  We examine a simple mechanism for the spatio-temporal evolution of transient,
slow slip. We consider the problem of slip on a fault that lies within an
elastic continuum and whose strength is proportional to sliding rate. This rate
dependence may correspond to a viscously deforming shear zone or the
linearization of a non-linear, rate-dependent fault strength. We examine the
response of such a fault to external forcing, such as local increases in shear
stress or pore fluid pressure. We show that the slip and slip rate are governed
by a type of diffusion equation, the solution of which is found using a Green's
function approach. We derive the long-time, self-similar asymptotic expansion
for slip or slip rate, which depend on both time $t$ and a similarity
coordinate $\eta=x/t$, where $x$ denotes fault position. The similarity
coordinate shows a departure from classical diffusion and is owed to the
non-local nature of elastic interaction among points on an interface between
elastic half-spaces. We demonstrate the solution and asymptotic analysis of
several example problems. Following sudden impositions of loading, we show that
slip rate ultimately decays as $1/t$ while spreading proportionally to $t$,
implying both a logarithmic accumulation of displacement as well as a constant
moment rate. We discuss the implication for models of post-seismic slip as well
as spontaneously emerging slow slip events.
