We consider the large time behavior of solutions to the porous medium equation with a Fisher–KPP type reaction term and nonnegative, compactly supported initial function in $L^{\infty }({\mathbb{R}}^N)\setminus \{0\}$:($\star$)$u_t=\mathrm{\Delta }{u}^m+u-u^2\text{ in }Q:={\mathbb{R}}^N\times {\mathbb{R}}_{+},u(\cdot ,0)=u_0\text{in }{\mathbb{R}}^N,$ 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 $t>0$ (whose boundary is called the free boundary), which is a main different feature of $(\star )$ to the corresponding semilinear case $m=1$. Similar to the corresponding semilinear case $m=1$, it is known that there is a minimal speed $c_{⁎}>0$ such that for any $c\ge c_{⁎}$, the equation admits a wavefront solution ${\mathrm{\Phi }}_c(r)$: For any $\nu \in {\mathbb{S}}^{N-1}$, $v(x,t):={\mathrm{\Phi }}_c(x\cdot \nu -ct)$ solves $v_t=\mathrm{\Delta }{v}^m+v-v^2$. When $m=1$, it is well known that the long-time behavior of the solution with compact initial support can be well approximated by ${\mathrm{\Phi }}_{c_{⁎}}(|x|-c_{⁎}t+\frac{N+2}{c_{⁎}}\log t+O(1))$, and the term $\frac{N+2}{c_{⁎}}\log t$ is known as the logarithmic correction term. When $m>1$, an analogous approximation has been an open question for $N\ge 2$. In this paper, we answer this question by showing that there exists a constant $c_{\mathrm{\#}}>0$ independent of the dimension N and the initial function $u_0$, such that for all large time, any solution of $(\star )$ is well approximated by ${\mathrm{\Phi }}_{c_{⁎}}(|x|-c_{⁎}t+(N-1)c_{\mathrm{\#}}\log t+O(1))$. This is achieved by a careful analysis of the radial case, where the initial function $u_0$ is radially symmetric, which enables us to give a formula for $c_{\mathrm{\#}}$ (involving integrals of ${\mathrm{\Phi }}_{c_{⁎}}(r)$), and to replace the $O(1)$ term by $C+o(1)$ with C a constant depending on $u_0$. The approximation for the general non-radial case is obtained by using the radial results and simple comparison arguments. We note that in sharp contrast to the $m=1$ case, when $N=1$, there is no logarithmic correction term for $(\star )$.

我们考虑具有Fisher-KPP型反应项和非负，紧致支持的初始函数的多孔介质方程解的长时间行为。 ${\mathrm{大号}}^{\infty }（{\mathbb{[R}}^{ñ}）\setminus \{0\}$：（$\star$）${ü}_{Ť}=\mathrm{\Delta }{ü}^{米}+ü-{ü}^2\text{ 在 }问：={\mathbb{[R}}^{ñ}\times {\mathbb{[R}}_{+}，ü（\cdot ，0）={ü}_0\text{在 }{\mathbb{[R}}^{ñ}，$ 与 $米>\mathrm{1个}$。众所周知，解决方案的空间支持$ü（\cdot ，Ť）$ 这个问题一直存在 $Ť>0$ （其边界称为自由边界），这是 $（\star ）$ 对应的半线性情况 $米=\mathrm{1个}$。类似于相应的半线性情况$米=\mathrm{1个}$，已知速度是最小的 $C_{⁎}>0$ 这样对于任何 $C\ge C_{⁎}$，该方程允许波前解 ${\mathrm{\Phi }}_C（\mathrm{[R}）$：对于任何 $\nu \in {\mathbb{小号}}^{ñ-\mathrm{1个}}$， $v（X，Ť）：={\mathrm{\Phi }}_C（X\cdot \nu -CŤ）$ 解决 $v_{Ť}=\mathrm{\Delta }{v}^{米}+v-v^2$。什么时候$米=\mathrm{1个}$，众所周知，具有紧凑初始支持的解决方案的长期行为可以很好地近似为 ${\mathrm{\Phi }}_{C_{⁎}}（|X|-C_{⁎}Ť+\frac{ñ+2}{C_{⁎}}\mathrm{日志}Ť+Ø（\mathrm{1个}））$，以及 $\frac{ñ+2}{C_{⁎}}\mathrm{日志}Ť$被称为对数校正项。什么时候$米>\mathrm{1个}$，对于 $ñ\ge 2$。在本文中，我们通过证明存在一个常数来回答这个问题$C_{\mathrm{＃}}>0$与维数N和初始函数无关${ü}_0$，这样，在很长一段时间内， $（\star ）$ 近似为 ${\mathrm{\Phi }}_{C_{⁎}}（|X|-C_{⁎}Ť+（ñ-\mathrm{1个}）C_{\mathrm{＃}}\mathrm{日志}Ť+Ø（\mathrm{1个}））$。这是通过仔细分析径向情况来实现的，其中初始函数${ü}_0$ 是径向对称的，这使我们能够给出一个公式 $C_{\mathrm{＃}}$ （涉及 ${\mathrm{\Phi }}_{C_{⁎}}（\mathrm{[R}）$），并替换 $Ø（\mathrm{1个}）$ 用 $C+Ø（\mathrm{1个}）$与Ç恒定取决于${ü}_0$。通过使用径向结果和简单的比较参数，可以得出一般非径向情况的近似值。我们注意到与$米=\mathrm{1个}$ 情况，何时 $ñ=\mathrm{1个}$，没有对数校正项 $（\star ）$。