Logarithmic corrections in Fisher-KPP type Porous Medium Equations

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=\Delta u^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 $\Phi_{c_*}$ 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 $\Phi_{c_*}$ 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 $$\lim_{t\to\infty}\left\{\sup_{x\in\mathbb R^N}\big|u(x,t)-\Phi_{c_*}(|x|-c_*t+(N-1)c^*\log t-r_0)\big|\right\}=0$$ 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 $c>c^*$.