STABILITY OF STATIONARY SOLUTIONS FOR A SCALAR NONLOCAL REACTION-DIFFUSION EQUATION

摘要

The stability of stationary solutions of the non-local reaction-diffusion equation u_t, = u_(xx)+f(u)-α∫_0~1f(u)dx with homogeneous Neumann boundary conditions is studied. Depending on a, bounds on the dimension of the unstable manifold of a stationary solution are given. In particular, it is shown that only constant or monotone stationary solutions may be stable. For the specific case of a cubic like f, the existence of a Hopf bifurcation is proven. Finally, some related equations are discussed.