Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions

In this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy-Littlewood-Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for J(mu)(u0) < 0 and decay rate of H-0(1) and L-2-norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.

Automated summary

In this paper, the initial boundary value problem for a nonlocal parabolic equation with the Hardy-Littlewood-Sobolev critical exponent on a bounded domain is studied. The long time behaviors of solutions are investigated for low, critical, and high initial energy. Global existence and blow-up of solutions are obtained using the modified potential well method for low or critical initial energy, and the global solutions are proved to be classical. An upper bound of blow-up time as well as decay rate of H-0(1) and L-2-norm of the global solutions are obtained when J(mu)(u0) < 0. Sufficient conditions for global existence and blow-up of solutions are derived for high initial energy. Furthermore, the asymptotic behavior of global solutions, which is similar to the Palais-Smale sequence of stationary equation, is considered.