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Stability and Boundedness of Solutions to Nonautonomous Parabolic Integrodifferential Equations.

1. Introduction and Statement of the Main Result

This paper is devoted to stability and boundedness of solutions to parabolic integrodifferential equations, that is, equations containing the first derivative in time, integral operators, and partial derivatives in spatial variables. Such equations play an essential role in numerous applications, in particular, in the transport theory [1], continuous mechanics [2], and radiation theory [3]. For other applications see [4].

The literature on the asymptotic properties of integrodifferential equations is rather rich, but mainly ordinary (linear and nonlinear) equations, that is, equations without partial derivatives, were investigated; compare [5-9] and references given therein. For important stability results on stochastic partial differential equations see the papers [10-12].

The parabolic autonomous integrodifferential equations are investigated considerably less than the ordinary ones. For the recent papers on stability and the asymptotic behaviour of solutions to autonomous parabolic integrodifferential equations see [13-16] and references therein.

Despite many important applications the stability properties of solutions of nonautonomous integrodifferential equations have not been not investigated. The motivation of the present paper is to particularly fill a gap between the developed theory for ordinary integrodifferential equations and almost nonexistence theory for nonautonomous parabolic integrodifferential equations.

We obtain the main result of the paper for differential-operator equation (1) in a Hilbert space. Based on that result we give explicit exponential stability conditions for the integrodifferential equations.

Let H be a Hilbert space with a scalar product (x, x), the norm [parallel] x [parallel] = [square root of (x, x)], and unit operator I. All the considered operators are assumed to be linear. For an operator A, [A.sup.*] is the adjoint one, [sigma](A) is the spectrum, [alpha](A) = sup Re [sigma](A), and Dom(A) denotes the domain.

Consider the equation

[??] = Su + B(t)u + F (t) (u = u (t), t > 0; [??] = du/dt), (1)

where S is a closed constant operator in H with a dense domain

[mathematical expression not reproducible]; (2)

B(t) is an operator uniformly bounded on [0, [infinity]), having a strong derivative uniformly bounded on [0, [infinity]) and commuting with S; F(x) : [0, [infinity]) [right arrow] Dom(S) satisfies the conditions pointed below.

An important example of (1) is the boundary values problem

[mathematical expression not reproducible], (3)

[mathematical expression not reproducible], (4)

wherec(x, x) : [0, [infinity]) x [a,b] [right arrow] R, k(x, x, x) : [0, [infinity]) x [[a,b].sup.2] [right arrow] R and f(x, x) : [0, [infinity]) x [a, b] x [0,1] [right arrow] R are given functions and u(x, x, x) is unknown. In our reasonings below, instead of [0,1] and [a, b], one can consider closed bounded Euclidean sets and more general differential operators.

In the present paper we suggest the conditions providing stability and boundedness of solutions to (1) with slowly varying operator B(t).

Certainly, (1) can be considered in some space as the equation [??] = A(t)u with an unbounded variable linear operator A(t). This identification which is a common device in the theory of partial differential equations when passing from a parabolic equation to an abstract evolution equation turns out to be useful also here. Observe that A(t) in the considered case has a special form: it is the sum of operators S and B(t). Besides, according to (3), B(t) has a special structure. These facts enable us to use the information about the coefficients more completely than the theory of differential equations containing an arbitrary operator A(t).

A solution to (1) for given [u.sub.0] [member of] Dom(S) is a function u : [0, [infinity]) [right arrow] Dom(S) having abounded measurable strong derivative and satisfying u(0) = [u.sub.0].

In particular, we will consider the homogeneous equation

[??] = Su + B(t)u (t > 0). (5)

Equation (5) is said to be exponentially stable, if there are constants M [greater than or equal to] 1, [member of] > 0, such that [parallel]u(t)[parallel] [less than or equal to] Mexp[- [member of] t] [parallel]u(0)[parallel] (t [greater than or equal to] 0) for any solution of (1). Condition (2) implies that S generates a strongly continuous semigroup [e.sup.St]; compare [17, Section I.4.4]. Since B(t) is bounded and commutes with S we can assert that (1) has solutions for any [u.sub.0] [member of] Dom(S).

