# Existence of Mild Solutions for the Elastic Systems with Structural Damping in Banach Spaces

Authors: Hongxia Fan; Yongxiang Li; Pengyu Chen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper deals with the existence and uniqueness of mild solutions for a second order evolution equation initial value problem in a Banach space, which can model an elastic system with structural damping. The discussion is based on the operator semigroups theory and fixed point theorem. In addition, an example is presented to illustrate our theoretical results.


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## Body

## 1. Introduction

Our aim in this paper is to study the existence and uniqueness of mild solutions for the semilinear elastic system with structural damping(1)u¨(t)+ρ𝒜u˙(t)+𝒜2u(t)=f(t,u(t)),0<t<a,u(0)=x0,u˙(0)=y0
in a Banach space 𝕏, where · means d/dt, ρ≥2 is a constant; 𝒜:D(𝒜)⊂𝕏→𝕏 is a closed linear operator and -𝒜 generates a C0-semigroup T(t)(t≥0) on 𝕏; f∈C([0,a]×𝕏,𝕏), x0∈D(𝒜), y0∈𝕏.In 1982, Chen and Russell [1] investigated the following linear elastic system described by the second order equation(2)u¨(t)+Bu˙(t)+Au(t)=0
in a Hilbert space H with inner (·,·), where A (the elastic operator) and B (the damping operator) are positive definite selfadjoint operators in H. They reduced (2) to the first order equation in H×H(3)ddt(A1/2uu˙)=(0A1/2-A1/2-B)(A1/2uu˙).
Let V=𝒟(A1/2), ℋ=V×H with the naturally induced inner products. Then, (2) is equivalent to the first order equation in ℋ(4)ddt(uu˙)=𝒜B(uu˙),
where
(5)𝒜B=(0I-A-B),D(𝒜B)=D(A)×[D(A1/2)∩D(B)].
Chen and Russell [1] conjectured that 𝒜B is the infinitesimal generator of an analytic semigroup on ℋ if
(6)D(A1/2)⊂D(B)
and either of the following two inequalities holds for some β1,β2>0:
(7)β1(A1/2v,v)≤(Bv,v)≤β2(A1/2v,v),v∈D(A1/2);β1(Av,v)≤(B2v,v)≤β2(Av,v),v∈D(A).
In the same paper they obtained some results in this direction. The complete proofs of the two conjectures were given by Huang [2, 3]. Then, other sufficient conditions for 𝒜B or its closure 𝒜B¯ to generate an analytic or differentiable semigroup on ℋ were discussed in [4–10], by choosing B to be an operator comparable with Aα for 0<α≤1, based on an explicit matrix representation of the resolvent operator of 𝒜B or 𝒜B¯.However, so far as we know, among the previous works, little is concerned with an elastic system with structural damping in a Banach space. Motivated by previous works, in this paper, we investigate the existence and uniqueness of mild solutions for the elastic system (1) in a frame of Banach spaces. To this end, we firstly introduce the concept of mild solutions for system (1), which is based on the discussion about associated linear system. Secondly, we prove the existence and uniqueness of mild solutions for the semilinear elastic system (1) in a Banach space 𝕏.The paper is organized as follows. In Section2, we discuss the associated linear elastic system and give its definition of mild solutions. In Section 3, we study the existence and uniqueness of mild solutions for the semilinear elastic system (1). An example to illustrate our theoretical results is given in Section 4.


