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 + infinity. First, we give a special sub-supersolution pair for some parabolic equations with nonlocal di ffusion 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.

Automated summary

This paper focuses on the existence, uniqueness, and long-time behavior of solutions for a parabolic equation with nonlocal diffusion, even in cases where the reaction term is not Lipschitz-continuous at 0 and grows superlinearly or exponentially at + infinity. Special sub-supersolution pairs are provided and the method of sub-supersolution is established. Using this method, the existence, uniqueness, and long-time behavior of positive solutions are proven, with numerical experiments presented for validation.