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On this paper, for an arbitrary order operator-differential equation with the weight $e^{\frac{-\alpha t}{2}}, \alpha \in (-\infty ,+ \infty )$, in the space $W^{n+m}_{2}(R_{+};H)$, we attain sufficient conditions for the well-posedness of a regular solvable of the boundary value problem. These conditions are provided only by the operator coefficients of the investigated equation where the leading part of the equation has multiple characteristics. We prove the connection between the lower bound of the spectrum of the higher-order differential operator in the main part and the exponential weight and also obtain estimations of the norms of operator intermediate derivatives. We apply the results of this paper to a mixed problem for higher-order partial differential equations (HOPDs).

Sufficient conditions for regular solvability of an arbitrary order operator-differential equation with initial-boundary conditions