Almost Global Existence of Solutions to the Kadomtsev-Petviashvili Equations

Abstract

We consider the Cauchy problem for the Kadomtsev-Petviashvili equations u_t + u_xxx + σ∂_x^–1u_yy = –(u²)_x, (x, y) ∈ R², t ∈ R, u(0, x, y) = u₀(x, y), (x, y) ∈ R², where σ = 1 or σ = –1, ∂_x^–1 = ∫_–∞^x dx′. We prove that the maximal existence time T is estimated from below as T ≥ exp(C/ε), where ε denotes the size of the initial data, C > 0 is a constant.