  A catalytic branching random walk on a multidimensional lattice, with
arbitrary finite number of catalysts, is studied in supercritical regime. The
dynamics of spatial spread of the particles population is examined, upon
normalization. The components of the vector random walk jump are assumed
independent (or close to independent) and have semi-exponential distributions
with, possibly, different parameters. A limit theorem on the almost sure
normalized positions of the particles at the population ``front'' is
established. Contrary to the case of the random walk increments with ``light''
distribution tails, studied by Carmona and Hu (2014) in one-dimensional setting
and Bulinskaya (2018) in multidimensional setting, the normalizing factor has a
power rate and grows faster than linear in time function. The limiting shape of
the front in the case of semi-exponential tails is non-convex in contrast to a
convex one in the case of light tails.
