An interface problem derived by a bistable reaction-diffusion system with the spatial average of an activator is studied on an -dimensional ball. We analyze the existence of the radially symmetric solutions and the occurrence of Hopf bifurcation as a parameter varies in two and three-dimensional spaces. 1. Introduction The study of interfacial patterns is important in several areas of biology, chemistry, physics, and other fields [1–4]. Internal layers (or free boundary), which separate two stable bulk states by a sharp transition near interfaces, are often observed in bistable reaction-diffusion equations when the reaction rate is faster than the diffusion effect. We consider a reaction-diffusion system with a sufficiently small positive constant [5, 6] where , , and are positive constants, is the ball in -dimensional space, and stands for the unit outward normal on the boundary . The nonlinear functions are where and denotes the spatial average, describing a global feedback effect, namely, The system (1.1) with (1.2) is a model for flow discharges proposed by et al. [7], in which is interpreted by the current density and by the voltage drop across the gas gap [7, 8] in a gas-discharge system. This system also exhibits a codimension-two Turing Hopf bifurcation [9], where the conditions of a spatial Turing instability [10] with a certain wavelength and a temporal Hopf bifurcation with a certain frequency are met simultaneously. Equation (1.1) determines the dynamics of an internal layer, and equation (1.1) together with (1.2) represents a basic model of globally coupled bistable medium which is relevant for current density dynamics in large area bistable semiconductor systems [11–14]. The internal layer has a physical reason as the current filament has a sharp profile with a narrow transition layer connecting flat on- and off-states. When in (1.1) is sufficiently small for the case of without the spatial average, the singular limit analysis is applied to show the existence and the stability of localized radially symmetric equilibrium solutions [15, 16]. In one-dimensional space for the case of without the spatial average, such equilibrium solutions should undergo certain instabilities, and the loss of stability resulting from a Hopf bifurcation produces a kind of periodic oscillation in the location of the internal layers [2, 17–19]. As the parameter varies, the stability of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions for the case of without the spatial average have been examined in [5, 6]. In this
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