# Stability and Boundedness of Solutions to Nonautonomous Parabolic Integrodifferential Equations

Authors: Michael Gil'
Journal: Mathematical Problems in Engineering
 (2016)

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## 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.


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## Body

## 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 ofnonautonomous 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.LetH be a Hilbert space with a scalar product (·,·), the norm ·=(·,·), and unit operator I. All the considered operators are assumed to be linear. For an operator A, A∗ is the adjoint one, σ(A) is the spectrum, αA=sup⁡Reσ(A), and Dom(A) denotes the domain.Consider the equation(1)u˙=Su+Btu+Ftu=ut,t>0;u˙=dudt, where S is a closed constant operator in H with a dense domain (2)DomS=DomS∗,γS≔supw∈DomS,w=1⁡S+S∗w,w2<∞;B(t) is an operator uniformly bounded on [0,∞), having a strong derivative uniformly bounded on [0,∞) andcommuting withS; F(·):[0,∞)→Dom(S) satisfies the conditions pointed below.An important example of (1) is the boundary values problem (3)∂ut,x,y∂t=∂2∂y2ut,x,y+ct,xut,x,y+∫abkt,x,sut,s,yds+ft,x,y,(4)ut,x,0=ut,x,1=0t>0;0≤y≤1;a≤x≤b, where c(·,·):[0,∞)×[a,b]→R, k(·,·,·):[0,∞)×[a,b]2→R and f(·,·):[0,∞)×[a,b]×[0,1]→R are given functions and u(·,·,·) 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 u˙=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 u0∈Dom(S) is a function u:[0,∞)→Dom(S) having a bounded measurable strong derivative and satisfying u(0)=u0.In particular, we will consider the homogeneous equation(5)u˙=Su+Btut>0. Equation (5) is said to be exponentially stable, if there are constants M≥1,ϵ>0, such that u(t)≤Mexp[-ϵt]u(0)  (t≥0) for any solution of (1). Condition (2) implies that S generates a strongly continuous semigroup eSt; compare [17, Section I.4.4]. Since B(t) is bounded and commutes with S we can assert that (1) has solutions for any u0∈Dom(S).We will assume that, for eachτ≥0, the operator B(τ)+γ(S)I is stable (Hurwitzian); namely, (6)qτ≔2∫0∞eBτ+γSIs2ds<∞. Now we are in a position to formulate the main result of this paper.Theorem 1.
Let conditions (2) and (6) and (7)supt≥0⁡q2tB′t<2 hold. Then (5) is exponentially stable. If, in addition, F(t) is bounded and measurable on [0,∞), then any solution of (1) is bounded on [0,∞).


