On self similar solutions well posedness and

On self-similar solutions, well-posedness and the conformal wave equation Fabrice Planchon Abstract We prove that the initial value problem for the conformally in- variant semi-linear wave equation is well-posed in the Besov space _ B 1 2 ;1 2 (R n ). This induces the existence of (non-radially symmetric) self-similar solutions for homogeneous data in such Besov spaces. Introduction We are interested in the Cauchy problem for the conformal semi-linear wave equation (1) 8 < : u = juj n+3 n1 ; u(x; 0) = u 0 (x); @ t u(x; 0) = u 1 (x): As usual, scaling plays an important role when looking for the lowest possible regularity on the data: it reads (2) 8 > < > : u 0 (x) ! u 0; (x) = n1 2 u 0 (x) u 1 (x) ! u 1; (x) = n+1 2 u 1 (x) u(x; t) ! u (x; t) = n1 2 u(x; t): Well-posedness

• wave equation


• equation has


• similar solution


• strichartz estimates


• conformal semi-linear


• super-conformal


• handle when dealing