Global infinite energy solutions of the critical semilinear wave equation

Abstract

We consider the critical semilinear wave equation $$ (NLW){2^-1} ;;; \left{ \begin{aligned} \square u + |u|^{2^-2} u & = 0 \ u{|t=0} & = u\_0 \ \partial\_t u\_{|t=0} & = u\_1 , ,, \end{aligned} \right. $$ 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.