KPP体-面系统在一般圆柱域内的传播
Propagation for KPP bulk-surface systems in a general cylindrical domain
摘要：
本文研究了具有一般截面和非均匀系数的圆柱形区域中KPP体-面系统的传播现象。对于标量KPP方程，我们证明了解的渐近传播速度可以用一族自伴椭圆算子的主特征值来计算。使用这种特性，我们分析的依赖性的传播速度的各种参数，包括扩散率和区域的部分的大小和形状。特别是，我们提供了新的理论结果，几个渐近制度，如小，高扩散率和小，大尺寸的部分。这些结果推广了早期的径向均匀情况下可用的。最后，我们数值研究的问题的形状优化的蔓延速度。通过计算它的形状导数，我们观察到，在齐次系数的情况下，一个磁盘要么最大化或最小化的速度，这取决于参数的问题，无论有或没有约束。我们还显示了非齐次系数的数值形状优化的结果，当磁盘不再是一个优化。
英文摘要：
In this paper, we investigate propagation phenomena for KPP bulk-surface systems in a cylindrical domain with general section and heterogeneous coefficients. As for the scalar KPP equation, we show that the asymptotic spreading speed of solutions can be computed in terms of the principal eigenvalues of a family of self-adjoint elliptic operators. Using this characterization, we analyze the dependence of the spreading speed on various parameters, including diffusion rates and the size and shape of the section of the domain. In particular, we provide new theoretical results on several asymptotic regimes like small and high diffusion rates and sections with small and large sizes. These results generalize earlier ones which were available in the radial homogeneous case. Finally, we numerically investigate the issue of shape optimization of the spreading speed. By computing its shape derivative, we observe, in the case of homogeneous coefficients, that a disk either maximizes or minimizes the speed, depending on the parameters of the problem, both with or without constraints. We also show the results of numerical shape optimization with non homogeneous coefficients, when the disk is no longer an optimizer.
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