# Global infinite energy solutions of the critical semilinear wave equation

### Pierre Germain

New York University, USA

## Abstract

We consider the critical semilinear wave equation

set in $R_{d}$, $d≥3$, with $2_{∗}=d−22d ⋅$ 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 $(u_{0},u_{1})∈H˙_{1}×L_{2}$), there exists a global solution such that $(u,∂_{t}u)∈C(R,H˙_{1}×L_{2})$. 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 $B˙_{2,∞}×B˙_{2,∞}$ 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 Calderón, and the method of Bourgain. These two methods give complementary results.

## Cite this article

Pierre Germain, Global infinite energy solutions of the critical semilinear wave equation. Rev. Mat. Iberoam. 24 (2008), no. 2, pp. 463–497

DOI 10.4171/RMI/543