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∂νAu+γ(x,u)−qβΔLBu=0, for x∈Ω⊂RNx∈Ω⊂RN and t⩾0t⩾0; the last equation holds on the boundary ∂Ω  . Here A={aij(x)}ijA={aij(x)}ij is a real, hermitian, uniformly positive definite N×NN×N matrix; β∈C(∂Ω)β∈C(∂Ω) with β>0β>0; γ:∂Ω×R→Rγ:∂Ω×R→R; q∈[0,∞)q∈[0,∞), ΔLBΔLB is the Laplace–Beltrami operator on the boundary, and ∂νAu 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,fA,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 AA, β, γ, q, and the initial condition f  . Moreover we cover the singular case of a problem with q=0q=0 which is approximated by problems with positive q.