Optimal decay estimates of a regularity-loss type system with constraint condition

In the paper [14], the authors formulated a new structural condition which includes the Kawashima-Shizuta condition, and analyzed the weak dissipative structure called the regularity-loss type for general systems which contain the Timoshenko system and the Euler-Maxwell system. However, this new structural condition can not cover all of dissipative systems. Indeed we introduce a dissipative system which does not satisfy the new condition and analyze the weaker dissipative structure in this paper. Precisely we first derive the L2 decay estimate of solutions and discuss the type of the corresponding regularity-loss structure. Moreover, in order to show the optimality of the decay estimate, we analyze the expansion for the corresponding eigenvalue of our problem and derive that the solution approaches the diffusion wave as time tends to infinity.