We will assume that, for each t > 0, the operator B(t) + y(S)I is stable (Hurwitzian); namely,

q([tau]) := 2 [[integral].sup.[infinity].sub.0] [[parallel][e.sup.(B([tau])+[gamma](S)I)s][parallel].sup.2] ds < [infinity]. (6)

Now we are in a position to formulate the main result of this paper.

Theorem 1. Let conditions (2) and (6) and

[mathematical expression not reproducible] (7)

hold. Then (5) is exponentially stable. If, in addition, [parallel]F(t)[parallel] is bounded and measurable on [0, [infinity]), then any solution of (1) is bounded on [0, [infinity]).

2. Proof of Theorem 1

Put C = S - [gamma](S)I and A(t) = B(t) + [gamma](S)I. Let V(t) be the Cauchy operator to the equation

[??](t) = A(t) v(t) (t [greater than or equal to] 0). (8)

That is, V(t)v(0) = v(t) for any solution of (8). Taking

u (t) = [e.sup.Ct] V (t) [u.sub.0] ([u.sub.0] [member of] Dom (S)) (9)

we have

[??] = (C + A(t))u = (S + B(t))u. (10)

Consequently, [e.sup.tC] V(t)[u.sub.0] really is a solution to (5). Since

[mathematical expression not reproducible], (11)

we have [parallel][e.sup.tC][parallel] [less than or equal to] 1, t [greater than or equal to] 0; compare [17, Theorem I.4.2]. Thus, [parallel]u(t)[parallel] [less than or equal to] [parallel]V(t)[u.sub.0][parallel]. So we have proved the following result.

Lemma 2. Under the hypothesis of Theorem 1, (5) is exponentially stable, provided (8) is exponentially stable.

Furthermore, recall that the equation

[A.sup.*.sub.0] Y + Y[A.sub.0] = E (12)

with given constant bounded stable operator [A.sub.0] (i.e., [alpha]([A.sub.0]) < 0) and a constant bounded operator E has a solution Y which is represented as

[mathematical expression not reproducible]. (13)

Compare [18, Section I.4.4]. Consequently, due to (2), the operator

[mathematical expression not reproducible] (14)

is a unique solution of the equation

[A.sup.*] (t)Q(t) + Q(t)A(t) = -2I (t [greater than or equal to] 0), (15)

[parallel]Q(t)[parallel] [less than or equal to] q(t). (16)

Lemma 3. Let condition (6) hold. Then Q(t) is differentiable and [parallel]Q'(t)[parallel] [less than or equal to] [q.sup.2] (t)[parallel]A'(t)[parallel].

Proof. Differentiating (15) we have

[A.sup.*] (t)Q' (t) + Q' (t)A(t) = -([A.sup.*] (t))' Q(t)-Q(t)A' (t) (t [greater than or equal to] 0). (17)

Hence, due to (15),

[mathematical expression not reproducible]. (18)

Thus,

[mathematical expression not reproducible]. (19)

Now (16) yields the result.

Lemma 4. Let

[mathematical expression not reproducible]. (20)

Then (Q(t)u(t), u(t)) [less than or equal to] (Q(0)u(0),u(0)) (t [greater than or equal to] 0), for a solution of (8).

Proof. Multiplying (8) by Q(t) and doing the scalar product, we can write (Q(t)u'(t),u(t)) = (Q(t)A(t)u(t),u(t)). Since

[mathematical expression not reproducible], (21)

it can be written as

[mathematical expression not reproducible]. (22)

Hence, condition (20) implies

d/dt (Q (t) u(t), u (t)) [less than or equal to] (-2 + [parallel]Q' (t)[parallel]) (u (t), u (t)) < 0. (23)

This proves the result.

Furthermore, for a stable operator [mathematical expression not reproducible]. Then [mathematical expression not reproducible] and

[mathematical expression not reproducible]. (24)

Hence

d([y.sub.1] (t), [y.sub.1] (t))/dt [greater than or equal to] [lambda] ([A.sub.0] + [A.sup.*.sub.0]) ([y.sub.1] (t), [y.sub.1] (t)) (25)

and therefore

[mathematical expression not reproducible], (26)

where [lambda]([A.sub.0] + [A.sup.*.sub.0]) is the smallest eigenvalue of [A.sub.0] + [A.sup.*.sub.0]. Recall that [A.sub.0] is stable, so [lambda]([A.sub.0] + [A.sup.*.sub.0]) < 0. Then due to (13) and (26) with E = -2I we get