## 2. Proof of Theorem1

PutC=S-γ(S)I and A(t)=B(t)+γ(S)I. Let V(t) be the Cauchy operator to the equation (8)v˙t=Atvtt≥0. That is, Vtv0=vt for any solution of (8). Taking (9)ut=eCtVtu0u0∈DomS we have (10)u˙=C+Atu=S+Btu. Consequently, etCVtu0 really is a solution to (5). Since (11)supw∈DomS,w=1⁡C+C∗w,w≤0, we have etC≤1,t≥0; compare [17, Theorem I.4.2]. Thus, u(t)≤V(t)u0. So we have proved the following result.Lemma 2.
Under the hypothesis of Theorem1, (5) is exponentially stable, provided (8) is exponentially stable.Furthermore, recall that the equation(12)A0∗Y+YA0=E with given constant bounded stable operator A0 (i.e.,  α(A0)<0) and a constant bounded operator E has a solution Y which is represented as (13)Y=-∫0∞eA0∗sEeA0sds. Compare [18, Section I.4.4]. Consequently, due to (2), the operator (14)Qt≔2∫0∞eA∗tseAtsds is a unique solution of the equation (15)A∗tQt+QtAt=-2It≥0,(16)Qt≤qt.Lemma 3.
Let condition (6) hold. Then Q(t) is differentiable and Q′(t)≤q2(t)A′(t).Proof.
Differentiating (15) we have(17)A∗tQ′t+Q′tAt=-A∗t′Qt-QtA′tt≥0. Hence, due to (15), (18)Q′t=∫0∞eA∗tsA∗t′Qt+QtA′teAtsds. Thus, (19)Q′t≤12qtA∗t′Qt+QtA′t≤qtQtA′t. Now (16) yields the result.Lemma 4.
Let(20)supt≥0⁡Q′t<2. Then (Q(t)u(t),u(t))≤(Q(0)u(0),u(0))(t≥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 (21)ddtQtut,ut=Qtu′t,ut+ut,Qtu′t+Q′tut,ut, it can be written as(22)ddtQtut,ut=QtAtut,ut+ut,QtAtut+Q′tut,ut=QtAt+A∗tQtut,ut+Q′tut,ut=-2ut,ut+Q′tut,ut. Hence, condition (20) implies (23)ddtQtut,ut≤-2+Q′tut,ut<0. This proves the result.Furthermore, for a stable operatorA0, put y1(t)=eA0tv(v∈H). Then y˙1=A0y1, and(24)dy1t,y1tdt=A0+A0∗y1t,y1t.Hence(25)dy1t,y1tdt≥λA0+A0∗y1t,y1tand therefore(26)eA0tv2≥etλA0+A0∗v2,where λ(A0+A0∗) is the smallest eigenvalue of A0+A0∗. Recall that A0 is stable, so λ(A0+A0∗)<0. Then due to (13) and (26) with E=-2I we get (27)Yh,h=2∫0∞eA0∗seA0sh,hds≥2∫0∞eλA0+A0∗sdsh2=2h2λA0+A0∗-1h∈H. Hence, for any continuous function u1:[0,∞)→H, we have (28)Qtu1t,u1t≥2u1t2λAt+A∗t-1. Now the previous lemma implies (29)ut,ut≤λAt+A∗tQ0u0,u0t≥0. But |λ(A(t)+A∗(t))| is uniformly bounded and therefore all the solutions of (8) are uniformly bounded (i.e., (8) is Lyapunov stable). Furthermore, substitute into (8) (30)ut=uϵte-ϵtϵ>0. Then (31)u˙ϵt=At+ϵIuϵt. Applying our above arguments to (31) can assert that (31) with small enough ϵ>0 is Lyapunov stable. So, due to (30), (8) is exponentially stable, provided (20) holds. Now Lemma 3 implies the following.Lemma 5.
Letsupt≥0q2(t)B′t<2. Then (8) is exponentially stable.Proof of Theorem1.
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

LetEj(j=1,2) be separable Hilbert spaces with scalar products 〈·,·〉j, the unit operators Ij, and the norms ·j=〈·,·〉j. Let H=E1⊗E2 be the tensor product of E1 and E2. This means that H is a collection of all formal sums of the form (32)u=∑jyj⊗hjyj∈E1,hj∈E2 with the understanding that (33)λy⊗h=λy⊗h=y⊗λh,y+y1⊗h=y⊗h+y1⊗h,y⊗h+h1=y⊗h+y⊗h1.Here y,y1∈E1;h,h1∈E2, and λ is a number. The scalar product in H is defined by (34)y⊗h,y1⊗h1H=y⊗h,y1⊗h1=y,y11h,h12y,y1∈E1,h,h1∈E2 and the norm ·≡·H∶=(·,·)H. The closure of H in the norm ·H is a Hilbert space; compare [19]. It is again denoted by H. In addition, the unit operator I=IH in H equals I1⊗I2.Furthermore, for a Hilbert-Schmidt operatorC in E1,N2(C) denotes the Hilbert-Schmidt norm: N2(C)=traceCC∗. Let S0 be a closed constant operator in E2 with a dense domain (35)DomS0=DomS0∗,γS0≔supw∈DomS0,w=1⁡S0+S0∗w,w2<∞. Let B0(t) be a linear operator in E1 uniformly bounded on [0,∞), having a strong derivative uniformly bounded on [0,∞). Put B(t)=B0(t)⊗I2, S=I1⊗S0 and assume that, for each τ≥0, the operator B0(τ)+γ(S0)I1 is stable: (36)q0τ≔2∫0∞eB0τ+γS0I1s2ds<∞τ≥0. Let (37)supt≥0⁡q02tB0′t<2. Then condition (7) holds and, therefore, (5) is exponentially stable. If, in addition, F(t) is bounded on [0,∞), then any solution of (1) is bounded on [0,∞).Furthermore, from (36), it follows that α(B0(τ))+γ(S0)<0. Assume that (38)B0t-B0∗tis a Hilbert-Schmidt operator∀t≥0, and put(39)gt≔12N22B0t-B0∗t-2∑k=1∞Im⁡λkB0t21/2, where λk(B0(t)) are nonreal eigenvalues of B0(t) taken with their multiplicities. Clearly,(40)gt≤N2B0t-B0∗t2. Due to [20, Example 7.10.3], (41)eB0τt≤expαB0τt∑k=0∞gkτtkk!3/2τ,t≥0. So (42)q0τ≤2∫0∞exp2αB0τ+γS0t∑k=0∞gkτtkk!3/22dt=2∫0∞exp2αB0τ+γS0t∑j,k=0∞gτtk+jj!k!3/2dt. Hence q0(τ)≤ψ(τ,γ(S0)), where (43)ψτ,γS0≔2∑j,k=0∞k+j!gj+kτ2αB0τ+γS0j+k+1j!k!3/2. This inequality, (37), and Theorem 1 imply the following.Corollary 6.
Let conditions (35) and (44)supt≥0⁡ψ2t,γS0B0′t<2 hold. Then (5) is exponentially stable.


