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# Dynamics of Partial Differential Equations

## Volume 17 (2020)

### Number 2

### On coupled systems of PDEs with unbounded coefficients

Pages: 129 – 163

DOI: https://dx.doi.org/10.4310/DPDE.2020.v17.n2.a3

#### Authors

#### Abstract

We study the Cauchy problem associated with parabolic systems of the form $D_t u = \mathcal{A} (t) u$ in $C_b (\mathbb{R}^d ; \mathbb{R}^m)$, the space of continuous and bounded functions $f : \mathbb{R}^d \to \mathbb{R}^m)$. Here $\mathcal{A} (t)$ is a coupled nonautonomous 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 $G(t, s)$ which governs the problem in $C_b (\mathbb{R}^d ; \mathbb{R}^m)$ and its positivity. The compactness of $G(t, s)$ in $C_b (\mathbb{R}^d ; \mathbb{R}^m)$ and some of its consequences are also studied. Finally, we extend the evolution operator $G(t, s)$ to the $L^p$-spaces related to the so called “evolution system of measures” and we provide conditions for the compactness of $G(t, s)$ in this setting.

#### Keywords

nonautonomous parabolic systems, unbounded coefficients, evolution operators, compactness, invariant subspaces, evolution systems of invariant measures

#### 2010 Mathematics Subject Classification

Primary 35K40. Secondary 35K45, 37L40, 46B50, 47A15.

The authors are members of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM). Work partially supported by the INdAM-GNAMPA Project 2017 “Equazioni e sistemi di equazioni di Kolmogorov in dimensione finita e non”.

Received 25 February 2019

Published 24 February 2020