UDC 517.941.9+517.946.9

MATHEMATICS

Yu. A. DUBINSKII

ON A CLASS OF HIGH-ORDER DIFFERENTIAL-OPERATOR EQUATIONS

(Presented by Academician I. G. Petrovskii, 11 IV 1969)

This paper studies a certain class of differential-operator (d.-o.) equations (1), called the class of partially hyperbolic d.-o. equations.* The cases \(s=1,2\) have been well studied; surveys of the main results and detailed bibliographies are available, for example, in \((^1,^2)\). Here we consider general boundary-value problems for d.-o. equations (1) of arbitrary order \(s \geqslant 1\).

Let \(H\) be a complex separable Hilbert space, and let \(u(t): I \to H\) be a function of the real variable \(t \in I \subset \mathbb{R}^1\), taking values in \(H\). Let, further, \(A: H \to H\) be a linear operator.

Consider the d.-o. equation

\[ \mathfrak{A}(u) \equiv P_s\left(\frac{d}{dt}\right)u + Au = h(t), \tag{1} \]

where

\[ P_s\left(\frac{d}{dt}\right)u \equiv \sum_{q=0}^{s} a_q u^{(q)}(t), \qquad a_q \in C^1,\qquad u^{(q)} \equiv d^q u/dt^q. \]

Definition. The operator \(\mathfrak{A}(u)\) is called partially hyperbolic if the following conditions are satisfied: 1) if \(s\) is odd, then \(a_s=i\) and the operator \(A\) is self-adjoint; 2) if \(s=2l\), then \(a_s=(-1)^{l-1}\) and the operator \(A\) is self-adjoint and semibounded from below.

Notation. a) \(H(l,0;\gamma)\), \(l \geqslant 0\), denotes the space of functions \(u(t): I \to H\) having finite norm

\[ \|u\|_{l,0;\gamma}^{2}=\int_I\left(\|u^{(l)}\|^{2}+\|u\|^{2}\right)e^{-\gamma t}\,dt, \]

where \(\|u\|\equiv\|u(t)\|\) is the norm in \(H\), and \(\gamma \in \mathbb{R}^1\) is a parameter. b) \(H(s,A;\gamma)\) denotes the space of functions \(u(t): I \to H\) having finite norm

\[ \|u\|_{s,A;\gamma}^{2}\equiv \|u\|_{s,0;\gamma}^{2}+\|Au\|_{0,0;\gamma}^{2}. \]

c) If \(A\) is semibounded from below, then \(\lambda_0\) denotes its first eigenvalue; if \(A\) is not bounded from below, then

\[ \lambda_0=\inf_{u\in D_A}\frac{|(Au,u)|}{\|u\|^2}, \]

where \(D_A\) is the domain of definition of the operator \(Au\).

I. The case of an operator \(A\) with point spectrum

§ 1. D.-o. equations on the entire axis (the case \(I \equiv \mathbb{R}^1\))

Theorem 1. Let the operator \(\mathfrak{A}(u)\) be partially hyperbolic. Then for every \(\gamma \ne 0\) there exists a number \(a(\gamma) \geqslant 0\) such that, for \(\lambda_0 \geqslant a(\gamma)\), equation (1) has a unique solution \(u(t)\in H(s,A;\gamma)\) for every right-

* Boundary-value problems for parabolic and elliptic equations of the form (1) are considered in \((^3)\).

part \(h(t)\in H(1,0;\gamma)\). In this case the estimate is valid

\[ \|u\|_{s,A;\gamma}\leqslant K(\gamma)\|h\|_{1,0;\gamma}, \tag{2} \]

where \(K(\gamma)>0\) is a constant.

Corollary 1. If the operator \(\mathfrak A(u)\) is partially hyperbolic, then equation (1) is normally solvable in the space \(H(s,A;\gamma)\). This means that, for any function \(h(t)\in H(1,0;\gamma)\) subject to a finite number of conditions*, equation (1) is solvable in \(H(s,A;\gamma)\). The homogeneous equation \((h(t)\equiv 0)\) has a finite number of linearly independent solutions.

§ 2. Differential-operator equations on the half-axis (the case \(I=R_{+}\equiv[0,\infty)\)).

