Global infinite energy solutions of the critical semilinear wave equation

Abstract

We consider the critical semilinear wave equation

set in ${\mathbb{R}}^d$, $d\ge 3$, with $2^{\ast }=\frac{2d}{d-2}\cdot$ 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 ${\dot{B}}_{2,\infty }^1\times {\dot{B}}_{2,\infty }^0$ is small enough. In this article, we build up global solutions of $(NLW{)}_{2^{\ast }-1}$ 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.