Sobolev regularity for t > 0 in quasilinear parabolic equations

We establish a regularity property for the solutions to the quasilinear par abolic initial-boundary value problem (1.4) below, showing that for t>0 the y belong to the same space to which the solutions of the second order hyper bolic problem (1.5), which is a singular perturbation of (1.4), belong. Thi s result provides another illustration of the asymptotically parabolic natu re of problem (1.5), and would be needed to establish the diffusion phenome non for quasilinear dissipative wave equations in Sobolev spaces.