Journal of Statistical Physics, Vol. 95, Nos.56, 1999 An Integrodifferential Model for Phase Transitions: Stationary Solutions in Higher Space Dimensions Peter W. Bates 1 and Adam Chmaj 1 Received March 5, 1998; final June 11, 1998 We study the existence and stability of stationary solutions of an integro- differential model for phase transitions, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. As such, this model is a nonlocal extension of the AllenCahn equation, which incorporates long-range interactions. We find that the set of stationary solutions for this model is much larger than that of the AllenCahn equation. KEY WORDS: Nonlocal AllenCahn equation; long-range interaction; pinning. 1. INTRODUCTION We study the integral equation ( J V u)( x )&u( x)&*f ( u( x ))=0, x # R n , n 1 (1.1) where J V u( x)(= R n J( x & y ) u( y ) dy ) is the convolution of J and u and * >0. We assume  R n J( x) dx=1 and f is bistable, e.g., f ( u )=u( u 2 &1). Solutions to (1.1) are stationary solutions of the evolution equation u t =J V u &u &*f ( u) (1.2) which may be thought of as a nonlocal version of the AllenCahn equa- tion. (1) We find that, in the special case where u +*f ( u ) is nonmonotone and * is sufficiently large, there exist stationary solutions having discontinuities across arbitrarily prescribed interfaces. We construct both stable and unstable solutions of this type. An important point is that some of our results allow for J to take negative as well as positive values. 1119 0022-4715990600-111916.000  1999 Plenum Publishing Corporation 1 Department of Mathematics, Brigham Young University, Provo, Utah 84602.