Degenerate fourth order parabolic equations with Neumann boundary conditions

Abstract

We study the generation property for a fourth order operator in divergence or in non divergence form with suitable Neumann boundary conditions. As a consequence we obtain the well posedness for the parabolic equations governed by these operators. The novelty of this paper is that the operators depend on a function a : [0, 1] → R+ that degenerates somewhere in the interval.

Fourth order differential operators with interior degeneracy and generalized Wentzell boundary conditions

“…In this section we recall some suitable weighted spaces and preliminary results given in [8], that will be crucial for the rest of the paper.…”

“…Thus, if u ∈ Z s (0, 1), u ′ is locally absolutely continuous in [0, 1] \ {x 0 } and not absolutely continuous in [0, 1] as for the weakly degenerate case; so equality (2.10) is not true a priori. For this reason in [8] we characterize the space Z s (0, 1). In particular, we introduce the space X := {u ∈ H 1 (0, 1) : u ′ is locally absolutely continuous in [0, 1] \ {x 0 }, au, au ′ ∈ H 1 (0, 1), au ′′ ∈ H 2 (0, 1), √ au ′′ ∈ L 2 (0, 1), (au (k) )(x 0 ) = 0, for all k = 0, 1, 2}.…”

“…Actually, Proposition 2.4 and Lemma 2.5.2 are proved in [8] under a weaker assumption (see [8,Hypothesis 3.1]; naturally, Hypothesis 2.1 implies this assumption).…”

“…Among these applications we can find dealloying (corrosion processes), population dynamics, bacterial films, thin film, tumor growth, clustering of mussels and so on (see [7,Introduction] for some detailed references). Moreover, operators of this type with suitable domains involving different boundary conditions arise in a natural way in several contexts as beam analysis and Euler-Bernoulli beam theory (see [8,Introduction] for some related references in this field).…”

Boundary controllability for a degenerate beam equation

“…where 𝛾 is a physical constant depending on the beam, on Young's modulus and on the second moment of inertia. If 𝛾 is a function that depends on the variable x and the external function acts only on the boundary, then we have exactly the equation of (1.1) (see, e.g., Camasta & Fragnelli [1] for other applications of (1.1)).…”