We consider the critical semilinear wave equation
\begin{equation*}
(NLW)_{2^*-1} \;\;\;
\left\{
\begin{aligned}
\square u + |u|^{2^*-2} u & = 0 \\
u_{|t=0} & = u_0 \\
\partial_t u_{|t=0} & = u_1 \, \,,
\end{aligned} \right.
\end{equation*}
set in $\mathbb{R}^d$, $d \geq 3$, with $2^* = \frac{2d}{d-2} \,\cdotp$ Shatah and
Struwe [Shatah, J. and Struwe, M.: Geometric wave equations. Courant Lecture Notes
in Mathematics 2. New York University, Courant Institute of Mathematical Sciences.
American Mathematical Society, RI, 1998] proved that, for finite energy initial
data (ie if $(u_0,u_1) \in \dot{H}^1 \times L^2$), there exists a global
solution such that $(u,\partial_t u)\in \mathcal{C}(\mathbb{R},\dot{H}^1 \times L^2)$.
Planchon [Planchon, F.: Self-similar solutions and semi-linear wave equations in Besov
spaces. J. Math. Pures Appl. (9) 79 (2000), no. 8, 809-820] showed that there also
exists a global solution for certain infinite energy initial data, namely, if the norm
of $(u_0,u_1)$ in $\dot{B}^1_{2,\infty} \times \dot{B}^0_{2,\infty}$ 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 Calderón,
and the method of Bourgain. These two methods give complementary results.