Partial Differential Equations
Asymptotic profiles of solutions to convection–diffusion equations
Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 369-374.

The large time behavior of zero-mass solutions to the Cauchy problem for the convection–diffusion equation u t -u xx +(|u| q ) x =0,u(x,0)=u 0 (x) is studied when q>1 and the initial datum u0 belongs to L 1 (,(1+|x|)dx) and satisfies u 0 (x)dx=0. We provide conditions on the size and shape of the initial datum u0 as well as on the exponent q>1 such that the large time asymptotics of solutions is given either by the derivative of the Gauss–Weierstrass kernel, or by a self-similar solution of the equation, or by hyperbolic N-waves.

Le comportement asymptotique des solutions de masse nulle du problème de Cauchy pour l'équation de convection–diffusion u t -u xx +(|u| q ) x =0,u(x,0)=u 0 (x) est étudié lorsque q>1 et la donnée initiale u0 appartient à L 1 (,(1+|x|)dx) et satisfait u 0 (x)dx=0. Nous donnons des conditions sur l'amplitude et la forme de la donnée initiale u0 et sur l'exposant q>1 sous lesquelles le comportement asymptotique des solutions est décrit par la dérivée première du noyau de Gauss–Weierstrass, ou par une solution auto-similaire de l'équation, ou par une N-onde hyperbolique.

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DOI: 10.1016/j.crma.2004.01.001
Benachour, Saı̈d 1; Karch, Grzegorz 2; Laurençot, Philippe 3

1 Institut Elie Cartan-Nancy, Université Henri Poincaré, BP 239, 54506 Vandœuvre lès Nancy cedex, France
2 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, and Institute of Mathematics, Polish Academy of Sciences, Warsaw (2002-2003), Poland
3 Mathématiques pour l'industrie et la physique, CNRS UMR 5640, Université Paul Sabatier-Toulouse 3, 118, route de Narbonne, 31062 Toulouse cedex 4, France
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Benachour, Saı̈d; Karch, Grzegorz; Laurençot, Philippe. Asymptotic profiles of solutions to convection–diffusion equations. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 369-374. doi : 10.1016/j.crma.2004.01.001. http://www.numdam.org/articles/10.1016/j.crma.2004.01.001/

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