Degenerate fourth order parabolic equations with Neumann boundary conditions

TL;DR

This paper investigates the generation and well-posedness of degenerate fourth order parabolic equations with Neumann boundary conditions, where the operator's degeneracy depends on a function that vanishes at some point.

Contribution

It introduces new results on the generation property and well-posedness for degenerate fourth order operators with boundary conditions, considering degeneracy depending on a variable function.

Findings

Established generation property for degenerate fourth order operators

Proved well-posedness of associated parabolic equations

Analyzed effects of operator degeneracy on solution behavior

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 $a: [0,1] \rightarrow \R_+$ that degenerates somewhere in the interval.