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

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...
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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
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

References

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