Stability of parabolic problems with nonlinear Wentzell boundary conditions


Of concern is the nonlinear uniformly parabolic problemut = div (A ∇ u), u (0, x) = f (x),ut + β ∂νA u + γ (x, u) - q β ΔLB u = 0, for x ∈ Ω ⊂ RN and t ≥ 0; the last equation holds on the boundary ∂Ω. Here A = {ai j (x)}i j is a real, hermitian, uniformly positive definite N × N matrix; β ∈ C (∂ Ω) with β > 0; γ : ∂ Ω × R → R; q ∈ [0, ∞), ΔLB is the Laplace-Beltrami operator on the boundary, and ∂νA u is the conormal derivative of u with respect to A: and everything is sufficiently regular. The solution of this wellposed problem depends continuously on the ingredients of the problem, namely, A, β, γ, q, f. This is shown using semigroup methods in [G.M. Coclite, A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian, Semigroup Forum 77 (1) (2008) 101-108]. Here we prove explicit stability estimates of the solution u with respect to the coefficients A, β, γ, q, and the initial condition f. Moreover we cover the singular case of a problem with q = 0 which is approximated by problems with positive q. © 2008 Elsevier Inc.

Publication Title

Journal of Differential Equations