Quasilinear parabolic equations with nonlinear Wentzell–Robin type boundary conditions

Let Ω ⊂ RN be a bounded domain with Lipschitz boundary, a ∈ C(Ω¯ ) with a > 0 on Ω¯. Let σ be the restriction to ∂Ω of the (N − 1)-dimensional Hausdorff measure and let B :∂Ω × R → [0,+∞] be σ-measurable in the first variable and assume that for σ-a.e. x ∈ ∂Ω, B(x, ·) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p ∈ (1,∞), the operator Ap := div(a|∇u|p−2∇u) with nonlinear Wentzell–Robin type boundary conditions Apu +b|∇u|p−2 ∂u ∂n +β(·,u) 0 on∂Ω, generates a nonlinear submarkovian C0-semigroup on suitable L2-space. Here n(x) denotes the unit outer normal at x and for σ-a.e. x ∈ ∂Ω the maximal monotone graph β(x, ·) denotes the subdifferential ∂B(x, ·) of the functional B(x, ·). We also assume that b ∈ L∞(∂Ω) and satisfies b(x) b0 > 0 σ-a.e. on ∂Ω for some constant b0. As a consequence we obtain that there exist consistence nonexpansive, nonlinear semigroups on suitable Lq -spaces for all q ∈ [1,∞). In the second part we show some domination results. © 2007 Elsevier Inc. All rights reserved.