  We present the growth dynamics of an island of particles $A$ injected from a
localized $A$-source into the sea of particles $B$ and dying in the course of
diffusion-controlled annihilation $A+B\to 0$. We show that in the 1d case the
island unlimitedly grows at any source strength $\Lambda$, and the dynamics of
its growth {\it 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 ({\it 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
the cases the numbers of surviving and dying $A$ particles are calculated, and
the scaling of the reaction zone is derived.
