Asymptotic profiles of solutions to convection–diffusion equations

Benachour, Saı̈d; Karch, Grzegorz; Laurençot, Philippe

Comptes Rendus. Mathématique, Tome 338 (2004) no. 5, pp. 369-374.

Le comportement asymptotique des solutions de masse nulle du problème de Cauchy pour l'équation de convection–diffusion $u_t-{\mathrm{u}}_{\mathrm{xx}}+{\left(\right|\mathrm{u}|}^q{)}_x=0,\mathrm{u}(\mathrm{x},0)={\mathrm{u}}_0\left(\mathrm{x}\right)$ est étudié lorsque q>1 et la donnée initiale u₀ appartient à $L^1(ℝ,(1+\left|\mathrm{x}\right|\left)\mathrm{d}\mathrm{x}\right)$ et satisfait ${\int }_{ℝ}{u}_0\left(\mathrm{x}\right)\mathrm{d}\mathrm{x}=0$. 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.

The large time behavior of zero-mass solutions to the Cauchy problem for the convection–diffusion equation $u_t-{\mathrm{u}}_{\mathrm{xx}}+{\left(\right|\mathrm{u}|}^q{)}_x=0,\mathrm{u}(\mathrm{x},0)={\mathrm{u}}_0\left(\mathrm{x}\right)$ is studied when q>1 and the initial datum u₀ belongs to $L^1(ℝ,(1+\left|\mathrm{x}\right|\left)\mathrm{d}\mathrm{x}\right)$ and satisfies ${\int }_{ℝ}{u}_0\left(\mathrm{x}\right)\mathrm{d}\mathrm{x}=0$. 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.

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