Abstract :
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.
