Regularity results for semilinear and geometric wave equations

Jalal Shatah

ArticleOriginal scientific text

Title

Regularity results for semilinear and geometric wave equations

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Affiliations

Courant Institute, 251 Mercer St., New York, N.Y. 10012, U.S.A.

J. Bergh and J. Löfström, Interpolation Spaces. Grundlehren 223, Springer-Verlag, 1976.
P. Brenner and W. von Wahl, Global classical solutions of nonlinear wave equations, Math. Z., 176:87-121, 1981.
T. Cazenave, J. Shatah, and A. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities for wave maps and Yang-Mills fields, preprint, 1995.
J. Ginibre and G. Velo, The Cauchy problem for the O(N), ℂP(N-1) and Gℂ(N,P) models, Ann. Physics, 142:393-415, 1982.
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189:487-505, 1985.
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Klein-Gordon equation revised, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 6:15-35, 1989.
M. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math., 132:485-509, 1990.
M. Grillakis, Classical solutions for the equivariant wave map in 1+2-dimensions, preprint, 1991.
M. Grillakis, A priori estimates and regularity for nonlinear waves, Communication at ICM 94, Zurich, 1994.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Math. Lib., volume 18. North-Holland, 1978.
K. Jörgens, Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen, Math. Zeit., 77:295-308, 1961.
L. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1:211-223, 1994.
S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math. Jour., 81:99-133, 1995.
H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Func. Anal., 130:357-426, 1995.
B. Marshall, W. Strauss, and S. Wainger, $L^{p} - L^{q}$ estimates for the Klein-Gordon equation, J. Math. Pures et Appl., 59:417-440, 1980.
H. Pecher, $L^{p}$ -Abschätzungen und klassische Lösungen für nichtlineare Wellegleichungen, I, Math. Z., 150:151-183, 1976.
G. Ponce and T. Sideris, Local regularity of nonlinear wave equations in threee space dimensions, Comm. Partial Diff. Eq., 18:169-177, 1993.
J. Rauch, I. The $u^{5}$ Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations, in H. Brézis and J. L. Lions, editors, Nonlinear Partial Differential Equations and Their Applications, volume 53, pages 335-364. Pitman, 1981.
I. Segal, Nonlinear Semi Groups, Ann. Math., 78:339-364, 1963.
J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. Math., 138:503-518, 1993.
J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, IMRN, 7:303-309, 1994.
J. Shatah and A. Tahvildar-Zadeh, Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math., XLV:947-971, 1992.
J. Shatah and A. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math, 47:719-754, 1994.
W. Strauss, Nonlinear invariant wave equations, in G. Velo and A. S. Wightman, editors, Invariant Wave Equations, pages 197-249. Springer-Verlag, Berlin, 1978.
W. Strauss, Nonlinear Scattering Theory at Low Energy, J. Funct. Anal., 41:110-133, 1981.
R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44:705-714, 1977.
M. Struwe, Globally regular solutions to the $u^{5}$ Klein-Gordon equation, Annali Sc. Norm. Sup. Pisa (Ser. 4), 15:495-513, 1988.
M. Struwe, Geometric Evolution Problems, Park City Geom. Ser. of the Amer. Math. Soc., Providence, R.I., 1992.

Title

Regularity results for semilinear and geometric wave equations

Affiliations

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