Stability of stationary solutions of nonlocal reaction-diffusion equations in $m$-dimensional space

Abstract

We consider nonlocal reaction--diffusion equations in $m$--dimensional space. An existence theory is established using standard techniques. It is shown that when local monotonicity conditions are imposed, the stationary solutions that can be stable are those that are stable for an auxiliary local problem. This contrasts with what happens in the general case, where more complex solutions may be stable. An example of such a case is given. These results are obtained using comparison techniques and a generalization of previous results to the $m$--dimensional case.