Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$begin{gathered} ( - 1)frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + sumlimits_{k = 0}^{m - 1} {frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)frac{{d^k u}}{{dt^k }} + sumlimits_{k = 1}^m {frac{{d^k }}{{dt^k }}} A_{2k} (t)frac{{d^k u}}{{dt^k }} + lambda _m A_0 (t)u = f, hfill t in ]0,t[,lambda _m geqslant 1, hfill {{d^i u} mathord{left/ {vphantom {{d^i u} {dt^i }}} right. kern-nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} mathord{left/ {vphantom {{d^j u} {dt^j }}} right. kern-nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., hfill end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$begin{gathered} mathop {lim}limits_{varepsilon to 0} Re(A_0 (t)B_varepsilon ^{ - 1} (t)(B_varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H geqslant c(A(t)u,u)_H hfill forall u in D(A_0 (t)),c > 0, hfill end{gathered} $$

, where the smoothing operators are defined by

$$B_varepsilon ^{ - 1} (t) = (I - varepsilon B(t))^{ - 1} ,(B_varepsilon ^{ - 1} (t)) * = (I - varepsilon B^ * (t))^{ - 1} ,varepsilon > 0.$$

.