We solve the Cauchy problem for a quasilinear parabolic equation in $[0,T]\times \mathbb R^n$ with quadratic nonlinearity in the gradient and with Hölder-continuous, not necessarily differentiable, initial datum. We get the same smoothing properties of linear parabolic equations, and we use them to improve the results now available in the literature on a class of stochastic forward-backward systems.
"Smoothing of quasilinear parabolic operators and applications to forward-backward stochastic systems." Adv. Differential Equations 10 (1) 65 - 88, 2005.