We consider the asymptotic behavior of solutions to the initial value problem for a certain class of hyperbolic-elliptic coupled systems. It will be proved that the solution is time asymptotically approximated by the superposition of diffusion waves constructed in terms of the self-similar solutions of generalized Burgers equations. We will give space-time decay estimates for the residual term through a pointwise estimate for the Green’s function of the linearized system.
"On space-time decay properties of solutions to hyperbolic-elliptic coupled systems." Hiroshima Math. J. 32 (2) 229 - 308, July 2002. https://doi.org/10.32917/hmj/1151007559