Stability and Boundedness of Solutions to Nonautonomous Parabolic Integrodifferential Equations

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.


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][6][7][8][9] and references given therein.For important stability results on stochastic partial differential equations see the papers [10][11][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][14][15][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 differentialoperator 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, () = sup Re (), and Dom() denotes the domain.
Consider the equation where  is a closed constant operator in  with a dense domain () is an operator uniformly bounded on [0, ∞), having a strong derivative uniformly bounded on [0, ∞) and commuting with ; (⋅) : [0, ∞) → Dom() satisfies the conditions pointed below.
Certainly, (1) can be considered in some space as the equation u = () 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 ().
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.

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.