Abstract

We prove global existence of smooth solutions of the 3D log-log energy-supercritical wave equation

with 0 < c < 8 ∕ 225 and smooth initial data (u(0) = u₀, ∂_tu(0) = u₁). First we control the L_t⁴L_x¹² 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 L_t^∞H²(R³) norm, with H²(R³) := Ḣ²(R³) ∩Ḣ¹(R³). 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^∞H²(R³) norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.

Keywords

Mathematical Subject Classification 2000

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