  Consider the Cauchy problem for a nonlinear diffusion equation
\begin{equation} \tag{P} \left\{ \begin{array}{ll} \partial_t u=\Delta
u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\
u(x,0)=\lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right.
\end{equation} where $m>0$, $\alpha\in(-\infty,1)$, $\lambda>0$ and $\varphi\in
BC({\bf R}^N)\,\cap\, L^r({\bf R}^N)$ with $1\le r<\infty$ and $\inf_{x\in{\bf
R}^N}\varphi(x)>-\lambda$. Then the positive solution to problem (P) behaves
like a positive solution to ODE $\zeta'=\zeta^\alpha$ in $(0,\infty)$ and it
tends to $+\infty$ as $t\to\infty$. In this paper we obtain the precise
description of the large time behavior of the solution and reveal the
relationship between the behavior of the solution and the diffusion effect the
nonlinear diffusion equation has.
