We establish the existence of positive self-similar solutions for the nonlinear homogeneous parabolic system given by \(\begin{aligned} {\left\{ \begin{array}{ll} u_t-\Delta u=\mu _1|u|^{2p}u+\beta |v|^{p+1}|u|^{p-1} u\ \ {\text {in}}\ \ (0,\infty )\times \mathbb {R}^N,\\ v_t-\Delta v=\mu _2|v|^{2p}v+\beta |u|^{p+1}|v|^{p-1} v\ \ {\text {in}}\ \ (0,\infty )\times \mathbb {R}^N,\\ u(0,x)=u_0(x),\ \ v(0,x)=v_0(x) \ \ {\text {in}}\ \ \mathbb {R}^N, \end{array}\right. } \end{aligned}\) where \(N=3\) , \(1<p<2\) , \(\mu _1,\mu _2,\beta >0\) , and \(u_0(x), v_0(x)\ge 0\) . Our approach involves demonstrating the existence of positive self-similar solutions with appropriate initial values. Additionally, we construct perturbations of the positive self-similar solution with more general initial values using a contraction mapping argument. Compared to previous works, we encounter new challenges due to the utilization of shooting methods. Fortunately, these challenges can be overcome by carefully studying the properties of the self-similar solutions.
