LOCAL STABILITY OF CRITICAL FRONTS IN NONLINEAR PARABOLIC PARTIAL-DIFFERENTIAL EQUATIONS

For the Ginzburg-Landau equation and similar nonlinear parabolic partial differential equations on the real line, we prove the nonlinear stability of the slowest monotonic front solution by computing explicitly the leading term in the asymptotic behaviour of a small perturbation as t --> infinity. The proof is based on the renormalization group method for parabolic equations.