On some nonlinear hyperbolic systems with damping boundary conditions

Abstract

IN RECENT years the effect of boundary damping on the solutions to interior initial boundary value problems for hyperbolic equations has been extensively studied. In [ 111, Quinn and Russell considered the wave equation with damping boundary condition on a part of the boundary. It was proved by control-theoretic method that if the boundary where the boundary condition was posed was star-shaped, then the solution of the wave equation had polynomial decay rate. Later on, Chen in [l] got the exponential decay of solutions to the wave equation under the more restrictive assumptions on the boundary. And also, Lagnese in [6] got the exponential decay of solutions to the three-dimensional elastic wave equations with damping boundary condition on a part of the boundary provided the geometry of the boundary is suitably restricted. As for the nonlinear problem, Greenberg and Li in 131 considered the nonlinear conservation system in one space dimension with damping boundary condition and proved the global existence of smooth solution when the initial data are small. Shen and Zheng [14] and Nagasawa [9] proved the global existence of classical solutions to the system of one-dimensional thermoelasticity and to the system of one-dimensional nonlinear problem in higher space dimension, respectively. For the nonlinear problem in higher space dimension, to our knowledge, the works were done by Qin [lo] and Zajaczkowski [26] only for the nonlinear wave equation with damping boundary conditions based on Chen’s results in [l]. In this paper our aim is to derive the decay rate of solutions to the linear hyperbolic system of second order with damping boundary condition without any assumptions on the geometry of the boundary by using the spectral analysis method (cf. Section 2 below), and then to prove the global existence of smooth solutions to the corresponding nonlinear system with applications to nonlinear elastodynamics and to nonlinear acoustic wave equations. Our approach to obtaining the rate of decay of solutions to linear problem is completely different from [l], [6] and [lo].