Global infinite energy solutions of the critical semilinear wave equation

We consider the critical semilinear wave equation (NLW )2∗−1 u+ |u|2∗−2u = 0 u|t=0 = u0 ∂tu|t=0 = u1 , set in R, d ≥ 3, with 2 = 2d d−2 · Shatah and Struwe [22] proved that, for finite energy initial data (ie if (u0, u1) ∈ Ḣ × L), there exists a global solution such that (u, ∂tu) ∈ C(R, Ḣ1×L2). Planchon [17] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u0, u1) in Ḃ 1 2,∞×Ḃ 2,∞ is small enough. In this article, we build up global solutions of (NLW )2∗−1 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.