In this paper, we are concerned with global existence and uniform decay for the energy of solutions of Klein-Gordon-Schrödinger equations:

where
is a bounded domain,
, with boundary
of class
, where
and
are two disjoint pieces of
each having nonempty interior and
are given functions. We will denote by
the unit outward normal vector to
.
stands for the Laplacian with respect to the spatial variables;
denotes the derivative with respect to time
. Here
is the normal displacement to the boundary at time
with the boundary point
.

The above equations describe a generalization of the classical model of the Yukawa interaction of conserved complex nucleon field with neutral real meson field. Here,
is complex scalar nucleon field while
and
are real scalar meson one.

In three dimension, [

1–

5] studied the global existence for the Cauchy problem to

Klein-Gordon-Schrödinger equations have been studied as many as ever by many authors (cf. [6–11], and a list of references therein). However, they did not have treated acoustic boundary conditions.

Boundary conditions of the fifth and sixth equations are called acoustic boundary conditions. Equation (1.1)_{5} (the fifth equation of (1.1)) does not contain the second derivative
, which physically means that the material of the surface is much more lighter than a liquid flowing along it. As far as
in (1.1)_{6} (the sixth equation of (1.1)) is concerned to the case of a nonporous boundary, (1.1)_{6} simulates a porous boundary when a function
is nonnegative. When general acoustic boundary conditions, which had the presence of
in (1.1)_{5}, are prescribed on the whole boundary, Beale [12–14] proved the global existence and regularity of solutions in a Hilbert space of data with finite energy by means of semigroup methods. The asymptotic behavior was obtained in [13], but no decay rate was given there. Recently, the acoustic boundary conditions have been treated by many authors (cf. [15–21] and a list of references therein). However, energy decay problem with acoustic boundary conditions was studied by a few authors. For instance, Rivera and Qin [22] proved the polynomial decay for the energy of the wave motion using the Lyapunov functional technique in the case of general acoustic boundary conditions and
. Frota and Larkin [23] considered global solvability and the exponential decay of the energy for the wave equation with acoustic boundary conditions, which eliminated the second derivative term for
. However, it is not simple to apply the semigroup theory as well as Galerkin's method in [23] because a system of corresponding ordinary equations is not normal and one cannot apply directly the Carathéodory's theorem. So they overcame this problem using the degenerated second order equation. And Park and Ha [20] studied the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-type equation by using the multiplier technique. Moreover, [20] proved the exponential and polynomial decay rates of solutions for all
.

In this paper, we prove the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-Schrödinger equations with acoustic boundary conditions and allow to apply the method developed in [23]. However, [23] did not treat the Klein-Gordon-Schrödinger equations.

This paper is organized as follows. In Section 2, we recall the notation and hypotheses and introduce our main result. In Section 3, using Galerkin's method, we prove the existence and uniqueness of solutions to problem (1.1). In Section 4, we prove the exponential energy decay rate for the solutions obtained in Section 3.