## 2. Preliminaries on Linear Elastic Systems

Let𝕏 be a Banach space, we consider the linear elastic system with structural damping
(8)u¨(t)+ρ𝒜u˙(t)+𝒜2u(t)=h(t),0<t<a,u(0)=x0,u˙(0)=y0,
where · means d/dt, ρ≥2 is a constant; 𝒜:D(𝒜)⊂𝕏→𝕏 is a closed linear operator, and -𝒜 generates a C0-semigroup T(t)(t≥0) on 𝕏; h:[0,a]→𝕏, x0∈D(𝒜), y0∈𝕏.For the second order evolution equation(9)u¨(t)+ρ𝒜u˙(t)+𝒜2u(t)=h(t),
it has the following decomposition
(10)(∂∂t+σ1𝒜)(∂∂t+σ2𝒜)u=h(t).
That is,
(11)∂2u∂t2+(σ1+σ2)𝒜∂u∂t+σ1σ2𝒜2u=h(t).
It follows from (9) and (11) that
(12)σ1+σ2=ρ,σ1σ2=1.
By (12), we have
(13)(i)ifρ>2,thenσ1=ρ+ρ2-42,σ2=ρ-ρ2-42,(ii)ifρ=2,thenσ1=σ2=1.
Let
(14)∂u∂t+σ2𝒜u=v(t),0≤t≤a,
which means
(15)v0:=v(0)=y0+σ2𝒜x0.
So we reduce the linear elastic system (8) to the following two abstract Cauchy problems in Banach space 𝕏:
(16)∂v∂t+σ1𝒜v=h(t),0<t<a,v(0)=v0,(17)∂u∂t+σ2𝒜u=v(t),0<t<a,u(0)=x0.
It is clear that (16) and (17) are linear inhomogeneous initial value problems for -σ1𝒜 and -σ2𝒜, respectively. Since -𝒜 is the infinitesimal generator of C0-semigroup T(t)(t≥0). Furthermore, for any ρ≥2, (13) yield σ1>0, σ2>0. Thus, by operator semigroups theory [11], -σ1𝒜 and -σ2𝒜 are infinitesimal generators of C0-semigroups, which implies initial value problems (16) and (17) are well-posed.Throughout this paper, we assume that-σ1𝒜 and -σ2𝒜 generate C0-semigroups S1(t)(t≥0) and S2(t)(t≥0) on 𝕏, respectively. Note that σ1>0, σ2>0 and -𝒜 is the infinitesimal generator of C0-semigroup T(t)(t≥0). It follows that
(18)S1(t)=T(σ1t),S2(t)=T(σ2t),t≥0.It is well known [12, Chapter 4], when h∈L1([0,a],𝕏), the linear initial value problem (16) has a mild solution v given by
(19)v(t)=S1(t)v0+∫0t‍S1(t-s)h(s)ds.
Similarly, if v∈C([0,a],𝕏), then the mild solution of the linear initial value problem (17) expressed by
(20)u(t)=S2(t)x0+∫0t‍S2(t-s)v(s)ds.
Substituting (19) into (20), we get
(21)u(t)=S2(t)x0+∫0t‍S2(t-s)S1(s)v0ds+∫0t‍∫0s‍S2(t-s)S1(s-τ)h(τ)dτds.
From the argument above, we obtain the following corollary.Corollary 1.
Ifh∈L1([0,a],𝕏), then the initial value problem (8) has at most one solution. If it has a solution, this solution is given by (21).For everyh∈L1([0,a],𝕏), the right-hand side of (21) is a continuous function on [0,a]. It is natural to consider it as a generalized solution of (8) even it is not differentiable and dose not strictly satisfy the equation. We therefore define the following.Definition 2.
Let-𝒜 is the infinitesimal generator of C0-semigroup T(t)(t≥0). Then a continuous solution u(t) of the integral equation
(22)u(t)=S2(t)x0+∫0t‍S2(t-s)S1(s)v0ds+∫0t‍∫0s‍S2(t-s)S1(s-τ)h(τ)dτds
is said to be a mild solution of the initial value problem (8). Where S1(t)(t≥0), S2(t)(t≥0) were defined in (18) and v0 was specified in (15).


