On coupled systems of PDEs with unbounded coefficients

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.

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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