Maximal regularity for abstract parabolic problems with inhomogeneous boundary data in $L_p$-spaces

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Existence and time-asymptotics of global strong solutions to dynamic Korteweg models (2014, unknown): “following from Sobolev embeddings and H1p(J ; Lp(Ω;E)) ∩ Lp(J ; H2p(Ω;E)) ↪→ Hθp(J ; H p (G;E)), θ ∈ (0, 1), H p (J ; Lp(Ω;E)) ∩ Lp(J ; H3p(Ω;E)) ↪→ H p (J ; H p (G;E)), which are a consequence of the mixed derivative theorem [23].”
Convergence to steady states of solutions of the Cahn–Hilliard and Caginalp equations with dynamic boundary conditions (2006, methodology): “Then, by Denk, Hieber, and Prüss [6, Theorem 8.2], Ap generates an analytic C0-semigroup with maximal regularity in Lp(Ω), for each p ∈ (1,∞).”
Strong solutions for a compressible fluid model of Korteweg type (2008, methodology): “For this, we draw on a general trace theorem, which can be found in [17] and [22], reading as follows.”
Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions (2006, background): “For ∂Ω = Rn−1, the first embedding is a consequence of Proposition 3 in [34] applied to (I−∆).”