The Cauchy problem for the rotation-modified Kadomtsev-Petviashvili type equation

This paper is devoted to studying the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) type equation partial derivative(x)(u(t) - beta partial derivative(3)(x)u + partial derivative(x)(u(2))) + beta'partial derivative(2)(y)u - gamma u = 0 in the anisotropic Sobolev spaces H-s1,H- s2 (R-2). When beta > 0 and gamma > 0, beta' < 0, we show that the Cauchy problem is locally well-posed in H-s1,H- s2 (R-2) with S-1 > -1/2 and s(2) >= 0. The main difficulty in establishing bilinear estimates related to nonlinear term of RMKP type equation is that the resonant function vertical bar 3 beta xi xi(1)xi(2) - gamma(xi(2)(1) - xi(1)xi(2) + xi(2)(2))/xi xi(1)xi(2) - beta'xi(1)xi(2)/xi (mu(1)/xi(1) - mu(2)/xi(2))(2)vertical bar may tend to zero since beta > 0, gamma > 0 and beta' < 0. When beta > 0 and gamma > 0 and beta' < 0, we also prove that the Cauchy problem for RMKP equation is ill-posed in H-S1,H-0 (R-2) with S-1 < -1/2 in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not C-3. (C) 2020 Elsevier Inc. All rights reserved.