Positive profiles for the perturbed nonlocal dispersal equations

In this paper, we study the following perturbed nonlocal dispersal equation integral(Omega)J(x-y)u(y)dy-u(x)+lambda u(x)+epsilon b(x)u(x)-a(x)u(p)(x)=0,x is an element of Omega<overline>, where Omega subset of R-N is a smooth bounded domain, p>1, lambda and epsilon>0 are parameters, the coefficient a(& sdot;) and the kernel function J(& sdot;) are nonnegative, while b(x) can be indefinite. We are interested in the asymptotic profiles and limiting behavior of patterns for the positive solutions. It is shown that the positive solution converges to the unique positive solution of nonlocal dispersal logistic equation as epsilon -> 0. However, we obtain that the new pattern appears when epsilon is large. Among others, we find that the positive solutions exhibit quenching or blow-up profiles. Our study reveals how the existence of new profiles of patterns is determined by the behavior of b(x).