Local Well‐Posedness for the Fourth‐Order Schrödinger Equation With Quadratic Derivative Nonlinearities in Fourier–Lebesgue Spaces

We investigate the Cauchy problem for the fourth‐order Schrödinger equation with quadratic nonlinearities involving second‐order derivatives: u xx u , u xx ū , ( u x ) 2 , uū xx , and | u x | 2 , where u = u ( x , t ) is a complex‐valued function defined on . The flow map of this Cauchy problem with the nonlinear terms u xx u or u xx ū fails to be C 2 differentiable at zero from the Sobolev space to for all . This motivates our choice of Fourier–Lebesgue (F–L) spaces as the functional framework. By applying the F–L‐type Bourgain spaces and the dyadic decomposition method, we derive crucial bilinear estimates. Therefore, the local well‐posedness of this Cauchy problem is established in the F–L spaces. MSC2020 Classification: 35G25, 35Q55, 42B35, 42B37