IJDEInternational Journal of Differential Equations1687-96511687-9643Hindawi Publishing Corporation94167910.1155/2011/941679941679Research ArticleGlobal Existence of the Higher-Dimensional Linear System of ThermoviscoelasticityMaZhiyongMigórskiStanisławCollege of Science, Shanghai Second Polytechnic UniversityShanghai 201209Chinasspu.cn20111207201120110405201128062011300620112011Copyright © 2011 Zhiyong Ma.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We obtain a global existence result for the higher-dimensional thermoviscoelastic equations. Using semigroup approach, we will establish the global existence of homogeneous, nonhomogeneous, linear, semilinear, and nonlinear, thermoviscoelastic systems.

1. Introduction

In this paper, we consider global existence of the following thermoviscoelastic model: utt-μΔu-(λ+μ)divu+μg*Δu+(λ+μ)g*divu+αθt=f,(x,t)Ω×(0,),θtt-Δθt-Δθ+βdivut=h,(x,t)Ω×(0,), where the sign “*” denotes the convolution product in time, which is defined by g*v(t)=-tg(t-s)v(x,s)ds with the initial data u(x,0)=u0(x),  ut(x,0)=u1(x),  θ(x,0)=θ0(x),xΩ,θt(x,0)=θ1(x),  u(x,0)-u(x,-s)=w0(x,s),(x,s)Ω×(0,) and boundary condition u=0,  θ=0,  (x,t)Γ×(0,). The body Ω is a bounded domain in Rn with smooth boundary Γ=Ω (say C2) and is assumed to be linear, homogeneous, and isotropic. u(x,t)=(u1(x,t),u2(x,t),,un(x,t)), and θ(x,t) represent displacement vector and temperature derivations, respectively, from the natural state of the reference configuration at position x and time t. λ,μ>0 are Lamé's constants and α,β>0 the coupling parameters; g(t) denotes the relaxation function, w0(x,s) is a specified “history,” and u0(x),u1(x),θ0(x) are initial data. Δ,  ,  div denote the Laplace, gradient, and divergence operators in the space variables, respectively.

We refer to the work by Dafermos . The following basic conditions on the relaxation function g(t) are

gC1[0,)L1(0,);

g(t)0,  g(t)0,t>0;

κ=1-0g(t)dt>0.

In what follows, we denote by · the norm of L2(Ω), and we use the notation v2=i=1nvi2,for  v=(v1,v2,,vn).

When f=g=h=0, system (1.1)–(1.4) is reduced to the thermoelastic system: utt-μΔu-(λ+μ)divu+αθt=0,(x,t)Ω×(0,),θtt-Δθt-Δθ+βdivut=0,(x,t)Ω×(0,),u=0,θ=0,(x,t)Γ×(0,),u(x,0)=u0(x),  ut(x,0)=u1(x),θ(x,0)=θ0(x),  xΩ. In the one-dimensional space case, there are many works (see e.g., ) on the global existence and uniqueness. Liu and Zheng  succeeded in deriving in energy decay under the boundary condition (1.4) orux=0=0,  σx=l=0,θxx=0,l=0,ux=0=0,  σx=l=0,  θx=0=0,θxx=l=0, orux=0=0,  σx=l=0,  θx=0,  l=0, and Hansen  used the method of combining the Fourier series expansion with decoupling technique to solve the exponential stability under the following boundary condition:ux=0=0,  σx=l=0,  θxx=0=0,  θx=l=0, where σ=ux-γθ is the stress. Zhang and Zuazua  studied the decay of energy for the problem of the linear thermoelastic system of type III by using the classical energy method and the spectral method, and they obtained the exponential stability in one space dimension, and in two or three space dimensions for radially symmetric situations while the energy decays polynomially for most domains in two space dimensions.

When α=β=0, f=h=0, system (1.1)-(1.4) is decoupled into the following viscoelastic system: utt-μΔu-(λ+μ)divu+μg*Δu+(λ+μ)g*divu=0,(x,t)Ω×(0,),u=0,(x,t)Γ×(0,),u(x,0)=u0(x),  ut(x,0)=u1(x),(x,t)Ω,u(x,0)-u(x,-s)=w0(x),(x,t)Ω×(0,), and the wave equation.

