| Abstract: |
| We consider the large time behaviour of solutions to the Porous Medium Equation with a Fisher-KPP type reaction term
\[
u_t=\Delta u^m+u-u^2\quad\text{in }\mathbb{R}^N\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0\quad\text{in }\mathbb{R}^N,
\]
$m>1$, for nonnegative, nontrivial, radially symmetric, bounded and compactly supported initial data $u_0$. It is well known that in spatial dimension one there is a minimal speed $c_*>0$ for which the equation admits a traveling wave profile $\Phi_{c_*}$ with a finite front. We prove that there exists a second constant $c^*>0$ independent of the dimension $N$ and the initial function $u_0$, such that
\[
\lim_{t\to\infty}\left\{\sup_{x\in\mathbb R^N}|u(x,t)-\Phi_{c_*}(|x|-c_*t+(N-1)c^*\log t-r_0)|\right\}=0
\]
for some $r_0\in\mathbb{R}$ (depending on $u_0$).
Moreover, the radius, $h(t)$, of the support of the solution at time $t$ satisfies
$$
\lim_{t\to\infty} \big[h(t)-c_*t+(N-1) c^*\log t\big]=r_0.
$$
Thus, in contrast with the semilinear case $m=1$, we have a logarithmic correction only for $N>1$. 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 $c>c^*$. |
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