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We prove that solutions to the critical wave equation (1.1) with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous Strauss conjecture about semilinear wave equations of the form u−* 2 t u+|u| p =0. The rest of the cases, the lower-dimensional case n 3, and the sub or super critical cases were settled many years earlier by the work of several authors.

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We verify the critical case p = p 0 (n) of Strauss' conjecture [31] concerning the blow-up of solutions to semilinear wave equations with variable coefficients in R n , where n ≥ 2. The perturbations of Laplace operator are assumed to be smooth and decay exponentially fast at infinity. We also obtain a sharp lifespan upper bound for solutions with compactly supported data when p = p 0 (n). The unified approach to blow-up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of Zhou [44] and Zhou & Han [46]: exponential "eigenfunctions" of the Laplacian [38] are used to construct the test function φq for linear wave equation with variable coefficients and John's method of iterations [14] is augmented with the "slicing method" of Agemi, Kurokawa and Takamura [1] for lower bounds in the critical case.