115 research outputs found
Symmetry of Traveling Wave Solutions to the Allen-Cahn Equation in \Er^2
In this paper, we prove even symmetry of monotone traveling wave solutions to
the balanced Allen-Cahn equation in the entire plane. Related results for the
unbalanced Allen-Cahn equation are also discussed
Propagating speeds of bistable transition fronts in spatially periodic media
This paper is concerned with the propagating speeds of transition fronts in
for spatially periodic bistable reaction-diffusion equations. The notion
of transition fronts generalizes the standard notions of traveling fronts.
Under the a priori assumption that there exist pulsating fronts for every
direction with nonzero speeds, we show some continuity and
differentiability properties of the front speeds and profiles with respect to
the direction . Finally, we prove that the propagating speed of any
transition front is larger than the infimum of speeds of pulsating fronts and
less than the supremum of speeds of pulsating fronts.Comment: Some modifications are made. Some notions are cited from
arXiv:1302.4817. Some auxiliary lemmas are cited from arXiv:1408.072
On the mean speed of bistable transition fronts in unbounded domains
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
Bistable transition fronts in R^N
This paper is chiefly concerned with qualitative properties of some reaction-diffusion fronts. The recently defined notions of transition fronts generalize the standard notions of traveling fronts. In this paper, we show the existence and the uniqueness of the global mean speed of bistable transition fronts in R^N. This speed is proved to be independent of the shape of the level sets of the fronts. The planar fronts are also characterized in the more general class of almost-planar fronts with any number of transition layers. These qualitative properties show the robustness of the notions of transition fronts. But we also prove the existence of new types of transition fronts in R^N that are not standard traveling fronts, thus showing that the notions of transition fronts are broad enough to include other relevant propagating solutions
Traveling front of polyhedral shape for a nonlocal delayed diffusion equation
This paper is concerned with the existence and stability of traveling fronts with convex polyhedral shape for nonlocal delay diffusion equations. By using the existence and stability results of V-form fronts and pyramidal traveling fronts, we first show that there exists a traveling front with polyhedral shape of nonlocal delay diffusion equation associated with . Moreover, the asymptotic stability and other qualitative properties of such traveling front are also established
Monotonicity of bistable transition fronts in R^N
International audienceThis paper is concerned with the monotonicity of transition fronts for bistable reaction-diffusion equations. Transition fronts generalize the standard notions of traveling fronts. Known examples of standard traveling fronts are the planar fronts and the fronts with conical-shaped or pyramidal level sets which are invariant in a moving frame. Other more general non-standard transition fronts with more complex level sets were constructed recently. In this paper, we prove the time monotonicity of all bistable transition fronts with non-zero global mean speed, whatever shape their level sets may have
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