Stability of solutions of quasilinear parabolic equations

We bound the difference between solutions u and v of ut=aΔu+divxf+h and vt=bΔv+divxg+k with initial data φ and ψ, respectively, by ‖u(t,⋅)−v(t,⋅)‖Lp(E)⩽AE(t)‖φ−ψ‖L∞(Rn)2ρp+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.