The initial value problem for the elliptic-hyperbolic Davey-Stewartson equation (Q1125247)

The author deals with the Davey-Stewartson equation, that is \[ \begin{cases} \partial_tu- i(\partial^2_x+ \partial^2_x) u= f(u)\quad &\text{in }(0,\infty)\times \mathbb{R}^2,\\ u(0,x,y)= u_0(x,y)\quad &\text{in }\mathbb{R}^2,\end{cases}\tag{1} \] where \(u(t,x,y)\) is \(\mathbb{C}\)-valued, \(i= \sqrt{-1}\). Under some natural conditions on the nonlinearity \(f(u)\), the author proves both local and global existence theorem of solutions to (1). The main difficulty in (1) is the absence of the classical energy estimates. The author overcomes this difficulty using the smoothing property of linear Schrödinger type equations.