We study the maximal displacement of branching random walks in a class of
time inhomogeneous environments. Specifically, binary branching random walks
with Gaussian increments will be considered, where the variances of the
increments change over time macroscopically. We find the asymptotics of the
maximum up to an $O_P(1)$ (stochastically bounded) error, and focus on the
following phenomena: the profile of the variance matters, both to the leading
(velocity) term and to the logarithmic correction term, and the latter exhibits
a phase transition.

Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in
\mathbb{T}}$ denote a branching random walk indexed by the vertices of the
tree, where the increments are i.i.d. and possess a logarithmic moment
generating function $\Lambda(\cdot)$. Let $m_n$ denote the minimum of the
variables $S_v$ over all vertices at the $n$th generation, denoted by
$\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a
constant, which for convenience may be taken to be 0.