There are many works (see, e.g., [9, 1215]) on exponential stability of energy and asymptotic stability of solution under different assumptions. The notation in this paper will be as follows. Lp,1p+,Wm,p,mN,H1=W1,2,H01=W01,2 denote the usual (Sobolev) spaces on Ω. In addition, ·B denotes the norm in the space B; we also put ·=·L2(Ω). We denote by Ck(I,B),  k0, the space of k-times continuously differentiable functions from JI into a Banach space B, and likewise by Lp(I,B),1p+, the corresponding Lebesgue spaces. Cβ([0,T],B) denotes the Hölder space of B-valued continuous functions with exponent β(0,1] in variable t.

2. Main Results

Let the “history space” L2(g,(0,),(H01(Ω))n) consist of ((H01(Ω))n)-valued functions w on (0,) for which wL2(g,  (0,),(H01(Ω))n)2=0g(s)w(s)(H01(Ω))n2ds<. Put H=(H01(Ω))n×(L2(Ω))n×H01(Ω)×L2(Ω)×L2(g,(0,),(H01(Ω))n) with the energy norm (u,v,θ,θt,w)H={κu(H01(Ω))n2+12(v2+αβθt2+θ)+0g(s)w(s)(H01(Ω))n2  ds}1/2, where κ denotes the positive constant in (H3), that is, κ=1-0g(t)dt>0. Thus we consider the following thermoviscoelastic system:utt-μΔu-(λ+μ)divu+μg*Δu+(λ+μ)g*divu+αθt=0,(x,t)Ω×(0,),θtt-Δθt-Δθ+βdivut=0,(x,t)Ω×(0,),u=0,        θ=0,(x,t)Γ×(0,),u(x,0)=u0(x),  ut(x,0)=u1(x),  θ(x,0)=θ0(x),  θt(x,0)=θ1(x),xΩ,u(x,0)-u(x,-s)=w0(x,s),(x,t)Ω×(0,). Let v(x,t)=ut(x,t),  w(x,t,s)=u(x,t)-u(x,t-s). Since ν-tg(t-s)u(s)ds=ν0g(s)u(t-s)ds=0g(s)ν(u(t)-w(t,s))ds=(1-κ)u(x,t)ν-0g(s)w(t,s)νds, System (2.5) can be written as follows: utt-κμΔu-κ(λ+μ)divu+αθt-μ0g(s)Δw(t,s)ds-(λ+μ)0g(s)divw(t,s)ds=0,(x,t)Ω×(0,),θtt-Δθt-Δθ+βdivut=0,(x,t)Ω×(0,),w(x,t,s)=u(x,t)-u(x,t-s),(x,t,s)Ω×(0,)×(0,),u=0,  θ=0,(x,t)Γ×(0,),u(x,0)=u0(x),  ut(x,0)=u1(x),θ(x,0)=θ0(x),  θt(x,0)=θ1(x),xΩ,w(0,s)=w0(s),(x,t)Ω×(0,). We define a linear unbounded operator A on by A(u,v,θ,θt,w)=(v,B(u,w)-αθt,θt,Δθt+Δθ-βdivv,v-ws), where ws=w/s and B(u,w)=κμΔu+κ(λ+μ)divu+μ0g(s)Δw(s)ds+(λ+μ)0g(s)divw(s)ds. Set v(x,t)=ut(x,t),w(x,t,s)=u(x,t)-u(x,t-s),Φ=(u,v,θ,θt,w),K=(0,f,0,h,0). Then problem (2.8) can be formulated as an abstract Cauchy problem dΦdt=AΦ+K, on the Hilbert space for an initial condition Φ(0)=(u0,u1,θ0,θ1,w0). The domain of A is given by D(A)={(u,v,θ,w)H:θH01(Ω),θtH01(Ω),  θ+θtH2(Ω)H01(Ω),v(H01(Ω))n,  κu+0g(s)w(s)ds(H2(Ω)H01(Ω))n,w(s)H1(g,(0,),(H01(Ω))n),w(0)=0}, where H1(g,(0,),(H01(Ω))n)={w:w,wsL2(g,(0,),(H01(Ω))n)}. It is clear that D(A) is dense in .

