, 1989. MR 1032250 (91g:35002)
- [17]
- R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-714. MR 0512086 (58:23577)
- [18]
- M. Struwe, Globally regular solutions to the
-Klein-Gordon equation, Ann. Sc. Norm. Sup. Pisa (Ser. 4) 15 (1988), 495-513. MR 1015805 (90j:35142) - [19]
- Y. Zheng, Concentration in sequences of solutions to the nonlinear Klein-Gordon equation, preprint, 1989. MR 1101227 (91m:35200)
- [20]
- L. V. Kapitanskii, The Cauchy problem for a semi-linear wave equation (1989).
- [21]
- J. Shatah and A. Tahvildar-Zadeh, Regularity of harmonic maps from Minkowski space into rotationally symmetric manifolds, Courant Institute, preprint, 1990. MR 1168115 (93c:58056)
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Additional Information:
DOI:
10.1090/S0273-0979-1992-00225-2
PII:
S 0273-0979(1992)00225-2
Copyright of article:
Copyright
1992,
American Mathematical Society
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