论文标题
反应扩散方程与异常扩散的繁殖加速度
Propagation acceleration in reaction diffusion equations with anomalous diffusions
论文作者
论文摘要
在本文中，我们对非局部方程的解决方案的属性感兴趣， $ 0 \ le u_0 <1 $是Heaviside类型功能，$Δ^S $代表分数laplacian，$ s \ in（0,1）$（0,1）$，$ f \ in C（[0,1]，\ Mathbb {r}^+）$是非负的非线性非线性的，是$ f（0）$ f（0）$ f（0）$ f（0）$ f（0）= 0 $ f（0）= 0 $ f（1）= 0 $ f（1）= 0 $ f（1）= 1）在这种情况下，众所周知，解决方案$ u（t，s）$在本地统一收敛至1，我们的目的是了解这种入侵过程的发生速度。 When $f$ is a Fisher-KPP type nonlinearity and $s \in (0,1)$, it is known that the level set of the solution $u(t,x)$ moves at an exponential speed whereas when $f$ is of ignition type and $s\in \left(\frac{1}{2},1\right)$ then the level set of the solution moves at a constant speed.在本文中，对于一般单稳定的非线性$ f $和（0,1）$中的任何$ s \ in（0,1）$，我们对解决方案$ u（t，x）$的级别集合的位置进行了通用估计，然后使我们能够更准确地描述此入侵过程的行为。特别是，我们在水平设置的“速度”上获得了代数通用上限，该速度突出了$ s $和$ f $的精致相互作用。当$ s \ in \ left（0，\ frac {1} {2} {2} \ right] $和$ f $是点火类型时，我们还完成了对$ u $行为的已知描述$ s \ in \ left（0，\ frac {1} {2} \ right）$，在关键情况下，$ s = \ frac {1} {2} $，尽管存在不行驶的前线，但级别集仍然在稳定的情况下，质量不变的情况下，级别仍然不变。非线性。
In this paper, we are interested in the properties of solution of the nonlocal equation $$\begin{cases}u_t+(-Δ)^su=f(u),\quad t>0, \ x\in\mathbb{R}\\ u(0,x)=u_0(x),\quad x\in\mathbb{R}\end{cases}$$ where $0\le u_0<1$ is a Heaviside type function, $Δ^s$ stands for the fractional Laplacian with $s\in (0,1)$, and $f\in C([0,1],\mathbb{R}^+)$ is a non negative nonlinearity such that $f(0)=f(1)=0$ and $f'(1)<0$. In this context, it is known that the solution $u(t,s)$ converges locally uniformly to 1 and our aim here is to understand how fast this invasion process occur. When $f$ is a Fisher-KPP type nonlinearity and $s \in (0,1)$, it is known that the level set of the solution $u(t,x)$ moves at an exponential speed whereas when $f$ is of ignition type and $s\in \left(\frac{1}{2},1\right)$ then the level set of the solution moves at a constant speed. In this article, for general monostable nonlinearities $f$ and any $s\in (0,1)$ we derive generic estimates on the position of the level sets of the solution $u(t,x)$ which then enable us to describe more precisely the behaviour of this invasion process. In particular, we obtain a algebraic generic upper bound on the "speed" of level set highlighting the delicate interplay of $s$ and $f$ in the existence of an exponential acceleration process. When $s\in\left (0,\frac{1}{2}\right]$ and $f$ is of ignition type, we also complete the known description of the behaviour of $u$ and give a precise asymptotic of the speed of the level set in this context. Notably, we prove that the level sets accelerate when $s\in\left(0,\frac{1}{2}\right)$ and that in the critical case $s=\frac{1}{2}$ although no travelling front can exist, the level sets still move asymptotically at a constant speed. These new results are in sharp contrast with the bistable situation where no such acceleration may occur, highlighting therefore the qualitative difference between the two type of nonlinearities.