Our hypotheses on f,  h can be stated as follows, which will be used in different theorems:

f=h=0;

f=f(x,t)C1([0,),(L2(Ω))n),  h=h(x,t)C1([0,),L2(Ω));

f(x,t)C([0,),(H01(Ω))n),h(x,t)C([0,),H2(Ω));

f(x,t)C([0,),(L2(Ω))n),  h(x,t)C([0,),L2(Ω)), and for any T>0, ftL1((0,T),(L2(Ω))n),  htL1((0,T),L2(Ω)).

We are now in a position to state our main theorems.

Theorem 2.1.

Suppose that condition (A1) holds. Relaxation function g satisfies (H1)–(H3). Then for any Φ(0)=(u0,u1,θ0,θ1,w0)D(A), there exists a unique global classical solution Φ=(u,v,θ,θt,w) to system (2.8) satisfying Φ=(u,v,θ,θt,w)C1([0,),)C([0,),D(A)).

Theorem 2.2.

Suppose that condition (A2) holds. Relaxation function g satisfies (H1)–( H3). Then for any Φ(0)=(u0,u1,θ0,θ1,w0), there exists a unique global classical solution Φ=(u,v,θ,θt,w) to system (2.8) satisfying Φ=(u,v,θ,θt,w)C1([0,),)C([0,),D(A)), that is, uC1([0,),(H01(Ω))n)C([0,),(H2(Ω)H01(Ω))n),vC1([0,),(L2(Ω))n)C([0,),(H01(Ω))n),θC1([0,),H01(Ω))C([0,),H2(Ω)H01(Ω)),θtC1([0,),L2(Ω))C([0,),H01(Ω)),wC1([0,),L2(g,(0,),(H01(Ω))n))C([0,),H1(g,(0,),(H01(Ω))n)).

Corollary 2.3.

Suppose that condition (A3)  or (A4) holds. Relaxation function g satisfies (H1)–(H3). Then for any Φ(0)=(u0,u1,θ0,θ1,w0)D(A), there exists a unique global classical solution Φ=(u,v,θ,θt,w)C1([0,),)C([0,),D(A)) to system (2.8).

Corollary 2.4.

If f(x,t) and h(x,t) are Lipschitz continuous functions from [0,T] into (L2(Ω))n and L2(Ω), respectively, then for any Φ=(u,v,θ,θt,w)D(A), there exists a unique global classical solution Φ=(u,v,θ,θt,w)C1([0,),)C([0,),D(A)) to system (2.8).

Theorem 2.5.

Suppose relaxation function g satisfies (H1)–(H3), f=f(Φ), and h=h(Φ),Φ=(u,v,θ,θt,w), and K=(0,f,0,h,0) satisfies the global Lipschitz condition on ; that is, there is a positive constant L such that for all Φ1,Φ2, K(Φ1)-K(Φ2)HLΦ1-Φ2H. Then for any Φ(0)=(u0,u1,θ0,θ1,w0), there exists a global mild solution Φ to system (2.8) such that ΦC([0,),), that is, uC([0,),(HΓ11(Ω))n),θC([0,),H01(Ω)),θtC([0,),L2(Ω)),vC([0,),(L2(Ω))n),wC([0,),L2(g,(0,),(H01(Ω))n).

Theorem 2.6.

Suppose f=f(Φ) and h=h(Φ),Φ=(u,v,θ,θt,w), and K=(0,f,0,h,0) is a nonlinear operator from D(A) into D(A) and satisfies the global Lipschitz condition on D(A); that is, there is a positive constant L such that for all Φ1,Φ2D(A), K(Φ1)-K(Φ2)D(A)LΦ1-Φ2D(A). Then for any Φ(0)=(u0,u1,θ0,θ1,w0)D(A), there exists a unique global classical solution Φ=(u,v,θ,θt,w)C1([0,),)C([0,),D(A)) to system (2.8).

3. Some Lemmas

In this section in order to complete proofs of Theorems 2.12.6, we need first Lemmas 3.13.5. For the abstract initial value problem, dudt+Bu=K,u(0)=u0, where B is a maximal accretive operator defined in a dense subset D(B) of a Banach space H. We have the following.

Lemma 3.1.

