The Mathematics of Porous Medium Combustion
Two partial differential equations arising from the theory of porous medium combustion are examined. While both equations possess a trivial steady solution, the form of the reaction rate, which is discontinuous as a function of the dependent variable, precludes bifurcation of non-trivial steady solutions from the branch of trivial solutions. A constructive approach to the existence theory for non-trivial global solution branches is developed. The method relies on finding an appropriate set of solution dependent transformations which render the problems in a form to which local bifurcation theory is directly applicable. Specifically, by taking a singular limit of the (solution dependent) transformation, an artificial trivial solution (or set of solutions) of the transformed problem is created. The (solution dependent) mapping is not invertible when evaluated at the trivial solution(s) of the transformed problem; however, for non-trivial solutions which exist arbitrarily close to the artificial trivial solution, the mapping is invertible. By applying local bifurcation theory to the transformed problem and mapping back to the original problem, a series expansion for the non-trivial solution branch is obtained.
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