An inhomogenous nonlinear boundary value problem in model reactive media

Abstract

We consider the differential equation uu′ + q λ(u)′ = βu″ + f( x) in R 1 subject to the conditions that u approaches constant values at x = ±∞ and u′(±∞) = 0. The problem arises from studying steady viscous solutions with sources in the Fickett-Majda model of detonation in the case where the chemical kinetics is governed by a reversible reaction R⇌ P , whose equilibrium curve is λ = λ(u), where λ is the mass fraction of the product species P and u is temperature. Both necessary and sufficient conditions for a solution are found in terms of the heat of reaction q > 0 and the L 1-norm ‖ f‖of the energy source term. Critical values of q are determined where bifurcations occur. Existence of solutions is shown by combining a comparison theorem for terminal boundary value problems and a continuation theorem for initial value problems.