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This paper deals with nonnegative solutions of the one dimensional degenerateparabolic equations with zero homogeneous Dirichlet boundary condition. Toobtain an existence result, we prove a sharp gradient estimate of |u_x|.Besides, we investigate the behaviors of nonnegative solutions such as thequenching phenomenon, and the finite speed of propagation. Our results of theDirichlet problem will be extended to the associated Cauchy problem. Inaddition, we show that the phenomenon of the instantaneous shrinking of compactsupport of the nonnegative solutions occurs if f satisfies some growthcondition.

Existence of a maximal solution of singular parabolic equations with absorptions: quenching phenomenon and the instantaneous shrinking phenomenon