Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation

Abstract

We prove global existence of smooth solutions of the 3D log-log energy-supercritical wave equationwith 0 < c < 8/225 and smooth initial data (u(0) = u 0 , ∂ t u(0) = u 1 ).First we control the L 4 t L 12x norm of the solution on an arbitrary size time interval by an expression depending on the energy and an a priori upper bound of its).The proof of this long time estimate relies upon the use of some potential decay estimates and a modification of an argument by Tao.Then we find an a posteriori upper bound of the L ∞ t H 2 ‫ޒ(‬ 3 ) norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.• time translation invariance: if u is a solution of (1-1) and t 0 is a fixed time then ũ(t, x) := u(t -t 0 , x) is also a solution of (1-1);• space translation invariance: if u is a solution of (1-1) and x 0 is a fixed point lying in ‫ޒ‬ 3 then ũ(t, x) := u(t, xx 0 ) is also a solution of (1-1);• time reversal invariance: if u is a solution to (1-1) then ũ(t, x) := u(-t, x) is also a solution.

Locations

• Analysis & PDE - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF