Non-Linear Elliptic Eigenvalue Problems, with Applications to Thermal Combustion

Abstract

A study is made of the equations of combustion for a steady – state thermal regime with non-linear (with temperature) heat generation in the material with a linearized radiation condition on the boundary. A mathematical model is discussed which gives rise to a (mildly) non-linear elliptic boundary value problem. The nature of the solution set of a class of non-linear elliptic boundary value problems as dependent on the form of the nonlinearity is studied. The conditions are discussed which determine the topological properties of the spectrum and for which there exist the critical points in spectrum The stability properties of the supremum of spectrum (under small perturbations of the operators involved) are studied. The definition of the critical explosion parameter is given and the question of its existence is discussed. The variational problem associated with the non-linear eigenvalue problem is studied, so that necessary conditions are established for the Wake and Rayner method of computation of the critical explosion parameter to be used in the general case. As a result, the mathematical theory of thermal combustion is developed - under a plausible heuristic hypothesis - in which the criticality parameters possess essentially the same properties as those in the Semenov qualitative theory of thermal combustion. The mathematical Justification of the widely used Frank - Kamenetskii approximation is obtained in the framework of this theory. Most of the results are extended to a broad class of problems with (mild) nonlinearities in, boundary conditions.