Stability of solutions of quasilinear parabolic equations ✩

We bound the difference between solutions u and v of ut = aΔu + divx f + h and vt = bΔv + divx g + k with initial data ϕ and ψ, respectively, by u(t, ·) − v(t, ·) Lp(E) AE(t) ϕ −ψ 2ρp L∞(Rn) +B(t) a −b ∞ + ∇x · f −∇x · g ∞ + fu −gu ∞ + h− k ∞ ρp |E|ηp . Here all functions a, f , and h are smooth and bounded, and may depend on u, x ∈ Rn, and t. The functions a and h may in addition depend on ∇u. Identical assumptions hold for the functions that determine the solutions v. Furthermore, E ⊂ Rn is assumed to be a bounded set, and ρp and ηp are fractions that depend on n and p. The diffusion coefficients a and b are assumed to be strictly positive and the initial data are smooth.  2005 Elsevier Inc. All rights reserved.