The critical case for a semilinear weakly hyperbolic equation

Abstract

We prove a global existence result for the Cauchy problem, in the three-dimensional space, associated with the equation u>tt - αλ (t)Δxu = -u|u|p(λ)-1 where αλ(t) ≥ 0 and behaves at (t - t0)λ close to some t0 > 0 with α(t0) = 0, and p(λ) = (3λ + 10) / (3λ + 2) with 3 ≤ p(λ) ≤ 5. This means that we deal with the superconformal, critical nonlinear case. Moreover we assume a small initial energy.