Global existence for a coupled wave system related to the Strauss conjecture

Abstract

A coupled system of semilinear wave equations is considered, and a small data global existence result related to the Strauss conjecture is proved. Previous results have shown that one of the powers may be reduced below the critical power for the Strauss conjecture provided the other power sufficiently exceeds such. The stability of such results under asymptotically flat perturbations of the space-time where an integrated local energy decay estimate is available is established.

Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times

“…Finally, interpolating (12) and (16) we get (6) which finishes the sketch. In the following subsections, we give the detailed proof for each part.…”

“…Recently, the similar argument was applied for the coupled system in [12], and it is proven that the solution is global when σ < 0 and p ι > 2, for any initial data which are sufficiently regular and small. It is to be remarked that the case p 1 = 2 and p 2 > 3.5 was excluded in [12], for which it is known to be admissible for global results on Minkowski space-time.…”

“…It remains to consider 2 < q < ∞, for which we setα 0 = qα/2, o 1 = o(q − 2)/(q − o), s 1 = qs/(q −2). Similarly to the proof in subsection 3.1, we have α 0 ≤ 1/2, 2 ≤ o 1 ≤ ∞ and 1/2 − 1/o 1 ≤ s 1 < n/2, by the conditions on q, o, α and s. Moreover, for θ = 1 − 2/q ∈ (0, 1)θ)0 + θs 1 = s.Firstly when q = o and s = 1/2 − 1/o, where o 1 = ∞ and s 1 = 1/2 − 1/o 1 , by(11) and(12) we obtain…”