On Coercive Solvability of Parabolic Equations with Variable Operator
In a Banach space E, the Cauchy problem vt(t)+ A(t)v(t)= f (t) (0 >t > 1), v(0) = v<sub>0</sub> is considered for a di erential equation with linear strongly positive operator A(t) such that its domain D = D(A(t)) is everywhere dense in E independently o t and A(t) generates an analytic semigroup exp{-sA(t)} (s < 0). Under some natural assumptions on A(t), we establish coercive solvability of the Cauchy problem in the Banach space C<sub>0</sub><sup>β,γ</sup> (E). We prove a stronger estimate of the solution compared to estimates known earlier, using weaker restrictions on f(t) and v<sub>0</sub>.