# Reference to Quasilinear Parabolic PDE

I'm studying the viscosity solutions to the Hamilton-Jacobi equations by Evans's book. If we apply the method of vanishing viscosity and add the regularizing term, we get the following quasilinear parabolic equation: $$u_t^{\varepsilon} + H(D_x u^\varepsilon, x) = \varepsilon \Delta u^\varepsilon \\ u^\varepsilon(x,0) = g$$ Evans simply puts it that these turn out to have smooth solutions. I've been wondering why that is true. I tried to find something in the book by Ladyzhenskaya "Linear and Quasilinear Equations of Parabolic Type", but seems like there the theory is much more general than I need, so it would take me a lot of time to pass through. Could anybody give me some reference or explanation of this fact? In this case $H$ is a smooth function, and $g$ is continuous.