The Biharmonic Heat Equation with General Dynamic Boundary Conditions

TL;DR

This paper studies the biharmonic heat equation with dynamic boundary conditions involving the bi-Laplace-Beltrami operator, analyzing its spectral properties and semigroup generation.

Contribution

It introduces a novel analysis of a fourth-order parabolic PDE with dynamic boundary conditions, establishing key spectral and semigroup properties.

Findings

Proves self-adjointness of the associated operator

Shows compactness of the resolvent and spectral properties

Analyzes the generated semigroup's analyticity, compactness, and positivity

Abstract

In this work, we initiate the study of the biharmonic heat equation in a spatial bounded domain subject to dynamic boundary conditions involving the bi-Laplace-Beltrami operator on the boundary. The boundary heat equation is coupled to the interior one via a normal derivative term. By combining the sesquilinear form method and semigroup theory, we establish substantial qualitative properties of the fourth-order parabolic equation; in particular, the self-adjointness of the associated operator, compactness of its resolvent, and further spectral properties. We also investigate the generation of a $C_0$-semigroup and analyze its main properties: analyticity, compactness, eventual positivity, and eventual $L^\infty$-contractivity.