A non-linear boundary value problem arising in the theory of thermal explosions

Abstract

When a heat-producing chemical re action takes place within a confined region, then under certain circumstances a thermal explosion will occur. In investigating from a theoretical viewpoint the conditions under which this happens, it is necessary to study the behaviour of the solution of a certain non-linear parabolic initial-boundary value problem . A frequently used approach is to study the problem indirectly, by investigating whether pos itive steady-state solutions exist ; the underlying assumption is that positive steady-state solutions exist if and only if a thermal explosion does not occur . The main theme of this t hesis is the development and application of an alternative direct approach to the problem, involving the construct ion of upper and lower solutions for the parabolic problem and the application of appropriate comparison theorems . The assumption here is that a thermal explosion will not occur if and only if the solution of the parabolic problem remains bounde d for all positive time . Following three chapters of introductory material, Chapter 4 c ont ains a survey of some of the important known results concerning the e xistence of positive steady-state solutions, especially those dealing with the effect on the theory of different assumptions as to the rate at which heat is produced in the reaction . The comparison theorems that are used in the alternative approach, which are modified versions of known results , are proved in Chapter 5 . In Chapter 6, the equivalence of the two criteria mentioned above for t he occurrence or non-occurrence of a thermal explos ion is established un der fairly general conditions . Also in this chapter , a critical value A* is defined for a parameter A appearing in the problem , s uch that a thermal explosion will not occur if the value of A is smaller than A*, but will occur if the value of A is greater than A*· In Chapter 7, upper and lower solutions are constructed for the t ime-dependent problem under a variety of assumptions as to the rate at which heat is produced in the react ion, and these are used to obtain a number of theorems concerning the behaviour of the solution of the problem, especially as the time variable tends to infinity . The information obtained from these theorems is related to and compared with t hat known from investigations of the existence of positive steady-st ate solutions . In conclus ion , a theorem is proved concerning the effect of re actant consumption on the theory. This is examined in the light of