Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions
Abstract
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
Recommended Citation
Coclite, G., Goldstein, G., & Goldstein, J. (2011). Well-posedness of nonlinear parabolic problems with nonlinear Wentzell boundary conditions. Advances in Differential Equations, 16 (9-10), 895-916. Retrieved from https://digitalcommons.memphis.edu/facpubs/6140
