Logarithmic Bramson correction for multi-dimensional periodic Fisher-KPP equations

Abstract

We study the long time behavior of solutions of periodic Fisher-KPP type equations in ${\mathbb{R}}^n$ that arise from compactly supported initial data. We prove that propagation along a fixed direction $e\in {\mathbb{S}}^{n-1}$ is completely determined by an associated minimizing direction $e^{\prime }$ which is unique and in a bijective correspondence with $e$. This correspondence allows us to determine the correct Bramson shift in higher dimensions and show that that the propagation along each $e$ occurs asymptotically at the speed $w_{\ast }(e)t-\frac{n/2+1}{{\lambda }_{\ast }(e^{\prime })e\cdot e^{\prime }}\log t+O(1)$.

44 pages, 4 figures

References in corpus (1)

• Exact solution and precise asymptotics of a Fisher-KPP type front