Nonlinear L2-stability under large disturbances

We derive time-asymptotic decay rates in L-2 for large disturbances to some important classes of solutions of the Cauchy problem for a number of uniformly parabolic equations, provided only that the disturbances belong to appropriate L-p spaces at initial time. Examples considered include the scalar nonlinear advection-diffusion equation u(t) + f(u)(x) = (b(u)u(x))(x) and the parabolic system u(t) + (u phi(|u|))(x) = (B(u)u(x))(x), where u(x, t) is an element of R-m, phi is a given scalar function and B(u) is a uniformly positive-definite diagonal matrix. (C) 1999 Elsevier Science B.V. All rights reserved.