Decay property of regularity-loss type for nonlinear hyperbolic–elliptic system in multi-dimensions

In this paper, we study the asymptotic behavior of solutions for a hyperbolic–elliptic system with a 2m-order elliptic part in multi-dimensional space. This system is a modified version of the simplest radiating gas model and it verifies Lp decay property of regularity-loss type in Rn. The global existence and Lp estimates of the solution to the Cauchy problem for the hyperbolic–elliptic system are obtained. Since the dissipative property of this system is so weak in high frequency region, the usual energy method does not work well in deriving the a priori estimates for global solutions to the nonlinear problem. Our analysis is based on a frequency decomposition and the method of combining the Green function with some time-weight energy estimates.