非周期缓变介质中反应扩散方程的最大和最小传播速度
Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media
摘要：
本文研究了非均匀介质中Fisher-KPP方程（$$\partial_t u = \partial_{xx} u + f(x,u),$$）解的渐近性态。我们考虑的典型非线性是$f(x,u) = μ_0 (ϕ(x)) u(1-u)$，其中$μ_0$是1周期函数，$ϕ$是满足$\lim_{x\to +\infty} ϕ(x) = +\infty$和$\lim_{x\to +\infty} ϕ' (x) = 0$的$\mathcal{C}^1$增函数。虽然非常具体，但选择这样的反应术语是出于其高度异质性的动机。我们表现出两种不同的行为$u$的大时代，根据$ϕ$在无穷大的收敛速度。如果$ϕ$增长足够慢，那么我们证明了$u$的扩展速度在两个不同的值之间振荡。如果$ϕ$增长迅速，那么我们明确计算出一个独特的和良好确定的传播速度$w_\infty$，所产生的限制问题的一个无限的时期。我们对这两种行为给出了启发式的解释。
英文摘要：
This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ associated with a compactly supported initial datum. A typical nonlinearity we consider is $f(x,u) = μ_0 (ϕ(x)) u(1-u)$, where $μ_0$ is a 1-periodic function and $ϕ$ is a $\mathcal{C}^1$ increasing function that satisfies $\lim_{x\to +\infty} ϕ(x) = +\infty$ and $\lim_{x\to +\infty} ϕ' (x) = 0$. Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for $u$ for large times, depending on the speed of the convergence of $ϕ$ at infinity. If $ϕ$ grows sufficiently slowly, then we prove that the spreading speed of $u$ oscillates between two distinct values. If $ϕ$ grows rapidly, then we compute explicitly a unique and well determined speed of propagation $w_\infty$, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.
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