Decay property for symmetric hyperbolic system with memory-type relaxation

Abstract

We consider the decay property for a symmetric hyperbolic system with memory-type relaxation. We assume that the memory kernel is a strongly positive definite kernel and prove that the system has the decay property of regularity-loss type. Our proof is based on a technical energy method in the Fourier space which makes use of the structural condition (called Craftsmanship condition or Condition (K)) and the modified properties of strongly positive definite kernels.