Let B be a linear operator defined in a Hilbert space H,B:D(B)HH. Then the necessary and sufficient conditions for B being maximal accretive are

Re(Bx,x)0,  for all    xD(B),

R(I+B)=H.

Proof.

We first prove the necessity. B is an accretive operator, so we have (x,x)=x2x+λBx2=(x,x)+2λRe(Bx,x)+λ2Bx2. Thus, for all λ>0, Re(Bx,x)-λ2Bx2. Letting λ0, we get (i). Furthermore, (ii) immediately follows from the fact that B is m-accretive.

We now prove the sufficiency. It follows from (i) that for all λ>0, x-y2Re(x-y,x-y+λB(x-y))x-yx-y+λ(Bx-By). Now it remains to prove that B is densely defined. We use a contradiction argument. Suppose that it is not true. Then there is a nontrivial element x0 belonging to orthogonal supplement of D(B)¯ such that for all xD(B), (x,x0)=0. It follows from (ii) that there is x*D(B) such that x*+Bx*=x0. Taking the inner product of (3.5) with x*, we deduce that (x*+Bx*,x*)=0. Taking the real part of (3.7), we deduce that x*=0, and by (3.6), x0=0, which is a contradiction. Thus the proof is complete.

Lemma 3.2.

Suppose that B is m-accretive in a Banach space H, and u0D(B). Then problem (3.1) has a unique classical solution u such that uC1([0,),H)C([0,),D(B)).

Lemma 3.3.

Suppose that K=K(t), and K(t)C1([0,),H),        u0D(B). Then problem (3.1) admits a unique global classical solution u such that uC1([0,),H)C([0,),D(B)) which can be expressed as u(t)=S(t)u0+0tS(t-τ)K(τ)dτ.

Proof.

Since S(t)u0 satisfies the homogeneous equation and nonhomogeneous initial condition, it suffices to verify that w(t) given by w(t)=0tS(t-τ)K(τ)dτ belongs to C1([0,),H)C([0,),D(B)) and satisfies the nonhomogeneous equation. Consider the following quotient of difference w(t+h)-w(t)h=1h(0t+hS(t+h-τ)K(τ)dτ-0tS(t-τ)K(τ)dτ)=1htt+hS(t+h-τ)K(τ)dτ+1h0t(S(t+h-τ)-S(t-τ))K(τ)dτ=1htt+hS(z)K(t+h-z)dz+1h0tS(z)(K(t+h-z)-K(t-z))dz. When h0, the terms in the last line of (3.13) have limits: S(t)K(0)+0tS(z)K(t-z)dzC([0,),H). It turns out that wC1([0,),H) and the terms in the third line of (3.13) have limits too, which should be S(0)K(t)-Bw(t)=K(t)-Bw(t). Thus the proof is complete.

Lemma 3.4.

Suppose that K=K(t), and K(t)C([0,),D(B)),        u0D(B). Then problem (3.1) admits a unique global classical solution.

Proof.

From the proof of Lemma 3.2, we can obtain w(t+h)-w(t)h=1htt+hS(t+h-τ)K(τ)dτ+1h0t(S(t+h-τ)-S(t-τ))K(τ)dτ=1htt+hS(t+h-τ)K(τ)dτ+1h0tS(t-τ)(S(h)-Ih)K(τ)dτ. When h0, the last terms in the line of (3.17) have limits: S(0)K(t)-0tS(t-τ)BK(τ)dτ=S(0)K(t)-B0tS(t-τ)K(τ)dτ=K(t)-Bw(t). Combining the results of Lemma 3.3 proves the lemma.

Lemma 3.5.

Suppose that K=K(t), and K(t)C([0,),H),        u0D(B), and for any T>0, KtL1([0,T],H). Then problem (3.1) admits a unique global classical solution.

Proof.

We first prove that for any K1L1([0,T],H), the function w given by the following integral: w(t)=0tS(t-τ)K1dτ belongs to C([0,T],H). Indeed, we infer from the difference w(t+h)-w(t)=0t+hS(t+h-τ)K1(τ)dτ-0tS(t-τ)K1(τ)dτ=(S(h)-I)w(t)+tt+hS(t+h-τ)K1(τ)dτ that as h0, w(t+h)-w(t)(S(h)-I)w(t)+tt+hK1(τ)dτ0, where we have used the strong continuity of S(t) and the absolute continuity of integral for K1L1[0,t].

