Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders

The paper is concerned with the long-time behaviour of the solutions of a certain class of semilinear parabolic equations in cylinders, which contains as a particular case the multidimensional thermo-diffusive model in combustion theory, We prove, under minimal conditions on the initial values, that the solutions eventually become monotone in the direction of the axis of the cylinder on every compact subset; this implies convergence to travelling fronts, This result is applied to propagation versus extinction problems: given a compactly supported initial datum, sufficient conditions ensuring that the solution will either converge to 0 or to a pair of travelling fronts are given, Additional information on the corresponding equations in finite cylinders is also obtained.