Long time behavior of solutions for a scalar nonlocal reaction-diffusion equation

In this paper we study the long time behavior of the solution for a scalar nonlocal reaction-diffusion equation, in which the nonlocal term acts to conserve the spatial integral of the power of the unknown function as time evolves. A class of initial data is found to guarantee the existence of positive global solutions and the convergence to some steady states. A sufficient condition for positive global solutions to be unbounded is also given.