Stability of stationary solutions for a degenerate parabolic system (Q2732513)

The authors investigate the stability of stationary solutions of the mathematical model of heat explosion in a two-phase medium. The model consists of the degenerate parabolic system of equations, where a parabolic equation is coupled with an ordinary differential equation. Spectral properties of the linearized problem are analyzed and used to study stability of continuous branches of solutions. For convex nonlinearities 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 are also valid for the corresponding scalar equations and they generalize the previous results obtained for heat explosion in the radially symmetric case.