In this paper we study the Cauchy problem for semilinear parabolic system with nonconstant coefficient singular initial data \(\begin{aligned} {\left\{ \begin{array}{ll} U_{t}-\Delta U=\mu _{1}|U|^{p-1}U+\beta |U|^{r-1}U|V|^{r+1},& x\in \mathbb {R}^{N},t>0, \\ V_{t}-\Delta V=\mu _{2}|V|^{p-1}V+\beta |U|^{r+1}|V|^{r-1}V,& x\in \mathbb {R}^{N},t>0, \\ U(x,0)=\lambda _{1} a(x/|x|)|x|^{-2/(p-1)},& x\in \mathbb {R}^{N}\setminus \{0\},\\ V(x,0)=\lambda _{2} b(x/|x|)|x|^{-2/(p-1)},& x\in \mathbb {R}^{N}\setminus \{0\},\\ \end{array}\right. } \end{aligned}\) where \(N\ge 2\) , \(p=2r+1\) , \(\mu _{1},\mu _{2},\beta >0\) , \(\lambda _{1},\lambda _{2}>0\) are constant parameters, \(a\ge 0\not \equiv 0\) , \(b\ge 0\not \equiv 0\) . We demonstrate that when \(2<N(p-1)<2(p+1)\) and \(p=(N+2)/(N-2)\) , the system has two positive self-similar solutions \((\underline{u}_{\lambda _{1}},\underline{v}_{\lambda _{2}})\) and \((\overline{u}_{\lambda _{1}},\overline{v}_{\lambda _{2}})\) if \(\lambda _{1}\) and \(\lambda _{2}\) are small enough. Additionally, there are no positive self-similar solutions if \(\lambda _{1},\lambda _{2}\) are sufficiently large.
