1. Multidimensional stability of V-shaped traveling fronts in time periodic bistable reaction–diffusion equations
- Author
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Wei-Jie Sheng, Department of Mathematics (HIT Harbin Institute of Technology), Harbin Institute of Technology (HIT), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and This paper was supported by NSF of China (11401134) and by China Scholarship Council for a one year visit of Aix Marseille Université.
- Subjects
Current (mathematics) ,Bistability ,010102 general mathematics ,Mathematical analysis ,Time periodic V-shaped traveling fronts ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Bistable ,Computational Mathematics ,Computational Theory and Mathematics ,Two-dimensional space ,Reaction diffusion equations ,Modeling and Simulation ,Stability theory ,Bounded function ,Reaction–diffusion system ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Initial value problem ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Multidimensional stability ,Mathematics - Abstract
International audience; This paper deals with the multidimensional stability of time periodic V-shaped traveling fronts in bistable reaction–diffusion equations. It is well known that time periodic V-shaped traveling fronts are asymptotically stable in two dimensional space. In the current study, we further show that such fronts are asymptotically stable under spatially decaying initial perturbations in Rn with n≥3. In particular, we show that the fronts are algebraically stable if the initial perturbations belong to L1 in a certain sense. Furthermore, we prove that there exists a solution oscillating permanently between two time periodic V-shaped traveling fronts, which implies that time periodic V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Finally we show that time periodic V-shaped traveling fronts are only time global solutions of the Cauchy problem if the initial perturbations lie between two time periodic V-shaped traveling fronts.
- Published
- 2016