[Paper Review] Logarithmic corrections in Fisher-KPP type Porous Medium Equations

Yihong Du, Fernando Quirós+1 more|arXiv (Cornell University)|Jun 6, 2018

Mathematical and Theoretical Epidemiology and Ecology Models29 references3 citations

TL;DR

This paper analyzes the large-time behavior of solutions to a porous medium equation with a Fisher-KPP type reaction term ($u_t = \Delta u^m + u - u^2$, $m > 1$). It establishes that in higher dimensions ($N \geq 2$), the solution's free boundary exhibits a logarithmic correction in its propagation speed, converging asymptotically to a traveling wave profile shifted by $(N-1)c^*\log t$, where $c^*$ is a universal constant independent of $N$ and initial data. This correction arises due to the degeneracy of the porous medium equation and is absent in the one-dimensional case.

ABSTRACT

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} ag{$\star$}u_t=Δu^m+u-u^2\quad ext{in }Q:=\mathbb{R}^N imes\mathbb{R}_+,\qquad u(\cdot,0)=u_0\quad ext{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 \[ \lim_{t o\infty}\left\{\sup_{x\in\mathbb R^N}\big|u(x,t)-Φ_{c_*}(|x|-c_*t+(N-1)c^*\log t-r_0)\big| ight\}=0 \] 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 $t o\infty$, $u(x,t)$ converges to $1$ uniformly in $\big\{|x|\leq c_*t-(N-1)c\log t\big\}$ for any $c>c^*$.

Motivation & Objective

• To characterize the precise large-time behavior of solutions to the porous medium equation with a Fisher-KPP reaction term in higher spatial dimensions.
• To determine how the free boundary of compactly supported solutions evolves over time when $ m > 1 $.
• To establish the existence of a universal logarithmic correction term in the propagation speed of the solution's support in $ \mathbb{R}^N $, $ N \geq 2 $.
• To extend the asymptotic stability of traveling wave solutions beyond the one-dimensional case, accounting for dimension-dependent logarithmic shifts.

Proposed method

• The authors use a comparison principle with carefully constructed subsolutions and supersolutions based on the traveling wave profile $ \Phi_{c_*} $, which connects the equilibria 1 and 0.
• They introduce a radial transformation to reduce the problem to a one-dimensional setting, enabling the use of known results on traveling wave solutions in one dimension.
• The key technique involves constructing a time-dependent shift $ k(t) = c_*t - (N-1)c^*\log t $, which accounts for the logarithmic correction in the front propagation.
• The proof relies on uniform convergence estimates in the moving frame $ x \mapsto |x| - c_*t + (N-1)c^*\log t $, using energy and comparison arguments.
• For non-radial initial data, the method applies the same comparison principle to show the free boundary lies within a spherical shell with logarithmic correction.
• The analysis uses weak solution theory and properties of degenerate parabolic equations, particularly the fact that $ \nabla u^m \in L^2_{\text{loc}} $.

Experimental results

Research questions

• RQ1 How does the free boundary of the solution to the Fisher-KPP type porous medium equation behave in higher dimensions ($ N \geq 2 $) as $ t \to \infty $?
• RQ2 What is the precise asymptotic form of the solution's support and its propagation speed when the initial data is compactly supported and non-radial?
• RQ3 Why does a logarithmic correction term $ (N-1)c^*\log t $ appear in the front propagation speed in higher dimensions but not in one dimension?
• RQ4 Can the asymptotic stability of the minimal speed traveling wave $ \Phi_{c_*} $ be extended to non-radial initial data in $ \mathbb{R}^N $, and if so, how is the shift parameter determined?
• RQ5 What is the role of the universal constant $ c^* > 0 $, independent of $ N $ and initial data, in the logarithmic correction of the front position?

Key findings

• In $ \mathbb{R}^N $ with $ N \geq 2 $, the solution $ u(x,t) $ converges uniformly to the traveling wave $ \Phi_{c_*}(|x| - c_*t + (N-1)c^*\log t - r_0) $ as $ t \to \infty $, for some $ r_0 \in \mathbb{R} $ depending on the initial data.
• The free boundary of the solution lies within a spherical shell $ \{ x : r_1 \leq |x| - c_*t + (N-1)c^*\log t \leq r_2 \} $ for all $ t \geq 1 $, with $ r_1, r_2 \in \mathbb{R} $ depending on $ u_0 $.
• For any $ c > c^* $, $ u(x,t) \to 1 $ uniformly in the ball $ \{ |x| \leq c_*t - (N-1)c\log t \} $ as $ t \to \infty $.
• The constant $ c^* > 0 $ is universal: it is independent of the spatial dimension $ N $ and the initial data $ u_0 $, and arises from the asymptotic behavior of the solution in the radial case.
• In one dimension, the solution converges to a shifted traveling wave without logarithmic correction, but in higher dimensions, the logarithmic term is essential for capturing the correct front position.
• The paper identifies and fills a gap in a prior result by Biró (2011), showing that the shift in the traveling wave is consistent with the front position, using non-degeneracy estimates and a refined time sequence construction.