On the Nonlinear Equations
نویسنده
چکیده
under the initial condition v\t^= a{x) and the boundary condition fl|dQ = 0. Here Q is a bounded domain in R whose boundary dQ is assumed to be sufficiently smooth. We assume that a = a(x) continuous in S. As I. M. Gel'fand [2] pointed out, these problems arise in the theory of thermal selfignition of a chemically active mixture of gases in a vessel. He showed, in the special case where Q is an m-dimensional ball of radius r, that for m = 1 or 2 there exists a critical radius rc such that BVP has two solutions, one solution or no solution, according as 0 rc. If m = 3, then the number of solutions of BVP can be 0, 1, 2, • • • , or oo, depending on r. I t would be quite interesting to extend Gel'fand's result to the case of general Q. However, we do not try to proceed in this direction. Instead, our main objective is to prove a certain relationship among solutions of BVP when they exist and to study the asymptotic stability of these solutions, i.e., the convergence of solutions of IVP to solutions of BVP as t—»+<*>. Below we describe some of our results. The other results and technical details will be published elsewhere together with generalizations including replacement of the function e by a general function ƒ =f(u); i?—»i? which is smooth, positive, increasing, and strictly convex. The author wishes to thank Professor Melvyn Berger who brought the author's attention to the present problem and provided the author with helpful preliminary information.
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