On Coercive Solvability of Parabolic Equations with Variable Operators

Abstract

In a Banach space E, the Cauchy problem

\( \upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\kern1em \upsilon (0)={\upsilon}_0, \)

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 \( {C}_0^{\beta, \upgamma} \) (E). We prove a stronger estimate for the solution compared with estimates known earlier, using weaker restrictions on f(t) and v₀.