Formulation of the problem. For a given function \(h(t)\), it is required to find a solution of equation (1) satisfying, at \(t=0\), the conditions

\[ R_\nu\left(\frac{d}{dt}\right)u\bigg|_{t=0}=0,\qquad \nu=0,\ldots,s-1-k, \tag{3} \]

where \(k=[s/2]\) if \(s\) is odd, and \(k=s/2-1\) if \(s\) is even (problem I); or \(k=[s/2]+1\) for any \(s\) (problem II).

Here

\[ R_\nu\left(\frac{d}{dt}\right)u\equiv \sum_{q=0}^{n_\nu}p_{\nu q}u^{(q)}(t),\qquad p_{\nu q}\in C^1,\quad p_{\nu n_\nu}=1,\quad n_\nu<s. \]

Suppose that the following conditions are satisfied.

Condition B. If \(\lambda>0\), then \(\det\|\mu_j^{n_\nu}(\lambda)\|\ne0\) \((\nu,j=0,1,\ldots,s-1-k)\), where \(\mu_j(\lambda)\) are roots of the equation \(a_s\mu^s+\lambda=0\) such that \(\operatorname{Re}\mu_j(\lambda)\leqslant0\) (for problem I).

Condition C. If \(\lambda>0\), then \(\det\|\mu_j^{n_\nu}(\lambda)\|\ne0\) \((\nu,j=0,\ldots,s-1-k)\), where \(\mu_j(\lambda)\) are roots of the equation \(a_s\mu^s+\lambda=0\) such that \(\operatorname{Re}\mu_j(\lambda)<0\) (for problem II).

Theorem 2. Let \(\gamma>0\), let the operator \(\mathfrak A(u)\) be partially hyperbolic, and let \(\lambda_0\geqslant0\) be sufficiently large \((\lambda_0\geqslant\alpha(\gamma))\). Then, if Condition B is satisfied, for any right-hand side \(h(t)\in H(1,0;\gamma)\) there exists a unique solution \(u(t)\in H(s,A;\gamma)\) of problem I. In this case the estimate (2) is valid.

Theorem 3. Let \(\gamma<0\), let the operator \(\mathfrak A(u)\) be partially hyperbolic, and let \(\lambda_0\geqslant\alpha(\gamma)\). Then, if Condition C is satisfied, for any right-hand side \(h(t)\in H(1,0;\gamma)\) there exists a unique solution \(u(t)\in H(s,A;\gamma)\) of problem II. In this case the estimate (2) is valid.

Corollary 2. Let \(\gamma>0\), let the operator \(\mathfrak A(u)\) be partially hyperbolic, and let Condition B be satisfied. Then problem I is normally solvable, i.e., has a finite-dimensional kernel and is solvable under a finite number of conditions on \(h(t)\). In this case the inequality is valid

\[ \|u\|_{s,A;\gamma}\leqslant K(\gamma)\bigl(\|h\|_{1,0;\gamma}+\|u\|_{0,0;\gamma}\bigr), \tag{4} \]

where \(K(\gamma)>0\) is a constant.

Corollary 3. Let \(\gamma<0\), let the operator \(\mathfrak A(u)\) be partially hyperbolic, and let Condition C be satisfied. Then problem II is normally solvable and inequality (4) is valid.

If \(s\) is even, then the following holds.

Theorem 4. If, for the operator \(\mathfrak A(u)\), where \(A\) is self-adjoint, problem I is normally solvable for every \(\gamma>0\), and problem II for every \(\gamma<0\), then the operator \(A\) is bounded below, i.e., the operator \(\mathfrak A(u)\) is partially hyperbolic.

§ 3. Differential-operator equations on an interval (the case \(I=[0,T]\)).

In this case it is required to find a solution of equation (1) under the following condi—

* These conditions have the character of orthogonality conditions of \(h(t)\) to a finite-dimensional subspace for all \(\gamma\in R^1\), excluding a discrete set. This also applies to [3], where in Theorems 2.1 and 3.1 it is inaccurately stated that \(\gamma\) is arbitrary.

forms:

\[ u(0)=0,\ldots,u^{(l)}(0)=0,\quad u'(T)=0,\ldots,u^{(l-1)}(T)=0,\quad s=2l\vee 1 . \tag{5} \]

(here the second group of conditions in (5) is absent if \(s=1,2\)).

