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Global solution to the two-dimensional Klein-Gordon equation. (English) Zbl 0741.35039
The existence of a global solution for the nonlinear Klein-Gordon equation is critical for the two-dimensional case. In this paper the authors prove global existence of solution for the nonlinear Klein-Gordon equation with small initial data in $$C_ 0^ \infty$$. It is needed a sharp $$L^ \infty-L^ \infty$$ estimate for the Cauchy problem involving the radial vector field $$L_ 0=t\partial_ t+x_ 1\partial_ 1+x_ 2\partial_ 2$$.
Reviewer: A.Tsutsumi (Sakai)

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35Q40 PDEs in connection with quantum mechanics 35L15 Initial value problems for second-order hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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##### References:
 [1] Bachelot A., Publications de 1’Universitk de Bordeaux I. 8406 (1984) [2] Bachelot A., Ann. Inst. Henri Poincaré (Physique théorique) 48 (1988) [3] Bachelot A., I, in: C. R. Acad. sc. pp 573– (1985) [4] Bachelot A., I 46, in: Ann. Inst. pp 45– (1985) [5] Bergh J., Interpolation spaces (1976) [6] DOI: 10.1007/BF02570764 · Zbl 0671.35052 [7] Georgiev V., Comp. Rend Acad. Sci. Bulg 42 pp 25– (1989) [8] Hanouzet B., Research Notes in Math. Pitman 89 pp 208– (1983) [9] Hanouzet B., and Publications de l’Université de Bordeaux i (1976) [10] Hanouzet B., C.R. Acad. Sc. Paris. 294 pp 745– (1982) [11] Hanouzet B., C.R. Acad. Sci. 301 (1985) [12] Hörmander, L., Lectures Lund 2 (1988) [13] Hörmander L., Saint–Jean-de-Monts. 2 pp 1– (1987) [14] DOI: 10.1002/cpa.3160340103 · Zbl 0453.35060 [15] kato T., Lecture Notes in Math 448 pp 25– (1975) [16] DOI: 10.1002/cpa.3160380305 · Zbl 0635.35059 [17] DOI: 10.1002/cpa.3160380512 · Zbl 0597.35100 [18] Klainerman S., Lectures in App. Math. 23 pp 293– (1986) [19] Lax P., SIAM Reg. Conf. Lecture #11 (1973) [20] Majda A., Math. Sciences 53 [21] Mwtivier G., Preprint 53 (1988) [22] DOI: 10.1016/0001-8708(76)90097-9 · Zbl 0344.35058 [23] DOI: 10.1007/BF01110149 · Zbl 0212.44201
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