# Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order In this paper we study the homogenization of degenerate quasilinear parabolic equations: $$\partial _{t} u - {\text{div}}a{\left( {\frac{t} {\varepsilon },\frac{x} {\varepsilon },u,\nabla u} \right)} = f{\left( {t,x} \right)},$$ where a(t, y, α, λ) is periodic in (t, y). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

, Volume 21 (1) – Jan 1, 2005

## Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 93–100 Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order 1,2 1,3 Xing-you Zhang , Yong Huang College of Mathematics and Physics, Chongqing University, 400044, China (E-mail: zhangxy@cqu.edu.cn) Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand Department of Applied Mathematics, Tsinghua University, Beijing, 100084, China (Email: huangyong04@mails.tsinghua.edu.cn) Abstract In this paper we study the homogenization of degenerate quasilinear parabolic equations: t x ∂ u − diva , ,u,∇u = f(t, x), ε ε where a(t, y, α, λ)isperiodic in(t, y). Keywords degenerate parabolic equations; homogenization; compensated compactness 2000 MR Subject Classiﬁcation 35B40; 35K57 1 Introduction and Main Results Let T> 0and letΩ ⊂ R be an open bounded domain with Lipschitz boundary. We consider the following initial-boundary value problem: t x ε ε ε ⎪ ∂ u − diva , ,u ,∇u = f (x, t), in Ω =Ω ×(0,T ) t T ε ε (P ) ε u (x, t)=0, on ∂Ω ×(0,T ) u (x, 0) = u (x) 1,p in the space X = L [0,T ; V ], where V = W (µ , Ω) is a weighted sobolev space, f ∈ ∗ p ∗ 2 X = L (0,T ; V ),u ∈ L (Ω) , and the degeneration is determined by a vector function µ (x)= µ(x/ε)=(µ ,µ ,··· ,µ ) with positive component µ in Ω satifying certain inte- ε 1 2 n i gerability assumptions. The existence and regularity result may be found in [3, 7]. Under a coerciveness condition of the type: a(t, y, α, λ)λ ≥ β|λ| (p> 1,β >0aconstant), the asymptotic behaviour for ε converging to zero of the problem (P ) has been widely studied by many authors, (see [1,5] and [8]). In this paper, we...

/lp/springer_journal/homogenization-for-degenerate-quasilinear-parabolic-equations-of-C0wNwUtjBZ
Publisher
Springer-Verlag
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0219-x
Publisher site
See Article on Publisher Site

### Abstract

In this paper we study the homogenization of degenerate quasilinear parabolic equations: $$\partial _{t} u - {\text{div}}a{\left( {\frac{t} {\varepsilon },\frac{x} {\varepsilon },u,\nabla u} \right)} = f{\left( {t,x} \right)},$$ where a(t, y, α, λ) is periodic in (t, y).

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

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