Theorem 5. Let the operator \(\mathfrak A(u)\) be partially hyperbolic and let the number \(\lambda_0\) be sufficiently large. Then for any function \(h(t)\in H(1,0;0)\) the problem (1), (5) has a unique solution \(u(t)\in H(s,A;0)\). Moreover, the inequality

\[ \|u\|_{s,A;0}\le K\|h\|_{1,0;0} \]

holds, where \(K>0\) is a constant.

Corollary 4. If the operator \(\mathfrak A(u)\) is partially hyperbolic, then the problem (1), (5) is normally solvable. Moreover, the inequality

\[ \|u\|_{s,A;0}\le K\bigl(\|h\|_{1,0;0}+\|u\|_{0,0;0}\bigr) \]

holds, where \(K>0\) is a constant.

II. The case of a self-adjoint operator \(A\) with arbitrary spectrum.

Everywhere in what follows \(s=2l\) (and, consequently, \(a_s=(-1)^{l-1}\)) and the operator \(A:H\to H\) is an arbitrary positive self-adjoint operator, i.e. \((Au,u)\ge \lambda_0(u,u)\), \(\lambda_0>0\).

Consider the following polynomial in the real variable \(\tau\):

\[ A(\gamma;\tau)\equiv \operatorname{Re}(\gamma/2-i\tau)\,[P_s(\gamma/2+i\tau)+\lambda_0]. \]

Obviously, for \(\gamma>0\) the polynomial \(A(\gamma;\tau)\) is semibounded below, and for \(\gamma<0\) it is semibounded above, i.e. there exists a constant \(\alpha(\gamma)\ge 0\) such that, for \(\gamma>0\) (\(\gamma<0\)), the polynomial

\[ A_\lambda(\gamma;\tau)\equiv \operatorname{Re}(\gamma/2-i\tau)\,[P_s(\gamma/2+i\tau)+\lambda+\lambda_0]>0 \quad (<0) \]

for any \(\lambda\ge \alpha(\gamma)\).

§ 1. D.-o. equations on the whole axis \(\mathbb R^1\). Consider the equation

\[ \mathfrak A_\lambda(u)\equiv \mathfrak A(u)+\lambda u=h(t),\qquad \lambda\ge \alpha(\gamma). \tag{6} \]

Theorem 6. Let the operator \(\mathfrak A(u)\) be partially hyperbolic and \(\gamma\ne 0\). Then for any right-hand side \(h(t)\in H(1,0;\gamma)\) there exists a unique solution of equation (6), \(u(t)\in H(s,A;\gamma)\). Moreover, the inequality

\[ \|u\|_{s,A;\gamma}\le K(\gamma)\|h\|_{1,0;\gamma} \tag{7} \]

holds, where \(K(\gamma)>0\) is a constant.

§ 2. D.-o. equations on a half-axis. Given a function \(h(t)\), it is required to find a solution of equation (6) under the conditions

\[ u(0)=0,\ldots,u^{(l)}(0)=0 \quad \text{(problem I),} \]

or under the conditions

\[ u'(0)=0,\ldots,u^{(l-1)}(0)=0 \quad \text{(problem II)} \tag{8} \]

(for \(s=2\) the conditions (8) are absent).

Theorem 7. Let the operator \(\mathfrak A(u)\) be partially hyperbolic and \(\gamma>0\) (\(\gamma<0\)). Then for any function \(h(t)\in H(1,0;\gamma)\) there exists a unique solution \(u(t)\in H(s,A;\gamma)\) of problem I (problem II) for equation (6). Moreover, the inequality (7) holds.

§ 3. D.-o. equations on an interval. Given a function \(h(t)\in H(1,0;0)\), it is required to find a solution of equation (6) on the interval \([0,T]\) under the conditions (5).

Theorem 8. If the operator \(\mathfrak A(u)\) is partially hyperbolic, then for any right-hand side \(h(t)\in H(1,0;0)\) problem (6), (5) has a unique solution \(u(t)\in H(s,A;0)\). Moreover, estimate (7) is valid for \(\gamma=0\).

