This paper is concerned with the multidimensional asymptotic stability of V-shaped traveling fronts in the Allen-Cahn equation under spatial decaying initial values.We frst show that V-shaped traveling fronts are asymptotically stable under the perturbations that decay at infnity.Then we further show that there exists a solution that oscillates permanently between two V-shaped traveling fronts,which indicates that V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations.Our main technique is the supersolutions and subsolutions method coupled with the comparison principle.
This paper is concerned with nonplanar traveling fronts for delayed reactiondiffusion equation with bistable nonlinearity in R^m(m≥3).By the comparison principle and super-and subsolutions technique,we establish the existence of pyramidal traveling fronts.
