Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations

The stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded interval J of the real line with homogeneous Dirichlet boundary conditions is studied. It is shown that it is possible to have stable stationary solutions which change sign once on J in the case of constant diffusion when the reaction term does not depend explicitly on the space variable. The problem of the possible types of stable solutions that may exist is considered. It is also shown that Matano's result on the lap-number is still true in the case of nonlocal problems. © 1994 Plenum Publishing Corporation.