论文标题
用强迫速度扩散和消失，以实现单一的反应扩散方程
Spreading and vanishing for a monostable reaction-diffusion equation with forced speed
论文作者
论文摘要
异质反应扩散方程的入侵现象在应用数学中是当代且具有挑战性的问题。在本文中，当反应术语为正的子域是以给定的速度$ c $转移/收缩时，我们对反应扩散方程的扩散问题感兴趣。这个问题尤其是在气候变化对人口动态的影响的建模中。通过将自己放置在适当的移动框架中，这使我们考虑了一个在空间反应项中具有异质性的反应扩散 - 添加方程，尺寸为$ n \ geq1 $。我们根据对流常数〜$ c $的价值研究解决方案$ u $的行为，这通常代表气候变化的速度。我们发现，当最初的基准被紧凑地支持时，恰好存在$ c $的三个范围，从而导致截然不同的情况。在较低的速度范围内，解决方案总是会扩散，而在上层范围内它总是消失。更令人惊讶的是，我们发现扩散和消失的速度可能发生在中间速度范围内。这两个结果之间的阈值总是很尖锐的，无论是$ c $还是初始条件。我们还简要考虑了指数下降的初始条件的情况，在这种情况下，我们将初始条件的降低速率与〜$ c $的值范围相关联，从而发生扩散。
Invasion phenomena for heterogeneous reaction-diffusion equations are contemporary and challenging questions in applied mathematics. In this paper we are interested in the question of spreading for a reaction-diffusion equation when the subdomain where the reaction term is positive is shifting/contracting at a given speed $c$. This problem arises in particular in the modelling of the impact of climate change on population dynamics. By placing ourselves in the appropriate moving frame, this leads us to consider a reaction-diffusion-advection equation with a heterogeneous in space reaction term, in dimension $N\geq1$. We investigate the behaviour of the solution $u$ depending on the value of the advection constant~$c$, which typically stands for the velocity of climate change. We find that, when the initial datum is compactly supported, there exists precisely three ranges for $c$ leading to drastically different situations. In the lower speed range the solution always spreads, while in the upper range it always vanishes. More surprisingly, we find that that both spreading and vanishing may occur in an intermediate speed range. The threshold between those two outcomes is always sharp, both with respect to $c$ and to the initial condition. We also briefly consider the case of an exponentially decreasing initial condition, where we relate the decreasing rate of the initial condition with the range of values of~$c$ such that spreading occurs.
