Existence and global behavior of the solution to a parabolic equation with nonlocal diffusion

In this paper, we are concerned with the existence, uniqueness and long-time behavior of the solutions for a parabolic equation with nonlocal diffusion even if the reaction term is not Lipschitz-continuous at 0 and grows superlinearly or exponentially at +∞. First, we give a special sub-supersolution pair for some parabolic equations with nonlocal diffusion and establish the method of sub-supersolution. Second, using the sub-supersolution method, we prove the existence, uniqueness and long-time behavior of positive solutions. Finally, some one-dimensional numerical experiments are presented.