  We consider diffusion-controlled evolution of a $d$-dimensional $A$-particle
island in the $B$-particle sea at propagation of the sharp reaction front
$A+B\to 0$ at equal species diffusivities. The $A$-particle island is formed by
a localized (point)$A$-source with a strength $\lambda$ that acts for a finite
time $T$. We reveal the conditions under which the island collapse time $t_{c}$
becomes much longer than the injection period $T$ (long-living island) and
demonstrate that regardless of $d$ the evolution of the long-living island
radius $r_{f}(t)$ is described by the universal law
$\zeta_{f}=r_{f}/r_{f}^{M}=\sqrt{e\tau|\ln\tau|}$ where $\tau=t/t_{c}$ and
$r_{f}^{M}$ is the maximal island expansion radius at the front turning point
$t_{M}=t_{c}/e$. We find that in the long-living island regime the ratio
$t_{c}/T$ changes with the increase of the injection period $T$ by the law
$\propto (\lambda^{2}T^{2-d})^{1/d}$ i.e. increases with the increase of $T$ in
the one-dimensional (1D) case, does not change with the increase of $T$ in the
2D case and decreases with the increase of $T$ in the 3D case. We derive the
scaling laws for particles death in the long-living island and determine the
limits of their applicability. We demonstrate also that these laws describe
asymptotically the evolution of the $d$-dimensional spherical island with a
uniform initial particle distribution generalizing the results obtained earlier
for the quasi-one-dimensional geometry. As striking results we present a
systematic analysis of the front relative width evolution for fluctuation,
logarithmically modified and mean-field regimes and demonstrate that in a wide
range of parameters the front remains sharp up to a narrow vicinity of the
collapse point.