## 3. Main Results

LetC(J,𝕏) be the Banach space of all continuous functions u:J→𝕏 with norm ∥u∥C=maxs∈J∥u(s)∥, J=[0,a]. Let ℒ(𝕏) be the Banach space of all linear and bounded operators on 𝕏. Note that S1(t)(t≥0) and S2(t)(t≥0) are C0-semigroups on 𝕏. Thus, there exist M1≥1 and M2≥1 such that
(23)M1=supt∈J∥S1(t)∥ℒ(𝕏),M2=supt∈J∥S2(t)∥ℒ(𝕏).In what follows, we firstly give the definition of a mild solution for the initial value problem (1) below.Definition 3.
Let-𝒜 is the infinitesimal generator of C0-semigroup T(t)(t≥0). Then a continuous solution u(t) of the integral equation
(24)u(t)=S2(t)x0+∫0t‍S2(t-s)S1(s)v0ds+∫0t‍∫0s‍S2(t-s)S1(s-τ)f(τ,u(τ))dτds
is said to be a mild solution of the initial value problem (1). Where S1(t)(t≥0), S2(t)(t≥0) were defined in (18) and v0 was specified in (15).Secondly, we consider the existence and uniqueness of mild solutions for (1). To this end, we make the following assumptions: (H1)
f:[0,a]×𝕏→𝕏 be continuous and there exists L>0, such that
(25)∥f(t,u2)-f(t,u1)∥≤L∥u2-u1∥,t∈[0,a],u1,u2∈𝕏.(H2)
f:[0,a]×𝕏→𝕏 be continuous and there exists a positive function μ∈L∞(J,ℝ+)(ℝ+=[0,+∞)) such that
(26)∥f(t,u)∥≤μ(t),t∈[0,a],u∈𝕏.(H3)
TheC0-semigroup T(t)(t≥0) is compact for t>0.Theorem 4.
Assume that(H1) holds, -𝒜 is the infinitesimal generator of C0-semigroup T(t)(t≥0). Then for every x0∈D(𝒜), y0∈𝕏 and ρ≥2, the initial value problem (1) has a unique mild solution u∈C([0,a],𝕏).Proof.
Define the operatorQ:C(J,𝕏)→C(J,𝕏) by
(27)(Qu)(t)=S2(t)x0+∫0t‍S2(t-s)S1(s)v0ds+∫0t‍∫0s‍S2(t-s)S1(s-τ)f(τ,u(τ))dτds.
It is obvious that the mild solution of the initial value problem (1) is equivalent to the fixed point of Q.
For anyu1,u2∈C(J,𝕏), (23), (27), and (H1) yield
(28)∥(Qu2)(t)-(Qu1)(t)∥≤∫0t‍∫0s‍∥S2(t-s)∥ℒ(𝕏)∥S1(s-τ)∥ℒ(𝕏)×∥f(τ,u2(τ))-f(τ,u1(τ))∥dτds≤LM1M2∫0t‍∫0s‍∥u2(τ)-u1(τ)∥dτds≤LM1M2∫0t‍∫0s‍∥u2-u1∥Cdτds≤LM1M2a22∥u2-u1∥C.
Using (27), (28), and induction on n it follows easily that
(29)∥Qnu2(t)-Qnu1(t)∥≤(LM1M2a2)n(2n)!∥u2-u1∥C.
Hence
(30)∥Qnu2-Qnu1∥C≤(LM1M2a2)n(2n)!∥u2-u1∥C.
Since
(31)(LM1M2a2)n(2n)!→0asn→∞.
Thus, for n large enough (LM1M2a2)n/(2n)!<1 and by well known extension of the contraction mapping principle, Q has a unique fixed point u∈C([0,a],𝕏). This fixed point is the desired solution of the integral equation (24).Theorem 5.
Suppose that assumptions(H2) and (H3) hold. Then for every x0∈D(𝒜), y0∈𝕏 and ρ≥2, the initial value problem (1) has at least one mild solution u∈C([0,a],𝕏).Proof.
Define the operatorQ:C(J,𝕏)→C(J,𝕏) as (27) and choose r>0 such that
(32)r≥M2∥x0∥+M1M2a∥v0∥+M1M2a2∥μ∥L∞(J,ℝ+).
Let Br={u∈C(J,𝕏):∥u∥C≤r}. We proceed in two main steps.
Step1. We show that Q(Br)⊂Br. For that, let u∈Br. Then for t∈J, we have
(33)∥(Qu)(t)∥≤∥S2(t)x0∥+∥∫0t‍S2(t-s)S1(s)v0ds∥+∥∫0t‍∫0s‍S2(t-s)S1(s-τ)f(τ,u(τ))dτds∥,
which according to (H2) and (23) gives
(34)∥(Qu)(t)∥≤M2∥x0∥+M1M2a∥v0∥+M1M2a2∥μ∥L∞(J,ℝ+).
