具有反常扩散的反应扩散方程的传播加速
Propagation acceleration in reaction diffusion equations with anomalous diffusions
摘要：
本文研究了非局部方程$$\begin{cases}u_t+(-Δ)^su=f(u),\quad t>0, \ x\in\mathbb{R}\\ u(0,x)=u_0(x),\quad x\in\mathbb{R}\end{cases}$$的解的性质，其中$0\le u_0<1$是Heaviside型函数，$Δ^s$是具有$s\in (0,1)$的分数拉普拉斯算子，$f\in C([0,1],\mathbb{R}^+)$是非负非线性项，使得$f(0)=f(1)=0$和$f'(1)<0$ .在这种情况下，它是已知的解决方案PDMATHAMT局部一致收敛到1，我们的目的是了解如何快速发生这种入侵过程。当$f$是Fisher-KPP型非线性且$s \in (0,1)$时，已知解$u(t,x)$的水平集以指数速度移动，而当$f$是点火型且$s\in \left(\frac{1}{2},1\right)$时，则解的水平集以恒定速度移动。在这篇文章中，对于一般的单稳态非线性$f$和任何$s\in (0,1)$，我们得到通用的估计的解决方案$u(t,x)$的水平集的位置，然后使我们能够更精确地描述这个入侵过程的行为。特别是，我们得到了一个代数通用的上限上的“速度”的水平集突出的微妙的相互作用的$s$和$f$的存在下，一个指数加速过程。当$s\in\left (0,\frac{1}{2}\right]$和$f$是点火型时，我们也完成了已知的描述的$u$的行为，并给出了一个精确的渐近速度的水平集在这种情况下。值得注意的是，我们证明了水平集加速时，$s\in\left(0,\frac{1}{2}\right)$和在临界情况下$s=\frac{1}{2}$，虽然没有旅行前可以存在，水平集仍然以恒定的速度渐近移动。这些新的结果是在鲜明的对比，没有这样的加速可能发生的情况下，因此突出了两种类型的非线性之间的定性差异。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
