Stability of Solutions of Quasilinear Parabolic Equations

Abstract. We bound the difference between solutions u and v of ut = a∆u+ divx f + h and vt = b∆v + 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‖∞)p |E| ηp . Here all functions a, f , and h are smooth and bounded, and may depend on u, x ∈ R, 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 ⊂ R 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.