Global well-posedness and scattering for the radial, defocusing, cubic   wave equation with initial data in a critical Besov space
  In this paper we prove that the cubic wave equation is globally well - posed and scattering for radial initial data lying in $B_{1,1}^{2} \times B_{1,1}^{1}$. This space of functions is a scale invariant subspace of $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. 
Benjamin Dodson
math.AP