Now it can be seen from the last line of (3.13) that for almost every t[0,T],dw/dt exists, and it equals S(t)K(0)+0tS(z)K(t-z)dz=S(t)K(0)+0tS(t-τ)K(τ)dτC([0,T],H). Thus, for almost every t, dwdt=-Bw+K. Since w and K both belong to C([0,T],H), it follows from (3.25) that for almost every t, Bw equals a function belonging to C([0,T],H). Since B is a closed operator, we conclude that wC([0,T],D(B))C1([0,T],H) and (3.25) holds for every t. Thus the proof is complete.

To prove that the operator A defined by (2.14) is dissipative, we need the following lemma.

Lemma 3.6.

If the function f:[0,)R is uniformly continuous and is in L1(0,), then limtf(t)=0.

Lemma 3.7.

Suppose that the relaxation function g satisfies (H1) and (H2). If wH1(g,(0,),(H01(Ω))n) and w(0)=0, then g(s)w(s)(H01(Ω))n2L1(0,),limsg(s)w(s)(H01(Ω))n2=0.

Proof.

See, for example, the work by Liu in .

Lemma 3.8.

Suppose relaxation function g satisfies (H1)–(H3). The operator A defined by (2.13) is dissipative; furthermore, 0ρ(A), where ρ(A) is the resolvent of the operator A.

Proof.

By a straightforward calculation, it follows from Lemma 3.7 that A(u,v,θ,θt,w),(u,v,θ,θt,w)H=κ(v,u)(H01(Ω))n+12(B(u,w)-αθt,v)+α2β(θt,θ)+α2β(Δθt+Δθ-divv,θt)+(v-ws,w)L2(g,(0,),(H01(Ω))n)=-α2βθt2+0g(s)w(s)(H01(Ω))n2ds0. Thus, A is dissipative.

To prove that 0ρ(A), for any G=(g1,g2,g3,g4,g5), consider AΦ=G, that is, v=g1,in  (H01(Ω))n,B(u,w)-αθt=g2,in  (L2(Ω))n,θt=g3,in  L2(Ω),Δθt+Δθ-βdivv=g4,in  L2(Ω),v-ws=g5,in  L2(g,(0,),(H01(Ω))n). Inserting v=g1 and θt=g3 obtained from (3.31), (3.33) into (3.34), we obtain Δθ=g4+βdivg1-Δg3L2(Ω). By the standard theory for the linear elliptic equations, we have a unique θH2(Ω)H01(Ω) satisfying (3.36).

We plug v=g1 obtained from (3.31) into (3.35) to get ws=g1-g5L2(g,(0,),(HΓ11(Ω))n). Applying the standard theory for the linear elliptic equations again, we have a unique wH1(g,(0,),(H01(Ω))n) satisfying (3.37). Then plugging θ and w just obtained from solving (3.36), (3.37), respectively, into (3.32) and applying the standard theory for the linear elliptic equations again yield the unique solvability of uD(A) for (3.32) and such that κu+0g(s)w(s)ds(H2(Ω)H01(Ω))n. Thus the unique solvability of (3.30) follows. It is clear from the regularity theory for the linear elliptic equations that ΦKG with K being a positive constant independent of Φ. Thus the proof is completed.

Lemma 3.9.

The operator A defined by (2.13) is closed.

Proof.

