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...

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Veröffentlicht in:Journal of Differential Equations 2000-11, Vol.167 (2), p.275-315
Hauptverfasser: Duro, Gema, Zuazua, Enrique
Format: Artikel
Sprache:eng
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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