Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λλ in equation. For λ≤λ1(Ω)λ≤λ1(Ω), the solution decays to zero as t→∞t→∞; while for λ>λ1(Ω)λ>λ1(Ω), the solution converges to the unique positive stationary solution as t→∞t→∞. In addition, we show that the solution blows up under some conditions.