Existence and uniqueness of solutions of parabolic Cauchy problem for $u_t -\epsilon \Delta (u) + f(x,t,u,\nabla u) = 0$

Question

Consider the Cauchy problem

$$u_t -\epsilon \Delta (u) + f(x,t,u,\nabla u) = 0,$$ $$u(x,0) = g(x)$$

$(x,t) \in\mathbb{R}^n \times (0,\infty)$, $u:\mathbb{R}^n \to \mathbb{R}$, with $f$ at least continuous.

Can you point out a reference that has a detailed exposition of existence, uniqueness, and estimates of classical solutions for this problem?

Answer 1

A decent starting point would be Lieberman's Second Order Parabolic Differential Equations. The latter half of the book is devoted to quasilinear and fully nonlinear parabolic equations.