Nonlinear L2-stability under large disturbances

We derive time-asymptotic decay rates in L2 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 Lp spaces at initial time. Examples considered include the scalar nonlinear advection-diffusion equation ut + f(u)x = (b(u)ux)x and the parabolic system ut + (ϕ(¦u¦))x = (B(u)ux)x, where u(x,t)∈Rm, ϕ is a given scalar function and B(u) is a uniformly positive-definite diagonal matrix.