Uniform global existence and parabolic limit for partially dissipative hyperbolic systems

This work concerns smooth solutions to the Cauchy problem for first-order partially dissipative hyperbolic systems with a small parameter. The systems are written in non-conservative form in several space variables. We introduce algebraic conditions on the structure of the systems. Under these conditions together with a partial dissipation condition and the Shizuta–Kawashima stability condition, we prove three main results around constant equilibrium states. These results are uniform global existence with respect to the parameter, global-in-time convergence of the systems to second-order nonlinear parabolic systems in a slow time variable, and global existence when the parameter is fixed. We also give examples of physical models to which the above results can be applied.