EVENTUAL MONOTONICITY AND CONVERGENCE TO TRAVELING FRONTS FOR THE SOLUTIONS OF PARABOLIC EQUATIONS IN CYLINDERS

The paper is concerned with the long-time behaviour of the solutions o f a certain class of semilinear parabolic equations in cylinders, whic h 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 d irection of the axis of the cylinder on every compact subset; this imp lies convergence to travelling fronts, This result is applied to propa gation versus extinction problems: given a compactly supported initial datum, sufficient conditions ensuring that the solution will either c onverge to 0 or to a pair of travelling fronts are given, Additional i nformation on the corresponding equations in finite cylinders is also obtained.