×

zbMATH — the first resource for mathematics

Renormalization group and asymptotics of solutions of nonlinear parabolic equations. (English) Zbl 0806.35067
Summary: We present a general method for studying long-time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusion-type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity. In an accompanying paper, [J. Bricmont and A. Kupiainen, Commun. Math. Phys. 150, No. 1, 193-208 (1992; Zbl 0765.35052)], the method is applied to systems of equations where some variables are “slaved,” such as the complex Ginzburg-Landau equation.

MSC:
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K40 Second-order parabolic systems
PDF BibTeX Cite
Full Text: DOI
References:
[1] Similarity, Self-Similarity and Intermediate Asymptotics, Consultants Bureau, New York. 1979.
[2] Bramson, Mem. Amer. Math. Soc. 44 pp 1– (1983)
[3] Brezis, Arch. Rational Mech. Anal. 95 pp 185– (1986)
[4] Brezis, J. Math. Pures Appl. 62 pp 73– (1983)
[5] Bricmont, Comm. Math. Phys. 150 pp 193– (1992)
[6] The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974.
[7] and , Instabilities and Fronts in Extended Systems, Princeton University Press. 1990. · Zbl 0732.35074
[8] Collet, Comm. Math. Phys. 145 pp 345– (1992)
[9] Collet, Helv. Phys. Acta 65 pp 56– (1992)
[10] Fujita, J. Fac. Sci. Univ. Tokyo 13 pp 109– (1966)
[11] Galaktionov, Math. USSR-Sb 54 pp 421– (1986)
[12] Gmira, J. Differential Equations 53 pp 258– (1984)
[13] Goldenfeld, Phys. Rev. Lett. 64 pp 1361– (1990)
[14] , and , Asymptotics of partial differential equations and the renormalisation group, pp. 375–383 in: Asymptotics Beyond All Orders, Proceedings of a NATO Advanced Research Workshop on Asymptotics Beyond All Orders, , and , eds., Plenum Press, New York, 1991.
[15] Goldenfeld, Phys. A 177 pp 213– (1991)
[16] Kamin, Ann. Scuola Norm. Sup. Pisa 12 pp 393– (1985)
[17] Kamin, Proc. Amer. Math. Soc. 95 pp 205– (1985)
[18] Levine, SIAM Rev. 32 pp 262– (1990)
[19] Functional Integration and Quantum Physics, Academic Press, New York, 1979. · Zbl 0434.28013
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.