Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations (Q1385309)

The author studies the singularly perturbed scalar parabolic equation \[ \partial_t u^\varepsilon + \sum_{i=1}^n A_i (t,x,u^\varepsilon) \partial_i u^\varepsilon - \varepsilon \Delta u^\varepsilon = 0, \] where \(t\in(0,T)\), \(x\in \Omega\), and \(\Omega\) is the halfspace \(\mathbb{R}_+\times \mathbb{R}^{d-1}\). A solution is subject to initial and Dirichlet boundary conditions. The main purpose of this paper is to construct and justify a complete asymptotic expansion of the solution with respect to the small parameter \(0<\varepsilon \ll 1\).