Lp-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space Lp(Rd;Rm)Lp(Rd;Rm)(d,m≥1)(d,m≥1) with p∈[1,+∞)p∈[1,+∞). Sufficient conditions for the associated evolution operator G(t,s)G(t,s) in Cb(Rd;Rm)Cb(Rd;Rm) to extend to a strongly continuous operator in Lp(Rd;Rm)Lp(Rd;Rm) are given. Some LpLp–LqLq estimates are also established together with LpLp gradient estimates.