标题： 空间异质反应扩散系统中的行进前沿解 显示英文标题

标题： Travelling front solutions in a spatially heterogeneous reaction-diffusion system

摘要： 我们研究了一个具有慢-快结构和空间变化系数$ f_1 $和$ f_2 $的两组分反应-扩散系统，这些系数出现在慢方程中。在$ f_1 $和$ f_2 $上的温和有界性和正则性条件下，该系统被证明以两种稳定的非均匀背景状态的形式表现出双稳态。这些背景状态可以通过静态和行进前沿解连接。行进前沿特征是一个界面，它以非均匀速度通过它所连接的静止空间变化背景状态移动。因此，与经典行进波不同，这些前沿在任何共动框架中都不是静止的。我们使用费尼切尔理论到非紧情况的扩展来构建背景状态和静态前沿。此外，我们建立了行进前沿解的存在性，并通过非自治空间动力学方法推导了移动界面动态位置的一阶表达式。这个表达式的形式是一个延迟微分方程，其准确性通过数值模拟得到了验证。我们工作的关键贡献在于对$ f_1 $和$ f_2 $的一般处理，这些量既不是（必然）渐近小的，也不限于周期性或局部结构等特定形式。此外，我们对前沿位置公式的推导避免了传统上对谱分析的依赖，使我们能够描述超出从静态前沿分岔之外的前沿动力学。这种方法有可能扩展到其他设置，在这些设置中，初始时刻的谱特性排除了传统的约简技术。 显示英文摘要

摘要： We investigate a two-component reaction-diffusion system with a slow-fast structure and spatially varying coefficients $ f_1 $ and $ f_2 $ appearing in the slow equation. Under mild boundedness and regularity conditions on $ f_1 $ and $ f_2 $ the system is shown to exhibit bi-stability in the form of two stable stationary heterogeneous background states. These background states can be connected by stationary and travelling front solutions. Travelling fronts feature an interface that moves with a non-uniform speed through the motionless spatially varying background states it connects. As a result, unlike classical travelling waves, these fronts are not stationary in any co-moving frame. We construct both the background states and stationary fronts using an extension of Fenichel theory to the non-compact case. Additionally, we establish the existence of travelling front solutions and derive a leading-order expression for the dynamic position of the moving interface through a non-autonomous spatial dynamics approach. This expression takes the form of a delay-differential equation, and its accuracy is validated through numerical simulations. A key contribution of our work lies in the general treatment of $ f_1 $ and $ f_2 $, which are neither (necessarily) asymptotically small nor restricted to specific forms such as periodic or localised structures. Furthermore, our derivation of the front position formula circumvents the traditional reliance on spectral analysis, enabling us to describe front dynamics beyond bifurcations from stationary fronts. This approach has the potential to be extended to other settings in which spectral properties at onset preclude conventional reduction techniques.