Kreten Florian
Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, Bonn, 53115, Germany.
J Math Biol. 2025 Feb 17;90(3):33. doi: 10.1007/s00285-025-02189-x.
We construct the traveling wave solutions of an FKPP growth process of two densities of particles, and prove that the critical traveling waves are locally stable in a space where the perturbations can grow exponentially at the back of the wave. The considered reaction-diffusion system was introduced by Hannezo et al. (Cell 171(1):242-255, 2017) in the context of branching morphogenesis: active, branching particles accumulate inactive particles, which do not react. Thus, the system features a continuum of steady state solutions, complicating the analysis. We adopt a result by Faye and Holzer (J Differ Equ 269(9):6559-6601, 2020) for proving the stability of the critical traveling waves, and modify the semi-group estimates to spaces with unbounded weights. We use a Feynman-Kac formula to get an exponential a priori estimate for the tail of the PDE, a novel and simple approach.
我们构建了两种粒子密度的FKPP增长过程的行波解，并证明了临界行波在一个扰动可以在波后呈指数增长的空间中是局部稳定的。所考虑的反应扩散系统是由Hannezo等人（《细胞》171(1):242 - 255, 2017）在分支形态发生的背景下引入的：活跃的分支粒子积累不发生反应的非活跃粒子。因此，该系统具有连续的稳态解，这使得分析变得复杂。我们采用Faye和Holzer（《微分方程杂志》269(9):6559 - 6601, 2020）的一个结果来证明临界行波的稳定性，并将半群估计修改到具有无界权重的空间。我们使用费曼 - 卡茨公式来得到偏微分方程尾部的指数先验估计，这是一种新颖且简单的方法。
