Sobolev regularity for t > 0 in quasilinear parabolic equations

We establish a regularity property for the solutions to the quasilinear parabolic initial-boundary value problem (1.4) below, showing that for t > 0 they belong to the same space to which the solutions of the second order hyperbolic problem (1.5), which is a singular perturbation of (1.4), belong. This result provides another illustration of the asymptotically parabolic nature of problem (1.5), and would be needed to establish the diffusion phenomenon for quasilinear dissipative wave equations in Sobolev spaces.