On logistic diffusion equations with nonlocal interaction terms

This paper is concerned with logistic diffusion equations with nonlocal interaction terms appearing in population biology. We intend to study effects of nonlocal terms and discuss the similarity and difference between local problems and nonlocal problems. Mainly, the stationary problem is investigated for a certain class of nonlocal terms. A constructive approach is proposed to look for positive stationary solutions and the unique existence of such a positive solution is established. The analysis of stationary solutions depends on the spectrum for the linearized operator around the stationary solution. However, the linearized operator contains a nonlocal term which makes the spectral analysis delicate and difficult. Putting some additional assumptions we will derive the asymptotic stability of the unique positive solution and, furthermore, its global attractivity. Finally, it will be seen that some arguments are valid to show the unique existence of a positive stationary solution for a considerably general class of nonlocal terms.