  The paper concerns the well-posedness and long-term asymptotics of
growth--fragmentation equation with unbounded fragmentation rates and
McKendrick--von Foerster boundary conditions. We provide three different
methods of proving that there is a strongly continuous semigroup solution to
the problem and show that it is a compact perturbation of the corresponding
semigroup with a homogeneous boundary condition. This allows for transferring
the results on the spectral gap available for the later semigroup to the one
considered in the paper. We also provide sufficient and necessary conditions
for the irreducibility of the semigroup needed to prove that it has
asynchronous exponential growth. We conclude the paper by deriving an explicit
solution to a special class of growth--fragmentation problems with
McKendrick--von Foerster boundary conditions and by finding its Perron
eigenpair that determines its long-term behaviour.