[mathematical expression not reproducible]. (27)

Hence, for any continuous function [u.sub.1] : [0, [infinity]) [right arrow] H, we have

(Q(t) [u.sub.1] (t), [u.sub.1] (t)) [greater than or equal to] 2 [[parallel][u.sub.1] (t)[parallel].sup.2] [[absolute value of [lambda](A(t) + [A.sup.*] (t))].sup.-1]. (28)

Now the previous lemma implies

(u (t), u (t)) [less than or equal to] [absolute value of (A (t) + [A.sup.*] (t))] (Q (0) u(0),u (0)) (t [greater than or equal to] 0). (29)

But [absolute value of [lambda](A(t) + [A.sup.*] (t))] is uniformly bounded and therefore all the solutions of (8) are uniformly bounded (i.e., (8) is Lyapunov stable). Furthermore, substitute into (8)

u (t) = [u.sub.[member of]] (t) [e.sup.-[member of]t] ([member of] > 0). (30)

Then

[[??].sub.[member of]] (t) = (A(t) + [member of]I) [u.sub.[member of]] (t). (31)

Applying our above arguments to (31) can assert that (31) with small enough [member of] > 0 is Lyapunov stable. So, due to (30), (8) is exponentially stable, provided (20) holds. Now Lemma 3 implies the following.

Lemma 5. Let [sup.sub.t [greater than or equal to] 0] [q.sup.2](t)[parallel]B'(t)[parallel] < 2. Then (8) is exponentially stable.

Proof of Theorem 1. The exponential stability of (5) immediately follows from Lemmas 2 and 5, and the equality A'(t) = B'(t). The rest of the proof is obvious.

3. Equations on a Tensor Product of Hilbert Spaces

Let [E.sub.j] (j = 1,2) be separable Hilbert spaces with scalar products [<x, x>.sub.j], the unit operators [I.sub.j], and the norms [[parallel] x [parallel].sub.j] = [square root of [<x, x>.sub.j]. Let H = [E.sub.1] [cross product] [E.sub.2] be the tensor product of [E.sub.1] and [E.sub.2]. This means that H is a collection of all formal sums of the form

u = [summation over (j)] [y.sub.j] [cross product] [h.sub.j] ([y.sub.j] [member of] [E.sub.1], [h.sub.j] [member of] [E.sub.2]) (32)

with the understanding that

[mathematical expression not reproducible]. (33)

Here y, [y.sub.1] [member of] [E.sub.1], h, [h.sub.1] [member of] [E.sub.2], and [lambda] is a number. The scalar product in H is defined by

[mathematical expression not reproducible] (34)

and the norm [parallel] x [parallel] [equivalent to] [[parallel] x [parallel].sub.H] := [square root of [(x, x).sub.H]]. The closure of H in the norm [[parallel] x [parallel].sub.H] is a Hilbert space; compare [19]. It is again denoted by H. In addition, the unit operator I = [I.sub.H] in H equals [I.sub.1] [cross product] [I.sub.2].

Furthermore, for a Hilbert-Schmidt operator C in [E.sub.1], [N.sub.2](C) denotes the Hilbert-Schmidt norm: [N.sub.2](C) = [square root of trace [CC.sup.*]]. Let S0 be a closed constant operator in [E.sub.2] with a dense domain

[mathematical expression not reproducible]. (35)

Let [B.sub.0](t) be a linear operator in [E.sub.1] uniformly bounded on [0, [infinity]), having a strong derivative uniformly bounded on [0, [infinity]). Put B(t) = [B.sub.0](t) [cross product] [I.sub.2], S = [I.sub.1] [cross product] [S.sub.0] and assume that, for each [tau] [greater than or equal to] 0, the operator [B.sub.0](t) + [gamma]([S.sub.0])[I.sub.1] is stable:

[mathematical expression not reproducible]. (36)

Let

[mathematical expression not reproducible]. (37)

Then condition (7) holds and, therefore, (5) is exponentially stable. If, in addition, [parallel]F(t)[parallel] is bounded on [0, [infinity]), then any solution of (1) is bounded on [0, [infinity]).

