Nonlinear L 2 - stability under large disturbances 1

We derive time-asymptotic decay rates in L 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 spaces at initial time. Examples considered include the scalar nonlinear advection–di usion equation ut + f(u)x =(b(u)ux)x and the parabolic system ut + (u’(|u|))x =(B(u)ux)x; where u(x; t)∈Rm; ’ is a given scalar function and B(u) is a uniformly positive-de nite diagonal matrix. c © 1999 Elsevier Science B.V. All rights reserved.