In view of the choice of r, we obtain
(35)∥Qu∥C≤r.
Step2. We prove that Q is completely continuous. Note that f:u→f(·,u(·)) is a continuous mapping from Br to C(J,𝕏). Thus, Q:Br→Br is continuous. Next, we show that Q is compact. To this end, we use the Ascoli-Arzela's theorem. For that, we first prove that {(Qu)(t):u∈Br} is relatively compact in 𝕏, for t∈J. Obviously, {(Qu)(0):u∈Br} is compact.
Lett∈(0,a]. For each ϵ∈(0,t) and u∈Br, we define the operator Qϵ by
(36)(Qϵu)(t)=S2(t)x0+∫0t-ϵ‍S2(t-s)S1(s)v0ds+∫0t-ϵ‍∫0s‍S2(t-s)S1(s-τ)f(τ,u(τ))dτds=S2(t)x0+S2(ϵ)∫0t-ϵ‍S2(t-ϵ-s)S1(s)v0ds+S2(ϵ)∫0t-ϵ‍∫0s‍S2(t-ϵ-s)S1(s-τ)×f(τ,u(τ))dτds.
Then the sets {(Qϵu)(t):u∈Br} are relatively compact in 𝕏 since by (H3) and (18), the semigroup S2(t)(t≥0) is compact for t>0 on 𝕏. Moreover, using (23) and (H2) we have
(37)∥(Qu)(t)-(Qϵu)(t)∥≤∥∫t-εt‍S2(t-s)S1(s)v0ds∥+∥∫t-εt‍∫0s‍S2(t-s)S1(s-τ)f(τ,u(τ))dτds∥≤M1M2∥v0∥ϵ+M1M2a∥μ∥L∞(J,ℝ+)ϵ.
Therefore, the set {(Qu)(t):u∈Br} is relatively compact in 𝕏 for all t∈(0,a] and since it is compact at t=0 we have the relatively compactness in 𝕏 for all t∈J.
Now, let us prove thatQ(Br) is equicontinuous. For 0≤t1<t2≤a, we have
(38)∥(Qu)(t2)-(Qu)(t1)∥≤∥S2(t2)x0-S2(t1)x0∥+∥∫0t1‍[S2(t2-s)-S2(t1-s)]S1(s)v0ds∥+∥∫t1t2‍S2(t2-s)S1(s)v0ds∥+∥∫0t1‍∫0s‍[S2(t2-s)-S2(t1-s)]×S1(s-τ)f(τ,u(τ))dτds∫0t1‍∥+∥∫t1t2‍∫0s‍S2(t2-s)S1(s-τ)f(τ,u(τ))dτds∥∶=I1+I2+I3+I4+I5,
where
(39)I1=∥S2(t2)x0-S1(t1)x0∥,I2=∥∫0t1‍[S2(t2-s)-S2(t1-s)]S1(s)v0ds∥,I3=∥∫t1t2‍S2(t2-s)S1(s)v0ds∥,I4=∥∫0t1‍∫0s‍[S2(t2-s)-S2(t1-s)]×S1(s-τ)f(τ,u(τ))dτds∫0t1‍∥,I5=∥∫t1t2‍∫0s‍S2(t2-s)S1(s-τ)f(τ,u(τ))dτds∥.
In fact, I1,I2,I3,I4 and I5 tend to 0 independently of u∈Br when t2-t1→0.
Note that the functionS2(t)x0 is continuous for t≥0. Thus, S2(t)x0 is uniformly continuous on [0,a] and thus limt2-t1→0I1=0.
From (23) and (H2), we have
(40)I2≤∫0t1‍∥S2(t2-s)-S2(t1-s)∥ℒ(𝕏)×∥S1(s)∥ℒ(𝕏)∥v0∥ds≤M1∥v0∥∫0a‍∥S2(t2-t1+τ)-S2(τ)∥ℒ(𝕏)dτ.I4≤∫0t1‍∫0s‍∥S2(t2-s)-S2(t1-s)∥ℒ(𝕏)×∥S1(s-τ)∥ℒ(𝕏)∥f(τ,u(τ))∥dτds≤M1a∥μ∥L∞(J,ℝ+)×∫0a‍∥S2(t2-t1+τ)-S2(τ)∥ℒ(𝕏)dτ.
Let ϕ(τ)=S2(t2-t1+τ)-S2(τ). By the compactness of T(·) and (18), we can easily conclude that S2(·) is compact and therefore S2(t) is continuous in the uniform operator topology for t>0. Then, ϕ(τ) is also continuous in the uniform operator topology on (0,a]. Thus ∥S2(t2-t1+τ)-S2(τ)∥ℒ(𝕏)→0 as t2-t1→0. Meanwhile, ϕ(τ) is bounded on [0,a]. Hence, using Lebesgue dominated convergence theorem we deduce that limt2-t1→0I2=limt2-t1→0I4=0.
Moreover, from (23) we have
(41)I3≤M1M2∥v0∥|t2-t1|,I5≤M1M2a∥μ∥L∞(J,ℝ+)|t2-t1|.
Hence, limt2-t1→0I3=limt2-t1→0I5=0.
In short, we have show, thatQ(Br) is relatively compact for t∈J, {Qu:u∈Br} is a family of equicontinuous functions. It follows from Ascoli-Arzela's theorem that Q is compact. By Schauder fixed point theorem Q has a fixed point u∈Br, which obviously is a mild solution to (1).


