Asymptotics of semigroups generated by operator matrices

TL;DR

This paper surveys known results on operator semigroups generated by operator matrices with applications to evolution equations with dynamic boundary conditions, providing new ill-posedness results for heat equations with Wentzell boundary conditions.

Contribution

It offers a comprehensive survey of operator semigroup theory for operator matrices and introduces new ill-posedness results for heat equations with general Wentzell boundary conditions.

Findings

Characterization of well- and ill-posedness for certain evolution equations

New ill-posedness results for heat equations with Wentzell boundary conditions

Application of abstract operator semigroup results to boundary value problems

Abstract

We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our ill-posedness results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vitillaro-V\'azquez.