arXiv 1806.020222018-06-06math.AP

Fisher-KPP型多孔介质方程的对数修正

Logarithmic corrections in Fisher-KPP type Porous Medium Equations

Yihong Du, Fernando Quiros, Maolin Zhou

摘要：

本文考虑了具有Fisher-KPP型反应项和非负紧支集初始函数的多孔介质方程解的大时间行为，并给出了$L^\infty(\mathbb{R}^N)\setminus\{0\}$：\begin{equation} \label{eq:abstract} \tag{$\star$}u_t=Δu^m+u-u^2\quad\text{in }Q:=\mathbb{R}^N\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0\quad\text{in }\mathbb{R}^N, \end{equation} with $m>1$ .众所周知，这个问题的解决方案$u(\cdot, t)$的空间支持仍然是有界的所有时间$t>0$。在空间维度一，它是已知的，有一个最小的速度$c_*>0$的方程承认波前解决方案$Φ_{c_*}$与有限的前面，它吸引的解决方案与初始功能表现得像一个Heaviside功能。在一维情况下，我们可以得到一个类似的稳定性结果的情况下，紧支持的初始数据。在更高的维度，我们表明，$Φ_{c_*}$仍然是有吸引力的，虽然发生了对数移位。更准确地说，如果另外假设\eqref{eq:abstract}中的初始函数是径向对称的，则存在独立于维度$N$和初始函数$u_0$的第二常数$c^*>0$，使得对于某些$r_0\in\mathbb{R}$（取决于$u_0$），\[ \lim_{t\to\infty}\left\{\sup_{x\in\mathbb R^N}\big|u(x,t)-Φ_{c_*}(|x|-c_*t+(N-1)c^*\log t-r_0)\big|\right\}=0 \]。如果初始函数不是径向对称的，则存在$r_1, r_2\in \mathbb{R}$，使得对于所有$t\ge1$，解$u(\cdot, t)$的空间支撑的边界包含在球壳$\{x\in\mathbb R^N: r_1\leq |x|-c_* t+(N-1)c^* \log t\leq r_2\}$中。此外，作为$t\to\infty$，对于任何$c>c^*$，$u(x,t)$在$\big\{|x|\leq c_*t-(N-1)c\log t\big\}$中一致地收敛到$1$。

英文摘要：

We consider the large time behaviour of solutions to the porous medium equation with a Fisher-KPP type reaction term and nonnegative, compactly supported initial function in $L^\infty(\mathbb{R}^N)\setminus\{0\}$: \begin{equation} \label{eq:abstract} \tag{$\star$}u_t=Δu^m+u-u^2\quad\text{in }Q:=\mathbb{R}^N\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0\quad\text{in }\mathbb{R}^N, \end{equation} with $m>1$. It is well known that the spatial support of the solution $u(\cdot, t)$ to this problem remains bounded for all time $t>0$. In spatial dimension one it is known that there is a minimal speed $c_*>0$ for which the equation admits a wavefront solution $Φ_{c_*}$ with a finite front, and it attract solutions with initial functions behaving like a Heaviside function. In dimension one we can obtain an analogous stability result for the case of compactly supported initial data. In higher dimensions we show that $Φ_{c_*}$ is still attractive, albeit that a logarithmic shifting occurs. More precisely, if the initial function in \eqref{eq:abstract} is additionally assumed to be radially symmetric, then there exists a second constant $c^*>0$ independent of the dimension $N$ and the initial function $u_0$, such that \[ \lim_{t\to\infty}\left\{\sup_{x\in\mathbb R^N}\big|u(x,t)-Φ_{c_*}(|x|-c_*t+(N-1)c^*\log t-r_0)\big|\right\}=0 \] for some $r_0\in\mathbb{R}$ (depending on $u_0$). If the initial function is not radially symmetric, then there exist $r_1, r_2\in \mathbb{R}$ such that the boundary of the spatial support of the solution $u(\cdot, t)$ is contained in the spherical shell $\{x\in\mathbb R^N: r_1\leq |x|-c_* t+(N-1)c^* \log t\leq r_2\}$ for all $t\ge1$. Moreover, as $t\to\infty$, $u(x,t)$ converges to $1$ uniformly in $\big\{|x|\leq c_*t-(N-1)c\log t\big\}$ for any $c>c^*$.

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。

摘要：

英文摘要：

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