  We consider the non-monotone degenerate diffusion equation with time delay.
Different from the linear diffusion equation, the degenerate equation allows
for semi-compactly supported traveling waves. In particular, we discover
sharp-oscillating waves with sharp edges and non-decaying oscillations. The
degenerate diffusion and the effect of time delay cause us essential
difficulties. We show the existence for both sharp and smooth traveling wave
solutions. Furthermore, we prove the oscillating properties of the waves for
large wave speeds and large time delay. Since the existing approaches are not
applicable, we develop a new technique to show the existence of the sharp,
smooth and oscillatory traveling waves.
