Global existence and blow up of solutions for a class of coupled parabolic systems with logarithmic nonlinearity

According to the difference of the initial energy, we consider three cases about the global existence and blow-up of the solutions for a class of coupled parabolic systems with logarithmic nonlinearity. The three cases are the low initial energy, critical initial energy and high initial energy, respectively. For the low initial energy and critical initial energy $ J(u_0, v_0)\leq d $, we prove the existence of global solutions with $ I(u_0, v_0)\geq 0 $ and blow up of solutions at finite time $ T < +\infty $ with $ I(u_0, v_0) < 0 $, where $ I $ is Nehari functional. On the other hand, we give sufficient conditions for global existence and blow up of solutions in the case of high initial energy $ J(u_0, v_0) > d $.