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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 u 0 belongs to L 1 ( R ,(1+|x|) d x) and satisfies ∫ R u 0 (x) d x=0 . We provide conditions on the size and shape of the initial datum u 0 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. To cite this article: S. Benachour et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004). Résumé 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 u 0 appartient à L 1 ( R ,(1+|x|) d x) et satisfait ∫ R u 0 (x) d x=0 . Nous donnons des conditions sur l'amplitude et la forme de la donnée initiale u 0 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. Pour citer cet article : S. Benachour et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004). References [1] S. Benachour, G. Karch, Ph. Laurençot, Asymptotic profiles of solutions to viscous Hamilton–Jacobi equations, submitted for publication [2] S. Benachour, H. Koch, Ph. Laurençot, Very singular solutions to a nonlinear parabolic equation with absorption. II – Uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, in press [3] S. Benachour Ph. Laurençot Global solutions to viscous Hamilton–Jacobi equations with irregular initial data Comm. Partial Differential Equations 24 1999 1999 2021 [4] S. Benachour Ph. Laurençot Very singular solutions to a nonlinear parabolic equation with absorption I. Existence Proc. Roy. Soc. Edinburgh Sect. A 131 2001 27 44 [5] M. Ben-Artzi H. Koch Decay of mass for a semilinear parabolic equation Comm. Partial Differential Equations 24 1999 869 881 [6] M. Ben-Artzi Ph. Souplet F.B. Weissler The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces J. Math. Pures Appl. 81 2002 343 378 [7] M. Escobedo J.L. Vázquez E. Zuazua Asymptotic behavior and source-type solutions for a diffusion–convection equation Arch. Rational Mech. Anal. 124 1993 43 65 [8] M. Escobedo E. Zuazua Large time behavior for convection–diffusion equations in R N J. Funct. Anal. 100 1991 119 161 [9] E. Feireisl Ph. Laurençot The L 1 -stability of constant states of degenerate convection–diffusion equations Asymptotic Anal. 19 1999 267 288 [10] B. Gilding M. Guedda R. Kersner The Cauchy problem for u t =Δ u +|∇ u | q J. Math. Anal. Appl. 284 2003 733 755 [11] G. Karch M.E. Schonbek On zero mass solutions of viscous conservation laws Comm. Partial Differential Equations 27 2002 2071 2100 [12] Y.J. Kim, An Oleinik type estimate for a convection–diffusion equation and convergence to N -waves, J. Differential Equations, in press [13] Ph. Laurençot Ph. Souplet On the growth of mass for a viscous Hamilton–Jacobi equation J. Anal. Math. 89 2003 367 383 [14] J. Smoller Shock Waves and Reaction–Diffusion Equations Grundlehren Math. Wiss. vol. 258 1983 Springer-Verlag New York

Asymptotic profiles of solutions to convection–diffusion equations