$L^p$-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

Abstract

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb{R}}^d;{\mathbb{R}}^m)$, $(d,m\ge 1)$ with $p\in [1,+\infty )$. Sufficient conditions for the associated evolution operator $\mathbf{G}(t,s)$ in $C_b({\mathbb{R}}^d;{\mathbb{R}}^m)$ to extend to a strongly continuous operator in $L^p({\mathbb{R}}^d;{\mathbb{R}}^m)$ are given. Some $L^p$-$L^q$ estimates are also established together with $L^p$ gradient estimates.