Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions


Of concern is the nonlinear parabolic problem with nonlinear dynamic boundary conditions for x ∈ Ω ⊂ ℝN and t ge; 0; the last equation holds on the boundary ∂Ω Here A = {aij(x)}ij is a real, Hermitian, uniformly positive definite N×N matrix; F ∈ C1 ℝN; ℝN is Lipschitz continuous; β ∈ C(∂Ω), with β > 0; γ: ∂Ω × R → R; q ≥ 0; and ∂Aνu is the conormal derivative of u with respect to A; everything is suciently regular. Here we prove the well-posedness of the problem. Moreover, we prove explicit stability estimates of the solution u with respect to the coecients A, F, β, γ, q, and the initial condition f. Our estimates cover the singular case of a problem with q = 0 which is approximated by problems with positive q.

Publication Title

Advances in Differential Equations

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