Qualitative properties of coupled parabolic systems of evolution equations

TL;DR

This paper investigates the well-posedness and qualitative behavior of coupled parabolic evolution equations using functional analysis, introducing a matrix formalism for sesquilinear mappings and applying it to various complex systems.

Contribution

It introduces a novel algebraic formalism for matrices of sesquilinear mappings and applies it to analyze diverse coupled parabolic systems.

Findings

Established well-posedness for multiple coupled parabolic systems.

Derived qualitative properties for systems in neurobiology and boundary conditions.

Unified framework for analyzing complex evolution equations.

Abstract

We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, a heat equation with dynamic boundary conditions, and a general semilinear Hodgkin--Huxley sytem.