  We present a full classification of the short-time behaviour of the
interfaces and local solutions to the nonlinear parabolic $p$-Laplacian type
reaction-diffusion equation of non-Newtonian elastic filtration \[
u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1<p<2, \beta >0 \] If the
interface is finite, it may expand, shrink, or remain stationary as a result of
the competition of the diffusion and reaction terms near the interface,
expressed in terms of the parameters $p,\beta, sign~b$, and asymptotics of the
initial function near its support. In some range of parameters, strong
domination of the diffusion causes infinite speed of propagation and interfaces
are absent. In all cases with finite interfaces we prove the explicit formula
for the interface and the local solution with accuracy up to constant
coefficients. We prove explicit asymptotics of the local solution at infinity
in all cases with infinite speed of propagation. The methods of the proof are
based on nonlinear scaling laws, and a barrier technique using special
comparison theorems in irregular domains with characteristic boundary curves. A
full description of small-time behaviour of the interfaces and local solutions
near the interfaces for slow diffusion case when $p>2$ is presented in a recent
paper {\it Abdulla \& Jeli, Europ. J. Appl. Math. 28, 5(2017), 827-853.}
