The global Cauchy problem for the nonlinear Klein-Gordon equation. (English) Zbl 0549.35108

This paper is devoted to the proof of existence and uniqueness of the solutions of the Cauchy problem for the nonlinear Klein-Gordon (NLKG) equation \(\square\phi +f(\phi)=0\), with \(\phi\) a complex function defined in space time \({\mathbb{R}}^{n+1}\) and f a nonlinear interaction satisfying suitable power bounds at infinity, typically \(f(\phi)=(\lambda_ 1|\phi |^{p_ 1-1}+\lambda_ 2|\phi |^{p_ 2- 1})\phi,\) with \(1\leq p_ 1\leq p_ 2\) and \(\lambda_ 2>0\). The result is proved for arbitrary initial data of finite energy, namely \((\phi,{\dot\phi })\in H^ 1\oplus L^ 2,\) for arbitrary space dimension. The authors rely for the existence of (weak) solutions on the known results of Lions, Segal, Strauss and others, and merely recall them briefly in a form close to that given by J. L. Lions [Quelques méthodes de résolution des problèmes aux limites non linéaires (1969; Zbl 0189.406)], with minor additions. The original part of the paper is the treatment of the uniqueness problem. It combines two ingredients. The first one is a proof that all solutions of NLKG with finite energy satisfy local (in time) space time integrability properties similar to those known to hold for the free wave equation. That proof proceeds through estimates of the solutions in homogeneous Sobolev spaces. The second one is a partial contraction method of suitable \(L^ r\)-norms of the solutions on bounded subsets of the energy space, and uses estimates of Strichartz and Peral on the free wave equation. The paper ends with a brief appendix on homogeneous Sobolev spaces. The authors have previously treated the nonlinear Schrödinger equation with similar results [see: The global Cauchy problem for the nonlinear Schrödinger equation revisited (preprint)].


35Q99 Partial differential equations of mathematical physics and other areas of application
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L70 Second-order nonlinear hyperbolic equations


Zbl 0189.406
Full Text: DOI EuDML


[1] Bergh, J., Löfström, J.: Interpolation spaces. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0344.46071
[2] Brenner, P.: OnL p-Lp ?, estimates for the wave equation. Math. Z.145, 251-254 (1975) · Zbl 0321.35052 · doi:10.1007/BF01215290
[3] Brenner, P.: On the existence of global smooth solutions of certain semilinear hyperbolic equations. Math. Z.167, 99-135 (1979) · Zbl 0395.35064 · doi:10.1007/BF01215117
[4] Brenner, P., von, Wahl, W.: Global classical solutions of non-linear wave equations. Math. Z.176, 87-121 (1981) · Zbl 0457.35059 · doi:10.1007/BF01258907
[5] Browder, F.E.: On nonlinear wave equations. Math. Z.80, 249-264 (1962) · Zbl 0109.32102 · doi:10.1007/BF01162382
[6] Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. Henri Poincaré (in press) · Zbl 0586.35042
[7] Glassey, R., Tsutsumi, M.: On uniqueness of weak solutions to semi linear wave equations. Commun. Partial Differ. Equations7, 153-195 (1982) · Zbl 0503.35059 · doi:10.1080/03605308208820221
[8] Heinz, E., von Wahl, W.: Zu einem Satz von F.E. Browder über nichtlineare Wellengleichungen. Math. Z.141, 33-45 (1975) · Zbl 0289.35076 · doi:10.1007/BF01236982
[9] Hörmander, L.: The analysis of linear partial differential operators. I.. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0521.35001
[10] Jörgens, K.: Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen. Math. Z.77, 295-308 (1961) · Zbl 0111.09105 · doi:10.1007/BF01180181
[11] Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod and Gauthier-Villars 1969
[12] Parenti, C., Strocchi, F., Velo, G.: A local approach to some nonlinear evolution equations of hyperbolic type. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.3, 443-500 (1976) · Zbl 0332.35040
[13] Pecher, H.:L p-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen I. Math. Z.150, 159-183 (1976) · Zbl 0347.35053 · doi:10.1007/BF01215233
[14] Pecher, H.: Ein nichtlinearer Interpolationssatz und seine Anwendung auf nichtlineare Wellengleichungen. Math. Z.161, 9-40 (1978) · Zbl 0384.35039 · doi:10.1007/BF01175610
[15] Pecher, H.: Nonlinear small data scattering for the wave and Klein-Gordon equation. Math. Z.185, 261-270 (1984) · Zbl 0538.35063 · doi:10.1007/BF01181697
[16] Pecher, H.: Low energy scattering for nonlinear Klein-Gordon equations., Preprint · Zbl 0588.35061
[17] Peral, J.C.:L p-estimates for the wave equations J. Funct. Anal.36, 114-145 (1980) · Zbl 0442.35017 · doi:10.1016/0022-1236(80)90110-X
[18] Reed, M.: Abstract nonlinear wave equations Berlin-Heidelberg-New York: Springer 1976 · Zbl 0317.35002
[19] Segal, I.E.: The global Cauchy Problem for a relativistic scalar field with power interaction. Bull. Soc. Math. Fr.91, 129-135 (1963) · Zbl 0178.45403
[20] Segal, I.E.: Nonlinear semigroups. Ann. Math.78, 339-364 (1963) · Zbl 0204.16004 · doi:10.2307/1970347
[21] Segal, I.E.: Space-time decay for solutions of wave equations., Adv. Math.22, 305-311 (1976) · Zbl 0344.35058 · doi:10.1016/0001-8708(76)90097-9
[22] Strauss, W.: Decay and asymptotics for ?u=f(u). J. Funct. Anal.2, 409-457 (1968) · Zbl 0182.13602 · doi:10.1016/0022-1236(68)90004-9
[23] Strauss, W.: On weak solutions of semilinear hyperbolic equations. An. Acad. Bras. Cienc.42, 645-651 (1970) · Zbl 0217.13104
[24] Strauss, W.: Nonlinear scattering theory at low energy. J. Funct. Anal.41, 110-133 (1981). Ibid. Strauss, W.: Nonlinear scattering theory at low energy. J. Funct. Anal.43, 281-293 (1981) · Zbl 0466.47006 · doi:10.1016/0022-1236(81)90063-X
[25] Strichartz, R.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J.44, 705-714 (1977) · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[26] Triebel, H.: Spaces of distributions of Besov type on Euclideann-space. Duality, interpolation. Ark. Mat.11, 13-64 (1973) · Zbl 0255.46026 · doi:10.1007/BF02388506
[27] Triebel, H.: Theory of function spaces. Basel: Birkhäuser 1983 · Zbl 0546.46028
[28] von Wahl, W.: Klassische Lösungen nichtlinearer Wellengleichungen im Grossen. Math. Z.112, 241-279 (1969) · Zbl 0177.36602 · doi:10.1007/BF01110242
[29] von Wahl, W.: Über nichtlineare Wellengleichungen mit zeitabhängigem elliptischem Hauptteil. Math. Z.142, 105-120 (1975) · Zbl 0299.35064 · doi:10.1007/BF01214943
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