On Coercive Solvability of Parabolic Equations with Variable Operators

In a Banach space E, the Cauchy problem υ′(t)+A(t)υ(t)=f(t)(0≤t≤1),υ(0)=υ0, is considered for a differential equation with linear strongly positive operator A(t) such that its domain D = D(A(t)) does not depend on t and is everywhere dense in E and A(t) generates an analytic semigroup exp{−sA(t)}(s ≥ 0). Under natural assumptions on A(t), we prove the coercive solvability of the Cauchy problem in the Banach space C0β,γ (E). We prove a stronger estimate for the solution compared with estimates known earlier, using weaker restrictions on f(t) and v 0 . © 2019, Springer Science+Business Media, LLC, part of Springer Nature.