existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium

Abstract

Abstract. The paper is devoted to analysis of an elliptic-algebraic system of equations describing heat explosion in a two phase medium filling a star-shaped domain. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as their behavior depending on the size of the domain and on the coefficient of heat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.Mathematics Subject Classification. 36J70, 80A20.

Stability of stationary solutions for a degenerate parabolic system

“…The main difference is that stationary solutions of (1.1)-(1.3) can lose their regularity. We show [7] that there are three types of solutions of (1.1)-(1.3): classical solutions, for which u 1 (x) < u * 1 inΩ; critical solutions, for which u 1 (x) 6 u * 1 inΩ, and the critical set Ω * = {x ∈ Ω : u 1 (x) = u * 1 } is not empty; and multi-valued solutions for which u 1 (x) can take values larger than u * 1 . It can easily be shown that the classical solutions are continuous, together with their second derivatives.…”

“…The existence of stationary solutions for this case and blow-up solutions have been studied in detail [8][9][10][11][12][13][14][15]. We show that stationary solutions of (1.1)-(1.3) converge to stationary solutions of (1.5) as α → ∞ [7].…”

“…Let w = (w 1 , w 2 ) be a classical stationary solution of (1.1)-(1.3). Then (see Barillon et al [7])…”

“…This model was introduced by Dik & Krainov [6] in the onedimensional case with a particular form of the nonlinearity, F(u 1 ) = e u 1 . We have studied [7] the existence and regularity of stationary solutions in the multi-dimensional case. This work is devoted to the stability of stationary solutions.…”

“…along with any point x ∈ Ω, the whole interval [0, x] belongs to Ω. We have studied the existence of stationary solutions of the problem (1.1)-(1.3) depending on the size of the domain [7]. For this we introduce a family of domains Ω L such that for any x ∈ Ω, xL ∈ Ω L .…”