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

Grenier, E.

Journ. Équ. Dériv. Partielles, St.-Jean-de-Monts 1996, Exp. No. 8, 9 p. (1996).

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

Citations:

Zbl 0831.34023