(untitled) (Journal of Partial Differential Equations, 1997)
In this paper, we continue to study the equation ◻Φ^I+f^I(Φ,∂Φ) = 0 where ◻ = -∂²_t + Δ denotes the standard D' Alembertian in R^2+1 and the nonlinear terms f have the form f^I = Σ_JKΓ^I_JK(Φ)Q_0(Φ^J,Φ^K) with Q_0(Φ,φ) = -∂_tΦ∂_tφ + Σ&sup_i=1∂_iΦ∂_tφ and Γ^I_JK being C^∞ function of Φ. In Y. Zhou [1], we showed that the initial value problem Φ(0,x) = Φ_0(x), ∂_tΦ(0,x) = Φ_1 (x) is locally well posed for Φ_0 ∈ H^s+1, Φ_1 ∈ H^s with s = 18. Here, we shall further prove that the initial value problem is locally well posed for any s > 0.