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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.

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K40 Second-order parabolic systems
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