## 4. An Example

In order to illustrate our main results, we consider the following initial-boundary value problem, which is a model for elastic system with structural damping(42)∂2u(x,t)∂t2-ρ∂3u(x,t)∂x2∂t+∂4u(x,t)∂x4=f(x,t,u(x,t)),(x,t)∈[0,1]×[0,a],u(0,t)=u(1,t)=0,t∈[0,a],u(x,0)=φ(x),∂∂tu(x,0)=ψ(x),x∈[0,1],
where a>0, ρ≥2 are all constants, f:[0,1]×[0,a]×ℝ→ℝ is continuous.Let𝕏=Lp([0,1],ℝ)(1<p<+∞), we define the linear operator 𝒜 in 𝕏 by
(43)𝒜u=-∂2u∂x2,u∈D(𝒜)=W2,p(0,1)∩W01,p(0,1).
It is well known from [13] that -𝒜 is the infinitesimal generator of a C0-semigroup T(t)(t≥0) on 𝕏.Letu(t)=u(·,t), f(t,u(t))=f(·,t,u(·,t)), then the initial-boundary value problem (42) can be reformulated as the following abstract second order evolution equation initial value problem in 𝕏:
(44)u¨(t)+ρ𝒜u˙(t)+𝒜2u(t)=f(t,u(t)),0<t<a,u(0)=φ,u˙(0)=ψ.In order to solve the initial-boundary value problem (42), we also need the following assumptions:(b1)
φ∈W2,p(0,1)∩W01,p(0,1), ψ∈Lp([0,1],ℝ).(b2)
The partial derivativefu′(x,t,u) is continuous.Theorem 6.
If the assumptions(b1) and (b2) are satisfied, then for any ρ≥2, the initial-boundary value problem (42) has a unique mild solution u∈C([0,a],Lp([0,1],ℝ)).Proof.
From the assumptions(b1) and (b2), it is easily seen that the conditions in Theorem 4 are satisfied. Hence, by Theorem 4, for any ρ≥2, the problem (44) has a unique mild solution u∈C([0,a],𝕏), which means u is a mild solution for initial-boundary value problem (42).



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