Global Existence, Uniqueness, and Stability for a Nonlinear Hyperbolic-Parabolic Problem in Pulse Combustion

A global existence theorem is established for an initial-boundary value problem, with time-dependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonlinear mixed hyperbolic-parabolic system. Using results previously established for the associated linear problem, a fixed point argument is employed to prove local existence for a regularized version of the nonlinear problem with artificial viscosity. Appropriate α-priori estimates are then derived which imply that the local existence result can be extended to a global existence theorem for the regularized problem. Finally, a different set of α priori estimates is generated which allows for taking the limit as the artificial viscosity parameter converges to zero; the corresponding solution of the regularized problem is then proven to converge to the unique solution of the initial-boundary value problem for the original, nonlinear, hyperbolic-parabolic system.