| 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 $L^\infty(\mathbb{R}^N)\setminus\{0\}$:
\[
(*)\;\;\;\;u_t=\Delta u^m+u-u^2\quad\text{in }Q:=\mathbb{R}^N\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0\quad\text{in }\mathbb{R}^N,
\]
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 traveling wave solution $\Phi_{c_*}$ with a finite front, and this traveling wave solution is asymptotically stable in the sense that if the initial function $u_0\in L^\infty(\mathbb R)$ satisfies $\liminf_{x\to -\infty} u_0(x)>0$ and $u_0(x)=0$ for all large $x$, then
$\lim_{t\to\infty} \left\{\sup_{x\in \mathbb R}|u(x,t)-\Phi_{c_*}(x-c_*t-x_0)|\right\}=0$ for some $x_0\in \mathbb R$. In dimension one we obtain an analogous stability result for the case of compactly supported initial data, not necessarily symmetric. In higher dimensions we show that $\Phi_{c_*}$ is still attractive, albeit that a logarithmic shifting occurs. More precisely, if the initial function in $(*)$ 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
\[
\lim_{t\to\infty}\left\{\sup_{x\in\mathbb R^N}|u(x,t)-\Phi_{c_*}(|x|-c_*t+(N-1)c^*\log t-r_0)|\right\}=0
\]
for some $r_0\in\mathbb{R}$ (depending on $u_0$).
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 $u(\cdot, t)$ for all large time $t$ is contained in the spherical shell $\{x\in\mathbb R^N: r_1\leq |x|-c_* t+(N-1)c^* \log t\leq r_2\}$, and for any $c\in (0, c^*)$,
$
\lim_{t\to\infty}u(x,t)=1 \mbox{ uniformly in } \big\{|x|\leq c_*t-(N-1)c\log t\big\}.
$ |
|