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International audience; This paper is concerned with the existence and qualitative properties of transition fronts for time periodic bistable reaction-diffusion equations in R-N. We first show that any almost-planar transition front is actually planar, regardless of the number of transition layers. Then we prove that all transition fronts admit a global mean speed gamma and it holds gamma = vertical bar c vertical bar, where c is the speed of the planar traveling front. Finally we establish the existence of a transition front in R-N that is not a standard traveling front. Such a front behaves like three moving time periodic planar fronts as time goes to -infinity and like a time periodic V-shaped traveling front as time goes to infinity.

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.

This paper is concerned with entire solutions of monotone bistable reaction–diffusion systems in $$\mathbb {R}^N$$ . We first show that there is a new entire solution for bistable reaction–diffusion systems which behaves as three moving planar fronts as time goes to $$-\,\infty $$ and as a V-shaped traveling front as time goes to $$\infty $$ . Then we address the speed of the interfaces corresponding to the entire solution by appealing to the super-sub solutions technique. Finally, we apply our results to two important models in biology.

International audience; This paper is concerned with the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, we show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique planar speed. We also give some geometrical and scaling conditions on the domain, either exterior or with multiple cylindrical branches, which guarantee that any transition front has a global mean speed.

This paper is concerned with the multidimensional stability of planar traveling fronts in bistable reaction–diffusion systems. It is first shown that planar traveling fronts are asymptotically stable under spatially decaying initial perturbations by appealing to the comparison principle and super-subsolution method. In particular, if the perturbations belong to L 1 ( R n − 1 ) in a certain sense, we obtain a convergence rate like t − n − 1 2 . Then we show that the solution of the Cauchy problem converges to the planar traveling front with rate t − n + 1 4 for a spatially non-decaying perturbation with the help of semigroup theory. Finally, we prove that there exists a solution oscillating permanently between two planar traveling fronts, which indicates that planar traveling fronts are not always asymptotically stable in multidimensional space under general bounded perturbations.

This paper is concerned with the existence and stability of time periodic traveling curved fronts for reaction–diffusion equations with bistable nonlinearity in \({\mathbb {R}}^3\). We first study the existence and other qualitative properties of time periodic traveling fronts of polyhedral shape. Furthermore, for any given \(g\in C^{\infty }(S^1)\) with \(\min \nolimits _{0\le \theta \le 2\pi }g(\theta )=0\) that gives a convex bounded domain with smooth boundary of positive curvature everywhere, which is included in a sequence of convex polygons, we show that there exists a three-dimensional time periodic traveling front by taking the limit of the solutions corresponding to the convex polyhedrons as the number of the lateral surfaces goes to infinity.

International audience; We study the existence of traveling wave solutions and spreading properties for single-layer delayed neural field equations. We focus on the case where the kinetic dynamics are of monos-table type and characterize the invasion speeds as a function of the asymptotic decay of the connectivity kernel. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed, which further can be explicitly characterized under a KPP type condition. We also investigate the case of algebraically decaying kernels where we prove the non-existence of traveling wave solutions and show the level sets of the solutions eventually locate in between two exponential functions of time. The uniqueness of traveling waves modulo translation is also obtained.