  Recent local tunneling data indicate strong nanoscale inhomogeneity of
superconducting gap in high temperature superconductors. Strong local nanoscale
inhomogeneity in the bosonic scattering mode has also been observed in the same
samples. We argue that these two inhomogeneities directly related to each
other. To address local boson scattering effects, we develop a local strong
coupling model of superconducting pairing in a coarse grained superconducting
state. Each patch is characterized by local coupling to the bosonic mode as
well as by local mode energy. We find that local gap value on each patch grows
with the local strength of electron-boson interaction. At the same time local
gap value decreases with the local boson mode energy, an observation consistent
with the tunneling experiments. We argue that features in the tunneling
spectrum due to boson scattering are consistent with experimentally observed
spectra. We also address the $^{16}O$ to $^{18}O$ isotope substitution. Since
both coupling constant and boson energy could change upon isotope substitution,
we prove that interplay between these two effects can produce results that are
very different from conventional BCS model.