## 4. Example

PutΩ=[a,b]×[0,1]. In this section E1=L2(a,b),  E2=L2(0,1), and H=E1⊗E2=L2(Ω) are the Hilbert spaces of real functions with usual scalar products.Consider problem (3), (4), assuming that, for almost all x,s∈[0,1], c(t,x) and k(t,x,s) have bounded measurable derivatives ct′(t,x) and kt′(t,x,s). In addition, the operators B0(t) and B0′(t) defined in L2(a,b) by (45)B0twx=ct,xwx+∫abkt,x,swsds,B0′twx=ct′t,xwx+∫abkt′t,x,swsdsx∈a,b;w∈L2a,b, respectively, are assumed to be bounded uniformly in t∈[0,∞). In addition, (46)g^t=12N2Bt-B0∗t=12∬abkt,x,s-kt,s,x2dsdx1/2<∞. Let S0=d2/dx2 with the domain(47)DomS0=w∈L20,1:d2wxdx2∈L20,1;w0=w1=0. Then S0 is self-adjoint with the eigenvalues -πk2(k=1,2,…). So γ(S0)=-π2. Assume that (48)supt≥0⁡αB0t-π2<0 and put (49)ψπτ≔2∑j,k=0∞k+j!g^j+kτ2αB0τ-π2j+k+1j!k!3/2. Now Corollary 6 implies the following.Corollary 7.
Let conditions (48) and (50)supt≥0ψπ2tB0′t<2 hold. Then the equation (51)∂ut,x,y∂t=∂2∂y2ut,x,y+ct,xut,x,y+∫abkt,x,sut,s,yds with condition (4) is exponentially stable. If, in addition, fyy′′(t,·,·)∈L2(Ω) and f(t,·,·)L2(Ω) is uniformly bounded on [0,∞), then any solution of (3) is uniformly bounded on [0,∞) in the norm of L2(Ω).To estimateα(B0(t)) one can apply various bounds for spectra of integral operators. For instance, consider the equation (52)∂ut,x,y∂t=∂2∂y2ut,x,y+ct,xut,x,y+∫axkt,x,sut,s,yds with condition (4). So B0(t)w(x)=c(t,x)w(x)+(V(t)w)(x), where (53)Vtwx=∫axkt,x,swsdsx∈a,b;w∈L2a,b. It is simple to check that in this case g(t)=N2(V(t)) and σ(B0(t))={c(t,x):x∈[a,b]}. Thus α(B0(t))=supx∈[0,1]c(x,t) and (54)ψπτ≔2∑j,k=0∞k+j!N2j+kVτ2supx∈0,1cx,τ-π2j+k+1j!k!3/2, provided (55)supx∈0,1cx,t-π2<0. 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.



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