The renormalization group and large-time behaviour of solutions of nonlinear parabolic partial differential equations.

摘要

The Renormalization Group method is used to study the asymptotic {dollar}(ttoinfty){dollar} behaviour of the solution of the Cauchy problem for the nonlinear parabolic partial differential equation{dollar}{dollar}dot usb{lcub}t{rcub} = Delta u + F(u,uspprime,usp{lcub}primeprime{rcub}), tge 1, x in {lcub}bf R{rcub}; u(x,1) = usb0(x), x in {lcub}bf R{rcub}.{dollar}{dollar}The existence of the bounded solution is established in the infinite strip R {dollar}times lbrack 1, infty){dollar} and this classical solution has the fundamental solution of the heat equation as its asymptotics in large time; it is also shown that, for some special nonlinear problems, further asymptotics of the classical solution can be obtained and any degree of asymptotic expansion of large-time behaviour of the solution can be made for a suitably chosen initial function {dollar}usb0.{dollar}