Blow-up criteria and large time behavior for a cooperative parabolic system in $ \mathbb{R}^3 $

We consider the parabolic system$ \begin{align} \begin{aligned} \left\{ \begin{array}{ll} u_t-\Delta u = \mu_1|u|^{2}u+\beta uv^{2}, &x\in\mathbb{R}^3, \ t>0, \\ v_t-\Delta v = \mu_2|v|^{2}v+\beta vu^{2}, &x\in\mathbb{R}^3, \ t>0, \ \end{array} \right. \end{aligned} \end{align} $where $ \mu_1, \mu_2, \beta>0 $. By the potential well method and forward similarity transform, we first derive a criterion for initial data, which guarantees either blow-up or global existence of its solutions. Then, for all global solutions $ (u, v) $ with suitable initial data, we show that $ ||u(t)||_{L^\infty(\mathbb{R}^3)}+||v(t)||_{L^\infty(\mathbb{R}^3)} $ goes to zero at most of order $ t^{-1/2} $ as $ t\to+\infty $. We directly study a new parabolic system by using the forward similarity transform rather than the original system, as the potential well method cannot be applied directly. This approach offers a novel perspective on the qualitative properties of parabolic systems.