Stability of stationary solutions for a degenerate parabolic system

Abstract

The paper is devoted to the stability of stationary solutions of an evolution system, describing heat explosion in a two-phase medium, where a parabolic equation is coupled with an ordinary differential equation. Spectral properties of the problem linearized about a stationary solution are analyzed and used to study stability of continuous branches of solutions. For the convex nonlinearity specific to combustion problems it is shown that solutions on the first increasing branch are stable, solutions on all other branches are unstable. These results remain valid for the scalar equation and they generalize the results obtained before for heat explosion in the radially symmetric case [1].