On coupled systems of PDEs with unbounded coefficients

2018/10/09 by Angiuli, Luciana, Lorenzi, Luca
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.04097

Abstract

We study the Cauchy problem associated to parabolic systems of the form D_t\boldsymbolu=\boldsymbol\mathcal A(t)\boldsymbol u in C_b(ℝ^d;ℝ^m), the space of continuous and bounded functions \boldsymbolf:ℝ^d→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 \boldsymbolG(t,s) which governs the problem in C_b(ℝ^d;ℝ^m) proving its positivity. The compactness of \boldsymbolG(t,s) in C_b(ℝ^d;ℝ^m) and some of its consequences are also studied. Finally, we extend the evolution operator \boldsymbolG(t,s) to the L^p- spaces related to the so called "evolution system of measures" and we provide conditions for the compactness of \boldsymbolG(t,s) in this setting.

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