Large Time Behavior for Convection-Diffusion Equations in [formula omitted]N with Periodic Coefficients
We describe the large time behavior of solutions of the convection-diffusion equationut−div(a(x)∇u)=d·∇(|u|q−1u)in(0, ∞)×RNwhere d∈RN and a=a(x) is a symmetric periodic matrix satisfying suitable ellipticity assumptions. We also assume that a∈W1, ∞(RN). First, we consider the linear problem (d=0) an...
Gespeichert in:
Veröffentlicht in: | Journal of Differential Equations 2000-11, Vol.167 (2), p.275-315 |
---|---|
Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | We describe the large time behavior of solutions of the convection-diffusion equationut−div(a(x)∇u)=d·∇(|u|q−1u)in(0, ∞)×RNwhere d∈RN and a=a(x) is a symmetric periodic matrix satisfying suitable ellipticity assumptions. We also assume that a∈W1, ∞(RN). First, we consider the linear problem (d=0) and prove that the large time behavior of solutions is given by the fundamental solution of the diffusion equation with a≡ah where ah is the homogenized matrix. In the nonlinear case, when q=1+1N, we prove that the large time behavior of solutions with initial data in L1(RN) is given by a uniparametric family of self-similar solutions of the convection-diffusion equation with constant homogenized diffusion a≡ah. When q>1+1N, we prove that the large time behavior of solutions is given by the fundamental solution of the linear-diffusion equation with a≡ah. |
---|---|
ISSN: | 0022-0396 1090-2732 |
DOI: | 10.1006/jdeq.2000.3796 |