On coupled systems of PDEs with unbounded coefficients

TL;DR

This paper investigates the existence, uniqueness, and properties of evolution operators for coupled parabolic PDE systems with unbounded coefficients, extending analysis to $L^p$ spaces and studying compactness.

Contribution

It establishes the existence, uniqueness, positivity, and compactness of the evolution operator for coupled PDE systems with unbounded coefficients, including extensions to $L^p$ spaces.

Findings

Proved existence and uniqueness of the evolution operator in $C_b$.

Established positivity and compactness of the evolution operator.

Extended the operator to $L^p$ spaces with conditions for compactness.

Abstract

We study the Cauchy problem associated to parabolic systems of the form $D_t\boldsymbol{u}=\boldsymbol{\mathcal A}(t)\boldsymbol u$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$, the space of continuous and bounded functions $\boldsymbol{f}:\mathbb{R}^d\to\mathbb{R}^m$. Here $\boldsymbol{\mathcal A}(t)$ is a weakly coupled elliptic operator acting on vector-valued functions, having diffusion and drift coefficients which change from equation to equation. We prove existence and uniqueness of the evolution operator $\boldsymbol{G}(t,s)$ which governs the problem in $C_b(\mathbb{R}^d;\mathbb{R}^m)$ proving its positivity. The compactness of $\boldsymbol{G}(t,s)$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$ and some of its consequences are also studied. Finally, we extend the evolution operator $\boldsymbol{G}(t,s)$ to the $L^p$- spaces related to the so called "evolution system of measures" and we provide conditions for the compactness of $\boldsymbol{G}(t,s)$ in this setting.