论文标题
带有对数的非线性的Fisher-KPP方程中的Bramson延迟
The Bramson delay in a Fisher-KPP equation with log-singular non-linearity
论文作者
论文摘要
我们考虑了Fisher-KPP类型的一类反应扩散方程，其中非线性（反应术语）$ f $仅是$ c^1 $ at $ u = 0 $，这是由于对数竞争术语。我们首先得出了（最小速度）波动波解决方案的渐近行为，即在无穷大时，我们获得了衰减至零的零元素的精确估计。然后，我们使用它来表征Bramson在局部初始数据和库奇问题的解决方案之间的bramson偏移。我们发现相位过渡，具体取决于单数$ f $接近$ u = 0 $，其行为完全不同。这与平滑的情况相反，即，当$ f \ in c^{1，δ} $中时，这些行为完全由$ f'（0）$确定。在奇异的情况下，出现了几个量表，需要新的技术才能理解。
We consider a class of reaction-diffusion equations of Fisher-KPP type in which the nonlinearity (reaction term) $f$ is merely $C^1$ at $u=0$ due to a logarithmic competition term. We first derive the asymptotic behavior of (minimal speed) traveling wave solutions that is, we obtain precise estimates on the decay to zero of the traveling wave profile at infinity. We then use this to characterize the Bramson shift between the traveling wave solutions and solutions of the Cauchy problem with localized initial data. We find a phase transition depending on how singular $f$ is near $u=0$ with quite different behavior for more singular $f$. This is in contrast to the smooth case, that is, when $f \in C^{1,δ}$, where these behaviors are completely determined by $f'(0)$. In the singular case, several scales appear and require new techniques to understand.