Furthermore, from (36), it follows that [alpha]([B.sub.0]([tau])) + [gamma]([S.sub.0]) < 0. Assume that

[B.sub.0] (t) - [B.sup.*.sub.0] (t) is a Hilbert-Schmidt operator [for all]t [greater than or equal to] 0, (38)

and put

[mathematical expression not reproducible], (39)

where [[lambda].sub.k]([B.sub.0](t)) are nonreal eigenvalues of [B.sub.0](t) taken with their multiplicities. Clearly,

g(t) [less than or equal to] [N.sub.2] ([B.sub.0] (t) - [B.sup.*.sub.0] (t))/[square root 2]. (40)

Due to [20, Example 7.10.3],

[mathematical expression not reproducible]. (41)

So

[mathematical expression not reproducible]. (42)

Hence [q.sub.0]([tau]) [less than or equal to] [psi]([tau], [gamma]([S.sub.0])),where

[mathematical expression not reproducible]. (43)

This inequality, (37), and Theorem 1 imply the following.

Corollary 6. Let conditions (35) and

[mathematical expression not reproducible] (44)

hold. Then (5) is exponentially stable.

4. Example

Put [OMEGA] = [a,b] x [0,1]. In this section [E.sub.1] = [L.sup.2](a,b), [E.sub.2] = [L.sup.2] (0,1), and H = [E.sub.1] [cross product] [E.sub.2] = [L.sup.2]([OMEGA]) are the Hilbert spaces of real functions with usual scalar products.

Consider problem (3), (4), assuming that, for almost all x, s [member of] [0,1], c(t, x) and k(t, x, s) have bounded measurable derivatives [c'.sub.t] (t, x) and [k'.sub.t] (t, x, s). In addition, the operators [B.sub.0](t) and [B'.sub.0](t) defined in [L.sup.2](a, b) by

[mathematical expression not reproducible], (45)

respectively, are assumed to be bounded uniformly in t [member of] [0, [infinity]). In addition,

[mathematical expression not reproducible].

Let [S.sub.0] = [d.sup.2]/d[x.sup.2] with the domain

[mathematical expression not reproducible]. (47)

Then [S.sub.0] is self-adjoint with the eigenvalues -[pi][k.sup.2] (k = 1,2, ...). So [gamma]([S.sub.0]) = -[[pi].sup.2]. Assume that

[mathematical expression not reproducible] (48)

and put

[mathematical expression not reproducible]. (49)

Now Corollary 6 implies the following.

Corollary 7. Let conditions (48) and

[mathematical expression not reproducible] (50)

hold. Then the equation

[mathematical expression not reproducible] (51)

with condition (4) is exponentially stable. If, in addition, [mathematical expression not reproducible] is uniformly bounded on [0, [infinity]), then any solution of (3) is uniformly bounded on [0, [infinity]) in the norm of [L.sup.2] ([OMEGA]).

To estimate [alpha]([B.sub.0](t)) one can apply various bounds for spectra of integral operators. For instance, consider the equation

[mathematical expression not reproducible] (52)

with condition (4). So [B.sub.0](t)w(x) = c(t, x)w(x) + (V(t)w)(x), where

[mathematical expression not reproducible]. (53)

It is simple to check that in this case g(t) = [N.sub.2](V(t)) and [alpha]([B.sub.0](t)) = {c(t,x) : x [member of] [a, b]}. Thus [alpha]([B.sub.0](t)) = [sup.sub.x [member of] [0,1]] c(x,t) and

[mathematical expression not reproducible], (54)

provided

[mathematical expression not reproducible]. (55)

Now we can directly apply the previous corollary. The general case of (51) can be considered as a perturbation of (52).

5. Conclusion

We have established the explicit stability test for linear parabolic integrodifferential equations in the case of slow varying in time coefficients. Stability of such equations has not been investigated in the available literature. As the example shows, the test is simply applicable and enables us to avoid the construction of the Lyapunov functionals in appropriate situations.

http://dx.doi.org/10.1155/2016/2468458

Competing Interests

The author declares that they have no competing interests.

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Michael Gil'

Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel

Correspondence should be addressed to Michael Gil'; gilmi@bezeqint.net

Received 20 January 2016; Revised 24 April 2016; Accepted 3 May 2016

Academic Editor: Quanxin Zhu
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