## Boundary layers of parabolic systems. (Couches limites de systèmes paraboliques.)(French)Zbl 0882.35012

Let $$\varepsilon>0$$ be a small parameter. Consider the quasilinear parabolic problem: $\partial_t u^\varepsilon+ \sum^d_{i=1} A_i(t,x,u^\varepsilon) \partial_i u^\varepsilon- \varepsilon\Delta u^\varepsilon= 0\quad\text{in }\Omega,$ $$u^\varepsilon= 0$$ on $$\partial\Omega$$, $$u^\varepsilon(0,x)= u^\varepsilon_0(x)$$, where $$u^\varepsilon$$ is a vector in $$\mathbb{R}^n$$, $$\Omega= \{(x_1,x_2,\dots,x_d)\in \mathbb{R}^d,\;x_1>0\}$$, and the matrices $$A_i$$ are symmetric and of class $$C^\infty$$ on $$\mathbb{R}\times \mathbb{R}^d\times\mathbb{R}^n$$. Under suitable assumptions on $$u^\varepsilon_0$$, the author builds asymptotic expansions for solutions $$u^\varepsilon$$, as $$\varepsilon\to 0$$. In the first (resp. second) section of the paper, the boundary $$\partial\Omega$$ is assumed to be uniformly characteristic, i.e. $$A_1(t,x,v)=0$$ for $$t\geq 0$$, $$x\in\partial\Omega$$, $$v\in\mathbb{R}^n$$ (resp. noncharacteristic). Related results have been obtained by O. Guès [Ann. Inst. Fourier 45, No. 4, 973-1006 (1995; Zbl 0831.34023)], when the matrices $$A_i$$ do not depend on $$u^\varepsilon$$.
Reviewer: D.Huet (Nancy)

### MSC:

 35B25 Singular perturbations in context of PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35C20 Asymptotic expansions of solutions to PDEs

### Keywords:

quasilinear parabolic problem

Zbl 0831.34023
Full Text: