# Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

Authors: Michael Gil'
Journal: International Journal of Partial Differential Equations
 (2013)

---

## Abstract

LetA be a generator of an exponentially stable
operator semigroup in a Banach space,
and let Ct t≥0 be a linear bounded variable operator.
Assuming that ∫0tCsds is sufficiently small in a certain sense for the equation
dx/dt=Ax+C(t)x, we derive exponential stability conditions.
Besides, we do not require  that for each t0≥0, the “frozen”
autonomous equation dx/dt=Ax+C(t0)x is stable. In particular, we consider
evolution equations with periodic operator coefficients.
These results are applied to partial differential equations.


---

## Body

## 1. Introduction and Statement of the Main Result

In this paper, we investigate stability of linear nonautonomous equations in a Banach space, which can be considered as integrally small perturbations of autonomous equations. The stability theory of evolution equations in a Banach space is well developed, compare and confare with [1] and references therein, but the problem of stability analysis of evolution equations continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems, because of the absence of its complete solution. One of the basic methods for the stability analysis is the direct Lyapunov method. By that method, many strong results were established, compare and confare with [2, 3]. But finding the Lyapunov functionals is usually a difficult mathematical problem. A fundamental approach to the stability of diffusion parabolic equations is the method of upper and lower solutions. A systematical treatment of that approach is given in [4]. In [5], stability conditions are established by a normalizing mapping. Note that a normalizing mapping enables us to use more complete information about the equation than a usual (number) norm. In [6], the “freezing” method for ordinary differential equations is extended to equations in a Banach space. About the recent results, see the interesting papers [7–11]. In particular, in [7] the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces is investigated. In the paper [8], a Rolewicz’s type theorem of in-solid function spaces is proved. Dragan and Morozan [9] established criteria for exponential stability of linear differential equations on ordered Banach spaces. Paper [10] deals with the stability and controllability of hyperbolic type abstract evolution equations. Pucci and Serrin [11] investigated the asymptotic stability for nonautonomous wave equations.Certainly, we could not survey the whole subject here and refer the reader to the previously listed publications and references given therein.LetX  be a complex Banach space with a norm ∥·∥X and the unit operator I. For a bounded operator K,  ∥K∥ is the operator norm.Everywhere belowA is a linear operator in X with a domain Dom(A), generating a strongly continuous semigroup T(t); that is,  A=limh↓0(1/h)(T(h)-I) in the strong topology, and C(t)(t≥0) is a linear bounded variable operator mapping  Dom(A) into itself.  Put B(t)=A+C(t). In the present paper, we establish stability conditions for the equation as follows:
(1)dudt=B(t)u,t≥0.
It should be noted that in the previously pointed papers it is assumed that for each t0≥0, the “frozen” autonomous equation dx/dt=B(t0)x is stable. We do not require that condition. The aim of this paper is to generalize the main result from the paper [12], which deals with finite dimensional equations.A solution of (1), for a given u0∈Dom(A) is a function u:[0.∞)→Dom(A) having a strong derivative, satisfying (1) and u(0)=u0. We will investigate (1) as a perturbation of the following equation:
(2)dvdt=Av,t≥0.
Put
(3)J(t):=∫0tC(s)ds,m(t):=∥AJ(t)-J(t)B(t)∥,(t≥0).
We say that (1) is exponentially stable if there is an  α=const>0, such that ∥u(t)∥X≤e-αt∥u(0)∥X  (t≥0) for any solution  u(t) with  u(0)∈Dom(A). Now we are in a position to formulate the main result of the paper.Theorem 1.
Let(4)∥T∥L1:=∫0∞∥T(t)∥dt<∞,(5)supt≥0(∥J(t)∥+∫0t∥T(t-s)∥m(s)ds)<1.
Then, (1) is exponentially stable.The proof of this theorem is divided into a series of lemmas which are presented in the next section. To the best of our knowledge, Theorem1 is new even in the case of bounded operators. In Section 3 we consider particular cases of Theorem 1. In Section 4, the previously pointed results are applied to a partial differential equation. For the brevity, we restrict ourselves by a scalar equation with the periodic boundary condition, but our results enable us to consider coupled systems of equations and other boundary conditions, for example, the Dirichlet condition.


## 2. Proofs

We need the following simple result.Lemma 2.
Letw(t),f(t), and v(t)(0≤t≤a≤∞) be functions whose values are bounded linear operators. Assume that w(t) is integrable and f(t) and v(t) have integrable derivatives on [0,a]. Then, with the notation  jw(t)=∫0tw(s)ds, one has
(6)∫0tf(s)w(s)v(s)ds=f(t)jw(t)v(t)-∫0t[f′(s)jw(s)v(s)+f(s)jw(s)v′(s)]ds,(t≤a).Proof.
Clearly,(7)ddtf(t)jw(t)v(t)=f′(t)jw(t)v(t)+f(t)w(t)v(t)+f(t)jw(t)v′(t).
Integrating this equality and taking into account that jw(0)=0, we arrive at the required result.LetV(t) be the Cauchy operator to (1); that is, V(t)u(0)=u(t) for a solution u(t) of (1).Lemma 3.
One has(8)(I-J(t))V(t)=T(t)+∫0tT(t-s)[AJ(s)-J(s)B(s)]V(s)ds.Proof.
As it is well known,(9)V(t)-T(t)=∫0tT(t-s)C(s)V(s)ds,
compare and confare with [13]. Thanks to the previous lemma, one has
(10)∫0t‍T(t-s)C(s)V(s)ds=T(0)J(t)V(t)-∫0t[(dT(t-s)ds)J(s)V(s)T(0)J(t)V(t)-+T(t-s)J(s)V′(s)(dT(t-s)ds)]ds.
But  dT(t-s)/ds=-AT(t-s). In addition, V′(s)=B(s)V(s). Thus,
(11)∫0tT(t-s)C(s)V(s)ds=J(t)V(t)+∫0tT(t-s)[AJ(s)-J(s)B(s)]V(s)ds.
Now, (9) implies the required result.Let,(12)ζ(t):=infh∈X;∥h∥=1∥(J(t)-I)h∥.Lemma 4.
Let condition(13)inft≥0ζ(t)>0
hold. Then,  ∥V(t)∥≤z(t), t≥0, where  z(t)  is a solution of the following equation:
(14)z(t)=1ζ(t)×[∥T(t)∥+∫0t∥T(t-s)∥m(s)z(s)ds],t≥0.Proof.
Thanks to the previous lemma,(15)∥(I-J(t))V(t)∥≤∥T(t)∥+∫0t∥T(t-s)∥m(s)∥V(s)∥ds.
Hence
(16)ζ(t)∥V(t)∥≤∥T(t)∥+∫0t∥T(t-s)∥m(s)∥V(s)∥ds.
Then by the well-known (comparison) Lemma 3.2.1 from [14] we have the required result.Let(17)η0:=supt≥01ζ(t)∫0t∥T(t-s)∥m(s)ds<1.
Then (14) implies
(18)suptz(t)≤supt≥0∥T(t)∥ζ(t)+suptz(t)η0.
Due to the previous lemma we get the following.Lemma 5.
Let conditions (13) and (17) hold. Then
(19)supt≥0∥V(t)∥≤supt≥0∥T(t)∥(1-η0)ζ(t).Proof of Theorem1.
Assume that(20)j(t):=∥J(t)∥≤q<1,(q=const;t≥0),
then ζ(t)≥1-j(t). If
(21)η1:=supt≥011-j(t)∫0t∥T(t-s)∥m(s)ds<1,
then η0≤η1<1 and thanks to the previous lemma, (1) is stable. But condition (5) implies that
(22)j(t)+∫0t∥T(t-s)∥m(s)ds<1
or
(23)11-j(t)∫0t‍∥T(t-s)∥m(s)ds<1,(t≥0).
Thus (5) implies the inequality  η1<1, and therefore, from (5), condition (21) follows. This proves the stability. To prove the exponential stability we use the well-known Theorem 4.1 [13, p. 116] (see also Theorem 2.44 [1, p. 49]). It asserts that the finiteness of the L1-norm of T implies the inequality
(24)∥T(t)∥≤Me-αt,t≥0,
where M=const≥1, α=const>0. So for 0<ϵ<α, the semigroup Tϵ(t) generated by A+Iϵ satisfies the inequality ∥Tϵ(t)∥≤Me-(α-ϵ)t, t≥0, and therefore it also has a finite L1-norm. Substitute the equality
(25)u(t)=y(t)e-ϵt
into (1). Then we obtain the equation
(26)y˙=(B(t)+Iϵ)y.
Denote the Cauchy operator of (26) by Vϵ(t). Repeating our above arguments with Vϵ(t) instead of V(t) and the equation x˙=(A+Iϵ)x instead of (2), due to Lemma 5 we can assert that  Vϵ(t) is bounded. Now (25) implies
(27)∥V(t)∥≤e-ϵtsupt≥0∥Vϵ(t)∥,t≥0.
This proves the theorem.


## 3. A particular Case of Theorem1

To illustrate Theorem1, consider the following equation:
(28)dudt=Au+c(t)C0u,
where C0 is a constant operator and c(t) is a scalar real piece-wise continuous function bounded on [0,∞). So,  C(t)=c(t)C0. Without any loss of generality, assume that
(29)supt|c(t)|=1,
and with the notation
(30)ic(t)=|∫0tc(s)ds|,
we obtain
(31)m(t)=∥AJ(t)-J(t)B(t)∥≤ic(t)∥AC0-C0(A+c(t)C0)∥≤ic(t)(∥AC0-C0A∥+|c(t)|∥C02∥)≤ic(t)(∥AC0-C0A∥+∥C02∥).
Due to (24),
(32)∫0t∥T(t-s)∥ds≤M∫0te-αsds≤Mα,(t≥0).
Thus, denoting
(33)θ0=supt|∫0tc(s)ds|,
due to Theorem 1, we arrive at the following result.Corollary 6.
If the inequality(34)θ0(∥C0∥+Mα(∥AC0-C0A∥+∥C02∥))<1
holds, then (28) is exponentially stable.For example, letc(t)=sin(ωt)  (ω>0).  Then,  ic(t)≤2/ω and
(35)m(t)≤2ω(∥AC0-C0A∥+∥C02∥).
Thus, (34) takes the following form:
(36)∥C0∥+Mα(∥AC0-C0A∥+∥C02∥)<ω2.


## 4. Equations with Periodic Boundary Conditions

Consider the problem(37)∂u(x,t)∂t=∂u(x,t)∂x+(-b0+c(t)a(x))u(x,t),(0≤x≤1),(38)u(0,t)=u(1,t),(t≥0),
with a positive constant b0 and a real differentiable function a(x); c(t) is the same as in the previous section.TakeX=L2(0,1), where L2=L2(0,1) is the Hilbert space of real functions f,h defined on [0,1] with the scalar product
(39)(f,h)=∫01f(x)h(x)dx,
and the norm ∥h∥=(h,h). Set
(40)Dom(A)={f∈L2:f′∈L2;f(0)=f(1)},(41)Au(x)=du(x)dx-b0u(x)(u∈Dom(A)),
then we have
(42)(Au,u)=∫01(du(x)dx-b0u(x))u(x)dx=∫01(du(x)dx-b0u(x))u(x)dx=∫01(12du2(x)dx-b0u2(x))dx=12(u2(1)-u2(0))-b0∫01u2(x)dx=-b0∫01u2(x)dx.
Let v=v(t,x) be a solution of (2) with A defined by (41). Then, we obtain
(43)ddt(v,v)=(v˙,v)+(v,v˙)=(Av,v)+(v,Av)=2(Av,v)≤-2b0(v,v).
Hence, (d/dt)∥v∥≤-b0∥v∥. Thus, ∥T(t)∥≤e-b0t. In addition, C0u(x)=a(x)u(x),
(44)(AC0-C0A)u(x)=d(a(x)u(x))dx-a(x)du(x)dx=a′(x)u(x),(u∈Dom(A)).
Due to condition (34), we obtain the following.Corollary 7.
If the inequality(45)θ0(|a(x)|+1b0(|a′(x)|+|a(x)|2))<1,(0≤x≤1)
holds, then (37) is exponentially stable.For example, letc(t)=sin(ωt)(ω>0). Then ic(t)≤2/ω and (45) takes the form
(46)|a(x)|+1b0(|a′(x)|+|a(x)|2)<ω2,(0≤x≤1).



---

*Source: 207581-2013-04-28.xml*