Stability of stationary solutions for a degenerate parabolic system

The paper is devoted to the stability of stationary solutions of an evoluti on system, describing heat explosion in a two-phase medium, where a parabol ic equation is coupled with an ordinary differential equation. Spectral pro perties of the problem linearized about a stationary solution are analyzed and used to study stability of continuous branches of solutions. For the co nvex nonlinearity specific to combustion problems it is shown that solution s on the first increasing branch are stable, solutions on all other branche s 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].