Stability of least energy patterns of the shadow system for an activator-inhibitor model

Stability of stationary solutions to the shadow system for the activator-in hibitor system proposed by Gierer and Meinhardt is considered in higher dim ensional domains. It is shown that a stationary solution with minimal "ener gy" is stable in a weak sense if the inhibitor reacts sufficiently fast, wh ile it is unstable whenever the reaction of the inhibitor is slow. Moreover , the loss of stability results in a Hopf bifurcation.