To prove that A is closed, let (un,vn,θn,θnt,wn)D(A) be such that (un,vn,θn,θnt,wn)(u,v,θ,θt,w)in  H,A(un,vn,θn,θtn,wn)(a,b,c,d,e)in  H. Then we have unu        in  (H01(Ω))n,vnv        in  (L2(Ω))n,θnθ        in  H01(Ω),θntθt        in  L2(Ω),wnw        in  L2(g,(0,),(H01(Ω))n),vna        in(H01(Ω))n,B(un,wn)-αθntb        in  (L2(Ω))n,θntc        in  H01(Ω),Δθnt+Δθn-βdivvnd        in  L2(Ω),vn-wnse        in  L2(g,(0,),(H01(Ω))n). By (3.40) and (3.44), we deduce vnvin  (H01(Ω))n,v=a(H01(Ω))n. By (3.42) and (3.46), we deduce θntθtin  H01(Ω),θt=cH01(Ω). By (3.47) and (3.49), we deduce Δθnt+Δθnd+βdivv        in    L2(Ω), and consequently, it follows from (3.41), that θnt+θnθt+θ        in  H2(Ω)H01(Ω), since Δ is an isomorphism from H2(Ω)H01(Ω) onto L2(Ω). It therefore follows from (3.47) and (3.54) that d=Δθnt+Δθn-βdivv,        θt+θH2(Ω)H01(Ω). By (3.43), (3.48), and (3.49), we deduce wnw        in  H1(g,(0,),(HΓ11(Ω))n),e=v-ws,wH1(g,(0,),(H01(Ω))n),  w(0)=0. In addition, it follows from (3.39), (3.43), (3.51) that B(un,wn)-αθntB(u,w)-αθt in the distribution. It therefore follows from (3.45) and (3.58) that b=B(u,w)-αθt,        B(u,w)(L2(Ω))n, and consequently, κu+0g(s)w(s)ds(H2(Ω)H01(Ω))n, since μΔ+(λ+μ)div is an isomorphism from H2(Ω)H01(Ω) onto L2(Ω). Thus, by (3.50), (3.52), (3.55), (3.57), (3.59), (3.60), we deduce A(u,v,θ,θt,w)=(a,b,c,d,e),        (u,v,θ,θt,w)D(A). Hence, A is closed.

Lemma 3.10.

Let A be a linear operator with dense domain D(A) in a Hilbert space H. If A is dissipative and 0ρ(A), the resolvent set of A, then A is the infinitesimal generator of a C0-semigroup of contractions on H.

Proof.

See, for example, the work by Liu and Zheng in  and by Pazy in .

Lemma 3.11.

Let A be a densely defined linear operator on a Hilbert space H. Then A generates a C0-semigroup of contractions on H if and only if A is dissipative and R(I-A)=H.

Proof.

See, for example, the work by Zheng in .

4. Proofs of Theorems <xref ref-type="statement" rid="thm2.1">2.1</xref>–<xref ref-type="statement" rid="thm2.3">2.5</xref>Proof of Theorem <xref ref-type="statement" rid="thm2.1">2.1</xref>.

By (2.2), it is clear that is a Hilbert space. By Lemmas 3.83.10, we can deduce that the operator A is the infinitesimal generator of a C0-semigroup of contractions on Hilbert space . Applying the result and Lemma 3.2, we can obtain our result.

Proof of Theorem <xref ref-type="statement" rid="thm2.2">2.2</xref>.

we have known that the operator A is the infinitesimal generator of a C0-semigroup of contractions on Hilbert space . Applying the result and Lemma 3.11, we can conclude that R(I-A)=H. If we choose operator B=-A, we can obtain D(B)=D(A) and D(B) is dense in . Noting that by (A2), we know that K=(0,f,0,h,0)C1([0,),); therefore, applying Lemma 3.1, we can conclude the operator B is the maximal accretive operator. Then we can complete the proof of Theorem 2.2 in term of Lemma 3.3.

Proof of Corollary <xref ref-type="statement" rid="coro2.1">2.3</xref>.

By (A3) or (A4), we derive that K=(0,f,0,h,0)C([0,),D(A)) or KC([0,),), and for any T>0,KtL1((0,T),). Noting that B=-A is the maximal accretive operator, we use Lemmas 3.4 and 3.5 to prove the corollary.

Proof of Corollary <xref ref-type="statement" rid="coro2.2">2.4</xref>.

We know that K(x,t)=(0,f,0,h,0) are Lipschitz continuous functions from [0,T] into . Moreover, by (2.2), it is clear that is a reflexive Banach space. Therefore, KtL1([0,T],H). Hence applying Lemma 3.5, we may complete the proof of the corollary.

Proof of Theorem <xref ref-type="statement" rid="thm2.3">2.5</xref>.

