论文标题
在空间周期性的双态反应扩散方程中的可允许速度
Admissible speeds in spatially periodic bistable reaction-diffusion equations
论文作者
论文摘要
空间周期性反应扩散方程通常允许脉动波，这些脉动波描述了从一个稳态到另一个稳定状态的过渡。由于异质性，通常情况下，这种方程并不是旋转不变的，因此脉动波的速度可能先验取决于其方向。但是，文献中实际上几乎没有知道它是否真正做到：令人惊讶的是，在一维单稳定的Fisher-KPP案例中，它甚至在相反的方向上的速度在相反的方向上是相同的。在这里，我们在双态案例中调查了这个问题，并表明一组可接受的速度实际上很大，这意味着传播的形状确实可能是不对称的。更确切地说，我们在任何空间维度上都可以显示一个人可以选择一个任意的大量方向，并找到一个空间周期性的可周期性双型方程，以在这些方向上实现任何速度组合，只要这些速度具有相同的符号。特别是，在空间维度1和与Fisher-KPP情况不同的情况下，任何对（非负或非阳性）的任何对右和向左波速都是可允许的。我们还表明，在可脉动波的速度方面的这些变化导致多稳态方程中强烈不对称的情况。
Spatially periodic reaction-diffusion equations typically admit pulsating waves which describe the transition from one steady state to another. Due to the heterogeneity, in general such an equation is not invariant by rotation and therefore the speed of the pulsating wave may a priori depend on its direction. However, little is actually known in the literature about whether it truly does: surprisingly, it is even known in the one-dimensional monostable Fisher-KPP case that the speed is the same in the opposite directions despite the lack of symmetry. Here we investigate this issue in the bistable case and show that the set of admissible speeds is actually rather large, which means that the shape of propagation may indeed be asymmetrical. More precisely, we show in any spatial dimension that one can choose an arbitrary large number of directions , and find a spatially periodic bistable type equation to achieve any combination of speeds in those directions, provided those speeds have the same sign. In particular, in spatial dimension 1 and unlike the Fisher-KPP case, any pair of (either nonnegative or nonpositive) rightward and leftward wave speeds is admissible. We also show that these variations in the speeds of bistable pulsating waves lead to strongly asymmetrical situations in the multistable equations.
