In this paper, we consider the Cauchy problem
$$\left\{ \begin{align}
& {{u}_{t}}=\Delta ({{u}^{m}}),\,\,\,\,\,x\in {{\mathbb{R}}^{N}},t>0,N\ge 3, \\
& u(x,0)={{u}_{0}}(x),\,\,\,\,\,x\in {{\mathbb{R}}^{N}}. \\
\end{align} \right.$$
We will prove that
(i) for
${{m}_{c}}\,<\,m,\,{{m}_{0}}\,<\,1,\,\left| u(x,\,t,m)-u(x,\,t,{{m}_{0}}) \right|\,\to \,0$
as
$m\,\to \,{{m}_{0}}$
uniformly on every compact subset of
${{\mathbb{R}}^{N}}\,\times \,{{\mathbb{R}}^{+}}$
, where
${{m}_{c}}\,=\,\frac{{{(N-2)}_{+}}}{N}$
;
(ii) there is a
${{C}^{*}}$
that explicitly depends on
$m$
such that
$${{\left\| u(\cdot ,\cdot ,m)-u(\cdot ,\cdot ,1) \right\|}_{{{L}^{2}}({{\mathbb{R}}^{N}}\times {{\mathbb{R}}^{+}})}}\le {{C}^{*}}\left| m-1 \right|.$$