  We consider a non-linear parabolic partial differential equation (PDE) on
$\mathbb R^d$ with a distributional coefficient in the non-linear term. The
distribution is an element of a Besov space with negative regularity and the
non-linearity is of quadratic type in the gradient of the unknown. Under
suitable conditions on the parameters we prove local existence and uniqueness
of a mild solution to the PDE, and investigate properties like continuity with
respect to the initial condition and blow-up times. We prove a global existence
and uniqueness result assuming further properties on the non-linearity. To
conclude we consider an application of the PDE to stochastic analysis, in
particular to a class of non-linear backward stochastic differential equations
with distributional drivers.
