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

On this paper, for an arbitrary order operator-differential equation with the weight e−αt2,α∈(−∞,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$e^{\frac{-\alpha t}{2}}, \alpha \in (-\infty ,+ \infty )$\end{document}, in the space W2n+m(R+;H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^{n+m}_{2}(R_{+};H)$\end{document}, 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).


Introduction
The theory of initial-boundary value problems of operator-differential equations in a Banach or Hilbert space is of great value and secures the possibility of looking at ordinary and partial differential operators [27]. It is worth mentioning that the principal parts of the investigated equations have multiple characteristics and thus appear in applications, for instance, by modeling the stability of the plates from the plastic and in particular, in the dynamics problems of arches and rings [25,26]. The solvability of initial boundary value problems for higher-order operator-differential equations has been researched by many authors, for example, A. A. Gasymov, S. S. Mirzoev, V. I. Gorbachuk, M. L. Gorbachuk, S. Yakubov, V. N. Pilipchuk, and their followers [7,11,[22][23][24][25]27].

Definition 1
If for any f (t) ∈ L 2,α (R + ; H), there exists a vector function u(t) ∈ W n+m 2,α (R + ; H) that satisfies (1) almost everywhere in R + , then it is called a regular solution of (1).

Definition 2
If for any function f (t) ∈ L 2,α (R + ; H), there exists a regular solution of (1) satisfying the boundary conditions (2) in the sense that [24] lim t→0 The studies of the last 60 years have enhanced the theory of operator-differential equations with considerable results. The theory of initial-value problems of operator-differential equations in a Banach or Hilbert space is useful because it facilitates studying equations of parabolic and elliptic differential operators with initial-boundary conditions, possibly, looking at ordinary and partial differential operators. Nowadays many papers concerning the study of initial value problems of the operator-differential equations in Banach spaces have been published. In both semiaxis and finite interval, second-order operatordifferential equations with zero weight exponential are studied [3,5,6,23,24]. Gasymov [8][9][10] analyzed both the solvability of operator-differential equations and the multiple completeness of some eigen-and associated vectors of corresponding operator pencils. His works are the most valuable as they motivated many papers, including the present one and others to be mentioned further. Moreover, the solvability of operatordifferential equations in Hilbert spaces with exponential weight has been extensively studied. Second-, third-, and fourth-order operator-differential equations with multiple characteristics with exponential weight have been studied on the semiaxis and the whole axis [2,22]. Moreover, general higher-order operator-differential equations with multiple characteristic in a Sobolev-type space with exponential weight have not been studied yet. In the present paper, we formulate sufficient conditions for the initial-boundary value problem to be regularly solvable.

Main results
From the theorem on intermediate derivatives [7,19] we have that if u(t) ∈ W n+m 2,α (R + ; H), then and the following inequalities are valid: In the present paper, for any natural number n, r 1 = r 2 = r 3 = · · · = r n = 1, and m = 1, we obtain d s u(0) dt s = 0, s = 0, n -1.
Using the Fourier transforms for the equation whereṽ(ξ ),g(ξ ) are the Fourier transforms of the functions v(t), g(t), respectively, and moreover Hence Then the integral operator is Taking iξ = ω, we have and for t > s, we get Similarly, for t < s, we have Using the spectral expansion of operator A [μ ∈ σ (A)], we get where R n = 2 (-n) 2 (k) (k)! and T n = 2 (-n) .
For v 0 (t), from equation (6) and so on, Then and v 0 (t) = - (iii) If m = 1, n = 4, then and v 0 (t) = - This case has not been researched so far.
Before we formulate exact conditions on regular solvability of problem (4)-(5), expressed only by its operator coefficients, we must estimate the norms of intermediate derivative operators participating in the second part of equation (4). It follows from Theorem 3 that the norm P 0 u L 2 (R + ;H) is equivalent to the norm u W n+1 2 (R + ;H) in the space W n+1 2 (R + ; H). Therefore by the norm P 0 u L 2 (R + ;H) the theorem on intermediate derivatives is valid as well.

Theorem 4 ([20])
Let the operators A j A -j , j = 1, n + 1, be bounded on H. Then in the case A j = 0 the operator P 1 is bounded from the space W n+1 2,α (R + ; H) to L 2,α (R + ; H).
Proof Since u(t) ∈ w n+1 2,α (R + ; H), from the theorem on intermediate derivatives [16,19] we have The theorem is proved. From Theorems 3 and 4 we get the following lemma.
Lemma 5 Let A j A -j , j = 1, n + 1, be bounded operators on H. Then in the case A j = 0 the operator P from the space W n+1 2 (R + ; H) to L 2 (R + ; H) is bounded [17].
Consider the following polynomial operator pencils depending on the real parameter β: Let us clarify the study of naturally arising pencils (9). Obviously, for u(x) ∈ W n+1 2 (R + ; H), we have that for estimating n j , j = 1, n, it is necessary to study some properties of pencils.
This case has not been researched yet.
. Then for any u(t) ∈ W n+1 2 (R + ; H), the following relation holds: on the values of the numbers n j , j = 1, n.
Theorem 3 implies that the norms W n+1 2 (R + ; H) and P 0 u L 2 (R + ;H) are equivalent on W n+1 2 (R + ; H). Then it follows from the theorem of intermediate derivatives that the following numbers are finite: A n-j+1 d j dt j u L 2 (R + ;H) P 0 u L 2 (R + ;H) , j = 1, n. Now let us calculate n j .

Proof
Since A is a self-adjoint positive operator, from the spectral theory we have
Hence the mixed partial differential equation has a unique solution.

Conclusion
In a Sobolev-type space with exponential weight, we found a solution of the initialboundary value problem of (n + 1)th-order operator-differential equation in the case that the second part equals zero. We obtained definite conditions for problem (4)-(5) to be regularly solvable; these conditions rely on the operator coefficients, the lower bound of the spectrum, and the weight exponent. The estimates of the norms of the intermediate derivatives of the differential operators in the substantial part of the given equation are provided and, in a similar way, for the second part of the above-mentioned equation.
We used the results of this paper to establish an application example of the initialboundary value problem (10)-(12) for mixed partial differential equations. The wellposedness of problem (4)-(5) is also proved using the polynomial operator pencils.