Global infinite energy solutions of the critical semilinear wave equation

We consider the critical semilinear wave equation (NLW)(2*-1) { square u + vertical bar u vertical bar(2*-2)u = 0 u vertical bar(t=0) = u(0) partial derivative tu vertical bar(t=0) = u(1,) set in R-d, d >= 3, with 2* = 2d/d-2 Shatah and Struwe [22] proved that, for finite energy initial data (ie if (u(0), u(1)) epsilon H-1 x L-2), there exists a global solution such that < u, partial derivative(t)u > epsilon C(R, H-1 x L-2). Planchon [17] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u(0), u(1)) in B-2,infinity(1) x B-2,infinity(0) is small enough. In this article, we build up global solutions of (NLW)2.-l for arbitrarily big initial data of infinite energy, by using two methods which enable to interpolate between finite and infinite energy initial data: the method of Calderon, and the method of Bourgain. These two methods give complementary results.