Initial boundary value problem for generalized Zakharov equations. (Q1941781)

The authors consider the following generalized Zakharov system \[ \begin{cases} iE_t + \Delta E - H^2\Delta ^2E - nE &= 0 \\ n_{tt} - \Delta n + H^2\Delta ^2n - \Delta | E| ^2 &= 0, \end{cases} \] where \(H\) is a dimensionless parameter, \(E\: \mathbb {R}^{+}\times \Omega \to \mathbb {C}^2\) and \(n\: \mathbb {R}^{+}\times \Omega \to \mathbb {R}\) with \(\Omega \subset \mathbb {C}^2\) a bounded smooth domain. The authors prove that the initial value problem \[ E| _{t = 0} = E_0(x), \;n| _{t=0} = n_0(x), \;n_t| _{t=0} = n_1(x) \] with boundary conditions \[ E| _{\partial \Omega } = \frac {\partial E}{\partial \nu }\Big | _{\partial \Omega } = \frac {\partial {(\nabla E)}} {{\partial \nu }}\Big | _{\partial \Omega } = 0 \qquad n| _{\partial \Omega } = \frac {\partial (\nabla n)}{\partial \nu }\Big | _{\partial \Omega } = 0 \] has a unique global generalized solution. The proof proceeds by Galerkin approximation and a priori integral estimates.