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is the homogeneous Sobolev space of order μ. We prove the global solvability of (1.1) in the critical Sobolev spacė Hn/2 with nonlinearityf of exponential type, a typical example of which is given by u2eλ|u| with λ > 0. There is a large literature on the Cauchy problem for the equation (1.1), see for instance [2,5,7,9,10,12,13,16,17,21,22]. It is well-known that the Cauchy problem (1.1) is locally well-posed in the usual Sobolev space Hμ(R) if μ > n/2 andf is any smooth function withf(0) = 0 [16], or if 1/2 ≤ μ < n/2 andf is given as a single power nonlinearity λ|u|p−1uwith λ ∈ Candp ≤ 1+4/(n−2μ) [9,13,17,22]. Moreover if p = 1+4/(n−2μ) and1/2 ≤ μ < n/2, then we have global̇ Hμ-solutions with the Cauchy

Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth