On the critical exponent pc of the 3D quasilinear wave equation [formula omitted] with short pulse initial data. I, global existence.

For the 3D quasilinear wave equation − (1 + (∂ t ϕ) p) ∂ t 2 ϕ + Δ ϕ = 0 with the short pulse initial data (ϕ , ∂ t ϕ) (1 , x) = (δ 2 − ε 0 ϕ 0 (r − 1 δ , ω) , δ 1 − ε 0 ϕ 1 (r − 1 δ , ω)) , where p ∈ N , p ≥ 2 , 0 < ε 0 < 1 , r = | x | , ω = x r ∈ S 2 , and δ > 0 is sufficiently small, under the outgoing constraint condition (∂ t + ∂ r) k ϕ (1 , x) = O (δ 2 − ε 0 − k max ⁡ { 0 , 1 − (1 − ε 0) p }) for k = 1 , 2 , we will establish the global existence of smooth large data solution ϕ when p > p c with p c = 1 1 − ε 0 being the critical exponent. In the forthcoming paper, when 1 ≤ p ≤ p c , we show the formation of the outgoing shock before the time t = 2 under the same outgoing constraint condition and the other suitable assumption of (ϕ 0 , ϕ 1). [ABSTRACT FROM AUTHOR]