§ 4. Examples. Let \(L(x,D)u\) be an elliptic self-adjoint operator, semibounded from below, in a domain \(G\subset \mathbb R^n\), with a coercive system of boundary conditions. In the examples we restrict ourselves to the case of the cylinder \(Q=\mathbb R_+^1\times G\).

1. Consider an equation of Schrödinger-equation type
   \[ \pm iu' + L(x,D)u = h(t,x). \]

For \(t=0\), either the initial value \(u(0,x)=0\) is prescribed (the mixed problem), or there is no initial condition (the free problem). We have
\[ A(\gamma;\tau)\equiv \gamma/2+\lambda_0, \]
whence it follows that, for \(\lambda_0\ge 0\), the mixed (free) problem is uniquely solvable for any \(\gamma>0\) \((\gamma<0)\).

1. Consider an equation of wave-equation type
   \[ u''+L(x,D)u=h(t,x), \]
   for which the problems studied are either the classical mixed problem or the free problem. We have
   \[ A(\gamma;\tau)\equiv (\gamma/2)(\gamma^2/4+\tau^2+\lambda_0), \]
   therefore, if \(\lambda_0\ge 0\), the mixed (free) problem is uniquely solvable for any \(\gamma>0\) \((\gamma<0)\). If \(\lambda_0<0\), unique solvability holds for
   \[ \gamma>2\sqrt{|\lambda_0|}\quad \bigl(\gamma<-2\sqrt{|\lambda_0|}\bigr) \]
   (cf. \((^4,^5)\)).

2. Consider an equation of the form
   \[ -u^{\mathrm{IV}}+L(x,D)u=h(t,x) \tag{9} \]
   under the conditions \(u(0,x)=0,\ u'(0,x)=0,\ u''(0,x)=0\) (problem I), or under the condition \(u(0,x)=0\) (problem II).

We have
\[ A(\gamma;\tau)\equiv(\gamma/2)(3\tau^4+\gamma^2\tau^2/2-\gamma^4/16+\lambda_0), \]
therefore, for \(\lambda_0>0\), problem I (problem II) for equation (9) is uniquely solvable for
\[ \gamma\in(0,2\sqrt[4]{\lambda_0})\quad \bigl(\gamma\in(-2\sqrt[4]{\lambda_0},0)\bigr). \]
For the remaining \(\gamma\) there is normal solvability. For \(\lambda_0\le 0\), both problems are only normally solvable.

Remark. Let
\[ P_s(\partial/\partial t,\partial/\partial x_1,\ldots,\partial/\partial x_n)u=h(t,x),\quad t\ge 0,\quad x\in\mathbb R^n, \tag{10} \]
be a homogeneous strictly hyperbolic equation of order \(s\). As is known (see \((^6–^8)\)), for equation (9) in Sobolev–Slobodetskii spaces with weight \(e^{-\gamma t}\), for \(\gamma>0\), the Cauchy problem is correctly posed. If, however, equation (10) is considered in Sobolev–Slobodetskii spaces with weight \(e^{-\gamma t}\) for \(\gamma<0\), then the problem without initial conditions is the correct one.

Moscow Power Engineering Institute

Received
27 III 1969

REFERENCES

\(^1\) J. Lions, Equations différentielles opérationnelles et problèmes aux limites, Berlin, 1962.
\(^2\) S. G. Krein, Linear Differential Equations in Banach Space, Moscow, 1967.
\(^3\) Yu. A. Dubinskii, Mat. Sb., 79 (121), 1 (1969).
\(^4\) O. A. Ladyzhenskaya, The Mixed Problem for Hyperbolic Equations, Moscow, 1953.
\(^5\) O. A. Ladyzhenskaya, Mat. Sb., 39 (81), 4, 383 (1956).
\(^6\) I. G. Petrovskii, Mat. Sb., 2 (44), 815 (1937).
\(^7\) J. Leray, Hyperbolic Differential Equations, Princeton, 1952.
\(^8\) L. Gårding, The Cauchy Problem for Hyperbolic Equations, IL, 1962.