Diffusion-controlled annihilation A+B-->0: the growth of an A-particle island from a localized A source in the B-particle sea.

We present the growth dynamics of an island of particles A injected from a localized A source into a sea of particles B and dying in the course of diffusion-controlled annihilation A+B-->0. We show that in the one-dimensional (1D) case the island grows unlimitedly at any source strength Lambda, and the dynamics of its growth does not depend asymptotically on the diffusivity of B particles. In the 3D case the island grows only at Lambda> Lambda(c), achieving asymptotically a stationary state (static island). In the marginal 2D case the island unlimitedly grows at any Lambda but at Lambda< Lambda(*) the time of its formation becomes exponentially large. For all cases the numbers of surviving and dying A particles are calculated, and the scaling of the reaction zone is derived.