By virtue of the proof of Theorem 2.2, we know that B=-A is the maximal accretive operator of a C0 semigroup S(t). On the other hand, K=(0,f,0,h,0) satisfies the global Lipschitz condition on . Therefore, we use the contraction mapping theorem to prove the present theorem. Two key steps for using the contraction mapping theorem are to figure out a closed set of the considered Banach space and an auxiliary problem so that the nonlinear operator defined by the auxiliary problem maps from this closed set into itself and turns out to be a contraction. In the following we proceed along this line.

Let ϕ(Φ)=S(t)Φ0+0tS(t-τ)K(Φ(τ))dτ,Ω={ΦC([0,+),H)supt0(Φ(t)e-kt)<}, where k is a positive constant such that k>L. In Ω, we introduce the following norm: ΦΩ=supt0(Φ(t)e-kt). Clearly, Ω is a Banach space. We now show that the nonlinear operator ϕ defined by (4.1) maps Ω into itself, and the mapping is a contraction. Indeed, for ΦΩ, we have ϕ(Φ)S(t)Φ0+0tS(t-τ)K(Φ)dτΦ0+0tK(Φ)dτΦ0+0t(LΦ(τ)+K(0))dτΦ0+C0t+Lsupt0Φ(t)e-kt0tekτdτΦ0+C0t+LkektΦΩ, where C0=K(0). Thus, ϕ(Φ)Ωsupt0[(Φ0+C0t)e-kt]+LkΦΩ<. that is, ϕ(Φ)Ω.

For Φ1,Φ2Ω, we have ϕ(Φ1)-ϕ(Φ2)Ω=supt0e-kt0tS(t-τ)(K(Φ1(τ))-K(Φ2(τ)))dτsupt0e-ktL0tΦ1-Φ2dτsupt0(e-ktLk(ekt-1))Φ1-Φ2ΩLkΦ1-Φ2Ω. Therefore, by the contraction mapping theorem, the problem has a unique solution in Ω. To show that the uniqueness also holds in C([0,),H), let Φ1,Φ2C([0,),H) be two solutions of the problem and let Φ=Φ1-Φ2. Then Φ(t)=0tS(t-τ)(K(Φ1)-K(Φ2))dτ,Φ(t)L0tΦ(τ)dτ. By the Gronwall inequality, we immediately conclude that Φ(t)=0; that is, the uniqueness in C([0,),H) follows. Thus the proof is complete.

Proof of Theorem <xref ref-type="statement" rid="thm2.4">2.6</xref>.

Since B is the maximal accretive operator, K=(0,f,0,h,0) satisfies the global Lipschitz condition on D(A). Let A1=D(B),  B1=B2:D(B1)=D(B2)A1. Then A1 is a Banach space, and B1=B2 is a densely defined operator from D(B2) into A1. In what follows we prove that B1 is m-accretive in A1=D(B).

Indeed, for any x,  yD(B2), since B is accretive in H, we have x-y+λ(Bx-By)D(B)=(x-y+λ(Bx-By)2+Bx-By+λ(B2x-B2y)2)1/2(x-y2+Bx-By2)1/2=x-yD(B). that is, B1 is accretive in A1. Furthermore, since B is m-accretive in H, for any yH, there is a unique xD(B) such that x+Bx=y. Now for any yA1=D(B), (4.10) admits a unique solution xD(B). It turns out that Bx=y-xD(B). Thus xD(B2); that is, B1 is m-accretive in A1. Let S1(t) be the semigroup generated by B1. If Φ0D(B2)=D(B1), then Φ(t)=S1(t)Φ0C([0,+),D(B2))C1([0,+),D(B)) is unique classical solution of the problem. On the other hand, Φ(t)=S1(t)Φ0 is also a classical solution in C([0,+),D(B))C1([0,+),H). This implies that S1(t) is a restriction of S(t) on A1. By virtue of the proof of Theorem 2.5, there exists a unique mild solution ΦC([0,+),A1). Since S1(t) is a restriction of S(t) on D(B), and moreover, we infer from K(Φ) being an operator from D(B) to D(B) and Lemma 3.4 that Φ is a classical solution to the problem. Thus the proof is complete.

Acknowledgment

This work was supported in part by Foundation of Shanghai Second Polytechnic University of China (no. A20XQD210006).

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