Stability and Boundedness of Solutions to Nonautonomous Parabolic Integrodifferential Equations

Abstract

We consider a class of linear nonautonomous parabolic integrodifferential equations. We will assume that the coefficients are slowly varying in time. Conditions for the boundedness and stability of solutions to the considered equations are suggested. Our results are based on a combined usage of the recent norm estimates for operator functions and theory of equations on the tensor product of Hilbert spaces.

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 be a Hilbert space with a scalar product , the norm , and unit operator . All the considered operators are assumed to be linear. For an operator , is the adjoint one, is the spectrum, , and denotes the domain.

Consider the equation where is a closed constant operator in with a dense domain is an operator uniformly bounded on , having a strong derivative uniformly bounded on and commuting with ; satisfies the conditions pointed below.

An important example of (1) is the boundary values problem where , and are given functions and is unknown. In our reasonings below, instead of and , 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 .

Certainly, (1) can be considered in some space as the equation with an unbounded variable linear operator . 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 in the considered case has a special form: it is the sum of operators and . Besides, according to (3), 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 solution to (1) for given is a function having a bounded measurable strong derivative and satisfying .

In particular, we will consider the homogeneous equation Equation (5) is said to be exponentially stable, if there are constants , such that    for any solution of (1). Condition (2) implies that generates a strongly continuous semigroup ; compare [17, Section I.4.4]. Since is bounded and commutes with we can assert that (1) has solutions for any .

We will assume that, for each , the operator is stable (Hurwitzian); namely, Now we are in a position to formulate the main result of this paper.

Theorem 1. Let conditions (2) and (6) and hold. Then (5) is exponentially stable. If, in addition, is bounded and measurable on , then any solution of (1) is bounded on .

2. Proof of Theorem 1

Put and . Let be the Cauchy operator to the equation That is, for any solution of (8). Taking we have Consequently, really is a solution to (5). Since we have ; compare [17, Theorem I.4.2]. Thus, . 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 with given constant bounded stable operator (i.e.,  ) and a constant bounded operator has a solution which is represented as Compare [18, Section I.4.4]. Consequently, due to (2), the operator is a unique solution of the equation

Lemma 3. Let condition (6) hold. Then is differentiable and .

Proof. Differentiating (15) we have Hence, due to (15), Thus, Now (16) yields the result.

Lemma 4. Let Then , for a solution of (8).

Proof. Multiplying (8) by and doing the scalar product, we can write . Since it can be written as Hence, condition (20) implies This proves the result.

Furthermore, for a stable operator , put . Then , andHenceand thereforewhere is the smallest eigenvalue of . Recall that is stable, so . Then due to (13) and (26) with we get Hence, for any continuous function , we have Now the previous lemma implies But is uniformly bounded and therefore all the solutions of (8) are uniformly bounded (i.e., (8) is Lyapunov stable). Furthermore, substitute into (8) Then Applying our above arguments to (31) can assert that (31) with small enough is Lyapunov stable. So, due to (30), (8) is exponentially stable, provided (20) holds. Now Lemma 3 implies the following.

Lemma 5. Let . Then (8) is exponentially stable.

Proof of Theorem 1. The exponential stability of (5) immediately follows from Lemmas 2 and 5, and the equality . The rest of the proof is obvious.

3. Equations on a Tensor Product of Hilbert Spaces

Let be separable Hilbert spaces with scalar products , the unit operators , and the norms . Let be the tensor product of and . This means that is a collection of all formal sums of the form with the understanding that Here , and is a number. The scalar product in is defined by and the norm . The closure of in the norm is a Hilbert space; compare [19]. It is again denoted by . In addition, the unit operator in equals .

Furthermore, for a Hilbert-Schmidt operator in denotes the Hilbert-Schmidt norm: . Let be a closed constant operator in with a dense domain Let be a linear operator in uniformly bounded on , having a strong derivative uniformly bounded on . Put , and assume that, for each , the operator is stable: Let Then condition (7) holds and, therefore, (5) is exponentially stable. If, in addition, is bounded on , then any solution of (1) is bounded on .

Furthermore, from (36), it follows that . Assume that and put where are nonreal eigenvalues of taken with their multiplicities. Clearly, Due to [20, Example 7.10.3], So Hence , where This inequality, (37), and Theorem 1 imply the following.

Corollary 6. Let conditions (35) and hold. Then (5) is exponentially stable.

4. Example

Put . In this section ,  , and are the Hilbert spaces of real functions with usual scalar products.

Consider problem (3), (4), assuming that, for almost all , and have bounded measurable derivatives and . In addition, the operators and defined in by respectively, are assumed to be bounded uniformly in . In addition, Let with the domain Then is self-adjoint with the eigenvalues . So . Assume that and put Now Corollary 6 implies the following.

Corollary 7. Let conditions (48) and hold. Then the equation with condition (4) is exponentially stable. If, in addition, and is uniformly bounded on , then any solution of (3) is uniformly bounded on in the norm of .

To estimate one can apply various bounds for spectra of integral operators. For instance, consider the equation with condition (4). So , where It is simple to check that in this case and . Thus and provided 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.

Competing Interests

The author declares that they have no competing interests.