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.
The Bramson correction for Fisher--KPP equations with nonlocal diffusion
我们建立了具有非局部扩散的Fisher-KPP方程解的位置的对数Bramson修正。对于显式常数$c_{*}$和$λ_{*}$，具有阶梯状初始数据的解通常类似于位置$c_{*} t - \frac{3}{2 λ_{*}} \log t + \mathcal{O}(1)$处的前沿。然而，某些奇异扩散表现出更奇异的行为。
We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice $\mathbb{Z}$. The level sets of solutions with step-like initial conditions are located at position $c_*t-\frac{3}{2λ_*}\ln t+\mathcal{O}(1)$ as $t\rightarrow+\infty$ for some explicit positive constants $c_*$ and $λ_*$. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed $c_*…
Refined long time asymptotics for Fisher-KPP fronts
研究了一维Fisher-KPP方程，初始条件$u_0(x)$与阶梯函数一致，但在紧集上不一致. M. Bramson指出，作为$t\to+\infty$，解收敛到位于位置$X(t)=2t-(3/2)\log t+x_0+o(1)$的行波，其位移$x_0$取决于$u_0$。联合Ebert和W. Van Saarloos已经正式推导出了对Bramson位移的修正，认为$X(t)=2t-(3/2)\log t+x_0-3\sqrtπ/\sqrt{t}+O(1/t)$。在这里，我们证明了这一结果确实持有，与大小$O(1/t^{1-γ})$的误差项，任何$γ>0$。这个渐近性的有趣之处在于，$1/\sqrt{t}$项前面的系数不依赖于$u_0$。
Logarithmic Bramson correction for Fisher--KPP equations on homogeneous trees
Universidad del Norte（北大学）；Facultad de Ciencias, Universidad de Cádiz（加的斯大学理学院）；Instituto de Matemáticas, Universidad de Talca（塔尔卡大学数学研究所）；Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins（奥希金斯大学工程科学研究所）
We show that long time solution dynamic for general reaction-advection-diffusion equations with KPP reactions is virtually linear in the following sense. Its leading order depends on the non-linear reaction only through its linearization at $u=0$, and it can also be recovered for general initial data by instead solving the PDE for restrictions of the initial condition to unit cubes on $\bbR^d$ (the latter means that non-linear interaction of these restricted solutions has only lower order effects on the overall solution dynamic). The result holds under a uniform bound on the advection coeffici…
We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are at distance $n+x$ from a position close to the tip of ${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$ as $n\to+\infty$. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a $1$-stable random variable. Our approach uses analytic techniques and is mot…
We prove stochastic homogenization for reaction-advection-diffusion equations with random space-time-dependent KPP reactions with temporal correlations that are decaying in an appropriate sense. We show that the limiting homogenized dynamic has the simple form of spreading with some deterministic direction-dependent speeds from the support of the initial datum. We obtain analogous results for G-equations with random flame speeds and incompressible background advections. Important ingredients in our proofs are a non-autonomous subadditive theorem and the principle of virtual linearity for KPP r…
We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength $β\in\mathbb{R}$. This equation exhibits a transition from pulled to pushed front behavior at $β_c=2$. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position $m_β(t)$ and study the asymptotics of the front location $m_β(t)$. When $β< 2$, it has the same form as for the standard Fisher-KPP equation established by Bramson \cite{Bramson1,Bramson2}: $m_β(t) = 2t - (3/2)\log(t) + x_\infty + o(1)$ as $t\to+\infty$. This form is typical of pulled…
The solution h to the Fisher-KPP equation with a steep enough initial condition develops into a front moving at velocity 2, with logarithmic corrections to its position. In this paper we investigate the value h(c t, t) of the solution ahead of the front, at time t and position c t, with c > 2. That value goes to zero exponentially fast with time, with a well-known rate, but the prefactor depends in a non-trivial way of c, the initial condition and the non-linearity in the equation. We compute an asymptotic expansion of that prefactor for velocities c close to 2. The expansion is surprisingly e…
