On weakly coupled systems of partial differential equations with different diffusion terms

TL;DR

This paper establishes maximal Schauder regularity and derivative estimates for solutions to nonautonomous weakly coupled elliptic systems with unbounded coefficients, extending regularity results to time-dependent coefficients in bounded continuous function spaces.

Contribution

It provides new regularity results and derivative estimates for weakly coupled elliptic systems with time-dependent coefficients, including unbounded cases, in the space of bounded continuous functions.

Findings

Maximal Schauder regularity for solutions in $C_b( ^d; ^m)$.

Derivative estimates up to third order for solutions.

Continuity properties of evolution operators and semigroups.

Abstract

We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled second-order elliptic operators $\bf{\mathcal A}$, with possibly unbounded coefficients and diffusion and drift terms which vary from equation to equation. We also provide estimates of the spatial derivatives up to the third-order and continuity properties both of the evolution operator ${\bf G}(t,s)$ associated to the Cauchy problem $D_t{\bf u}=\bf{\mathcal A}(t){\bf u}$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$, and, for fixed $\overline t$, of the semigroup ${\bf T}_{\overline t}(\tau)$ associated to the autonomous Cauchy problem $D_{\tau}{\bf u}={\bf{\mathcal A}}(\overline t){\bf u}$ in $C_b(\mathbb{R}^d;\mathbb{R}^m)$. These results allow us to deal with elliptic problems whose coefficients also depend on time.