Global relaxation of bistable solutions for gradient systems in one unbounded spatial dimension

This paper is concerned with spatially extended gradient systems of the form $$u_t=-\nabla V (u) + \mathcal{D} u_{xx}\,,$$ where spatial domain is the whole real line, state-parameter $u$ is multidimensional, $\mathcal{D}$ denotes a fixed diffusion matrix, and the potential $V$ is coercive at infinity. "Bistable" solutions, that is solutions close at both ends of space to stable homogeneous equilibria, are considered. For a solution of this kind, it is proved that, if the homogeneous equilibria approached at both ends belong to the same level set of the potential and if an appropriate (localized in space) energy remains bounded from below when time increases, then the solution approaches, when time approaches infinity, a pattern of stationary solutions homoclinic or heteroclinic to homogeneous equilibria. This result provides a step towards a complete description of the global behaviour of all bistable solutions that is pursued in a companion paper. Some consequences are derived, and applications to some examples are given.Comment: 69 pages, 15 figure

Global behaviour of radially symmetric solutions stable at infinity for gradient systems

This paper is concerned with radially symmetric solutions of systems of the form $$u_t = -\nabla V(u) + \Delta_x u$$ where space variable $x$ and and state-parameter $u$ are multidimensional, and the potential $V$ is coercive at infinity. For such systems, under generic assumptions on the potential, the asymptotic behaviour of solutions "stable at infinity", that is approaching a spatially homogeneous equilibrium when $|x|$ approaches $+\infty$, is investigated. It is proved that every such solutions approaches a stacked family of radially symmetric bistable fronts travelling to infinity. This behaviour is similar to the one of bistable solutions for gradient systems in one unbounded spatial dimension, described in a companion paper. It is expected (but unfortunately not proved at this stage) that behind these travelling fronts the solution again behaves as in the one-dimensional case (that is, the time derivative approaches zero and the solution approaches a pattern of stationary solutions).Comment: 52 pages, 14 figures. arXiv admin note: substantial text overlap with arXiv:1703.01221. text overlap with arXiv:1604.0200

Competition between stable equilibria in reaction-diffusion systems: the influence of mobility on dominance

This paper is concerned with reaction-diffusion systems of two symmetric species in spatial dimension one, having two stable symmetric equilibria connected by a symmetric standing front. The first order variation of the speed of this front when the symmetry is broken through a small perturbation of the diffusion coefficients is computed. This elementary computation relates to the question, arising from population dynamics, of the influence of mobility on dominance, in reaction-diffusion systems modelling the interaction of two competing species. It is applied to two examples. First a toy example, where it is shown that, depending on the value of a parameter, an increase of the mobility of one of the species may be either advantageous or disadvantageous for this species. Then the Lotka-Volterra competition model, in the bistable regime close to the onset of bistability, where it is shown that an increase of mobility is advantageous. Geometric interpretations of these results are given.Comment: 43 pages, 10 figure

Retracting fronts for the nonlinear complex heat equation

The "nonlinear complex heat equation" $A_t=i|A|^2A+A_{xx}$ was introduced by P. Coullet and L. Kramer as a model equation exhibiting travelling fronts induced by non-variational effects, called "retracting fronts". In this paper we study the existence of such fronts. They go by one-parameter families, bounded at one end by the slowest and "steepest" front among the family, a situation presenting striking analogies with front propagation into unstable states.Comment: 21 pages, 6 figure

A variational proof of global stability for bistable travelling waves

We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully applied to more general parabolic or hyperbolic gradient-like systems.Comment: 21 pages, 4 figure

Siegel Disks and Periodic Rays of Entire Functions

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk such that f restricts to a homeomorphism of the boundary. We show that the Siegel disk is bounded. Using a result of Herman, we deduce that if additionally the rotation number of the Siegel disk is Diophantine, then its boundary contains a critical point of f. Suppose furthermore that all singular values of f lie in the Julia set. We prove that, if f has a Siegel disk $U$ whose boundary contains no singular values, then the condition that f is a homeomorphism of the boundary of U is automatically satisfied. We also investigate landing properties of periodic dynamic rays by similar methods.Comment: 22 pages, 4 figures. A problem with the image quality of some of the figures was fixed. Some minor corrections were also made. Final versio