We develop techniques to capture the effect of transport on the long-term dynamics of small, localized initial data in nonlinearly coupled reaction-diffusion-advection equations on the real line. It is well-known that quadratic or cubic nonlinearities in such systems can lead to growth of small, localized initial data and even finite time blow-up. We show that, if the components exhibit different velocities, then quadratic or cubic mix-terms, i.e. terms with nontrivial contributions from both components, are harmless. We establish global existence and diffusive Gaussian-like decay for exponentially and algebraically localized initial conditions allowing for quadratic and cubic mix-terms. Our proof relies on a nonlinear iteration scheme that employs pointwise estimates. The situation becomes very delicate if other quadratic or cubic terms are present in the system. We provide an example where a quadratic mix-term and a Burgers'-type coupling can compensate for a cubic term due to differences in velocities.

中文翻译：

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不同速度下非线性耦合反应-扩散-平流方程的全局存在和衰减

我们开发了技术来捕捉传输对实线上非线性耦合反应-扩散-平流方程中小的局部初始数据的长期动力学的影响。众所周知，此类系统中的二次或三次非线性会导致小的、局部初始数据的增长，甚至是有限时间的爆炸。我们表明，如果分量表现出不同的速度，那么二次或三次混合项，即来自两个分量的非平凡贡献的项，是无害的。我们为允许二次和三次混合项的指数和代数局部初始条件建立全局存在和扩散高斯式衰减。我们的证明依赖于采用逐点估计的非线性迭代方案。如果系统中存在其他二次或三次项，情况就会变得非常微妙。我们提供了一个示例，其中二次混合项和 Burgers 型耦合可以补偿由于速度差异而导致的三次项。