An exercise in the textbook "Partial Differential Equations (Second Edition by Emmanuele DiBenedetto), page 172: Show that if $m> 1$, possible solutions to the Cauchy problem \begin{align}\nonumber %\label{eqn: T1 and f1 relation} u_t-\Delta u^m&=0 \;\,\text{in}\;\, \mathbb{R}^N\times (0,T],\ \ u\ge0\\ \notag u(\cdot,0)&=u_0\in C(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N) \notag \end{align} cannot be represented as the convolution of $\Gamma_m$ with the initial datum $u_0$. Here $\Gamma_m$ is the fundamental solution to the porous media equation $u_t=\Delta u^m$. How can we prove this?
When $m=1$, the porous media equation is reduced to the heat equation $u_t=\Delta u$ and the representation formula for the solution is
\begin{equation}
u(x,t)=\frac{1}{\sqrt{4\,\pi\,t}}\int_{\mathbb{R}^N} u_0(y)\,e^{-\frac{|y-x|^2}{4\,t}}\,dy.
\end{equation}
It is easy to see that $\lim_{|x|\to\infty}u(x,t)=0$ for $t>0$. However, when $m\neq1$, the Cauchy problem cannot have a representation formula for the solution. How can we show that $\lim_{|x|\to\infty}u(x,t)=0$ for $t>0$ for $m\neq1$? Any reference, suggestion, idea, or comment is welcome. Thank you!