Large time behavior for convection-diffusion equations in \(\mathbb{R}{} ^ N\). (English) Zbl 0762.35011

Summary: We describe the large time behavior of solutions of the convection- diffusion equation \[ u_ t-\Delta u=a\cdot\nabla(| u|^{q- 1}u)\quad\text{in } (0,\infty)\times\mathbb{R}^ N \] with \(a\in\mathbb{R}^ N\) and \(q\geq 1+1/N\), \(N\geq 1\). When \(q=1+1/N\), we prove that the large time behavior of solutions with initial data in \(L^ 1(\mathbb{R}^ N)\) is given by a uniparametric family of self-similar solutions. The relevant parameter is the mass of the solution that is conserved for all \(t\). Our result extends to dimensions \(N>1\) well known results on the large time behavior of solutions for viscous Burgers equations in one space dimension. The proof is based on La Salle’s invariance principle applied to the equation written in its self-similarity variables.
When \(q>1+1/N\) the convection term is too weak and the large time behavior is given by the heat kernel. In this case, the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem.


35B40 Asymptotic behavior of solutions to PDEs
35K05 Heat equation
35K15 Initial value problems for second-order parabolic equations
76R99 Diffusion and convection
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI


[2] Aguirre, J.; Escobedo, M.; Zuazua, E., Une équation elliptique dans \(\textbf{R}^N\) provenant d’un problème parabolique avec convection, C. R. Acad. Sci. Paris, 307, 235-237 (1988) · Zbl 0696.35055
[3] Aguirre, J.; Escobedo, M.; Zuazua, E., Existence de solutions à moyenne donnée pour un problème elliptique dans \(\textbf{R}^N \), C. R. Acad. Sci. Paris, 307, 463-466 (1988) · Zbl 0669.35033
[4] Aguirre, J.; Escobedo, M.; Zuazua, E., Self-similar solutions of a convection diffusion equation and related elliptic problems, Comm. Partial Differential Equations, 15, No. 2, 139-157 (1990) · Zbl 0708.35030
[5] Brezis, H., Analyse Fonctionnelle: Théorie et Applications (1983), Masson: Masson Paris · Zbl 0511.46001
[6] Chern, I. L.; Liu, T. P., Convergence to Diffusion Waves of Solutions for Viscous Conservation Laws, Comm. Math. Phys., 110, 503-517 (1987) · Zbl 0635.76071
[7] Escobedo, M.; Kavian, O., Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. T. M. A., 11, No. 10, 1103-1133 (1987) · Zbl 0639.35038
[8] Escobedo, M.; Kavian, O., Asymptotic behavior of positive solutions of a nonlinear heat equation, Houston J. Math., 13, No. 4, 39-50 (1987) · Zbl 0666.35046
[9] Escobedo, M.; Zuazua, E., Comportement asymptotique des solutions d’une équation de convection-diffusion, C. R. Acad. Sci. Paris, 309, 329-334 (1989) · Zbl 0787.35015
[11] Giga, Y.; Kambe, T., Large time behavior of the vorticity of two-dimensional viscous flows and its applications to vortex formation, Comm. Math. Phys., 117, 549-568 (1988) · Zbl 0661.76018
[12] Gmira, A.; Veron, L., Large time behavior of the solutions of a semilinear parabolic equation in \(\textbf{R}^N \), J. Differential Equations, 53, 258-276 (1984) · Zbl 0529.35041
[13] Goodman, J., Stability of viscous scalar shock fronts in several dimensions, Trans. Amer. Math. Soc., 311, No. 2, 683-695 (1988) · Zbl 0693.35112
[14] Hormander, L., Linear Partial Differential Operators (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0321.35001
[15] Il’in, A. M.; Oleĭnick, O., Behavior of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of time, (American Math. Soc. Transl., Vol. 42 (1964), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 19-23, No. 2 · Zbl 0148.34004
[16] Kajikiya, R.; Miyakawa, T., On \(L^2\) decay of weak solutions of the Navier-Stokes equations in \(\textbf{R}^N \), Math. Z., 192, 135-148 (1986) · Zbl 0607.35072
[17] Kamin, S.; Peletier, L. A., Large time behavior of the porous media equation with absorption, Israel J. Math., 55, No. 2, 129-146 (1986) · Zbl 0625.35048
[18] Kamin, S.; Peletier, L. A., Source type solutions of degenerate diffusion equations with absorption, Israel J. Math., 50, 219-230 (1985) · Zbl 0581.35035
[19] Kavian, O., Remarks on the large time behavior of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4, No. 5, 423-452 (1987) · Zbl 0653.35036
[20] Kawashima, S., Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, (Doctoral thesis (1983), Kyoto Univ)
[21] Liu, T. P., Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc., 56, No. 328, 1-108 (1985) · Zbl 0617.35058
[22] Liu, T. P.; Pierre, M., Source solutions and asymptotic behavior in conservation laws, J. Differential Equations, 51, 419-441 (1984) · Zbl 0545.35057
[23] Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0153.13602
[24] Sachdev, P. L.; Nair, K. R.C; Tikekar, V. G., Generalized Burgers equations and Euler-Painlevé transcendents, I, J. Math. Phys., 27, No. 6, 1506-1522 (1986) · Zbl 0638.35072
[25] Schonbek, M. E., Large time behavior of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations, 11, 733-763 (1986) · Zbl 0607.35071
[26] Schonbek, M. E., Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations, 5, No. 5, 449-473 (1980) · Zbl 0476.35012
[27] Schonbek, M. E., Uniform decay rates for parabolic conservation laws, Nonlinear Anal. T. M. A., 10, No. 9, 943-953 (1986) · Zbl 0617.35060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.