Time-Asymptotic Limit of Solutions of a Combustion Problem

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A combustion model which captures the interacting among nonlinear convection, chemical reaction and radiative heat transfer is studied. New phenomena are found with radiative heat transfer present. In particular, there is a detonation wave solution in which there is a sonic point inside the reaction zone. As a consequence, the traveling wave speed cannot be determined before the problem is solved. The shooting method is used to prove the existence of the traveling wave. The condition we shoot at is the compatibility condition at the sonic point. Furthermore, the speed of the detonation wave decreases as the heat loss coefficient increases, as expected physically. We study the time-asymptotic limit of solutions of initial value problem for the same problem. We prove that the solution exists globally and the solution converges uniformly, away from the shock, to a shifted traveling wave solution as t → + ∞ for certain “compact support” initial data.

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