# Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation

Authors: Jin-soo Hwang
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

We consider a strongly damped quasilinear membrane equation with Dirichlet boundary condition. The goal is to prove the well-posedness of the equation in weak and strong senses. By setting suitable function spaces and making use of the properties of the quasilinear term in the equation, we have proved the fundamental results on existence, uniqueness, and continuous dependence on data including bilinear term of weak and strong solutions.


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## Body

## 1. Introduction

LetΩ be an open bounded set of Rn with the smooth boundary Γ. We set Q=0,T×Ω, Σ=0,T×Γ for T>0. The nonlinear equation of the longitudinal motion of vibrating membrane surrounding Ω with clamped boundary is described by the following Dirichlet boundary value problem:(1)∂2y∂t2-∇·∇y1+∇y2-μΔ∂y∂t=finQ,y=0onΣ,y0,x=y0x,∂y∂t0,x=y1xinΩ,where y is the height of a membrane, μ>0, f is a forcing function, and · denotes the Euclidean norm on Rn. A brief physical background of (1) is given in our previous paper [1].For damped linear or semilinear systems, there are many books and articles about the well-posedness with applications to various dynamic system’s topics (cf. [2–4], etc.) with semigroup or unified variational treatments. However, the quasilinear cases like (1) require more manipulations in the analysis of systems, because the systems like (1) are very much model-dependent due to the strong nonlinearity.Equation (1) is proposed in Kobayashi et al. [5] and the well-posedness of strongly regular solutions is studied by using the resolvent estimates of linearized operators in a modified Banach space. Besides, the well-posedness of less regular solutions is proved in [1], called weak solutions in the framework of the variational method in Dautray and Lions [3]. Based on these results, we have treated the associated optimal control and identification problems in [6] and [7], respectively. Furthermore, in [8] we have extended the results in [1] to more general quasilinear nonautonomous wave equation with strong damping term.In this paper, our concerned model is given by the following problem:(2)∂2y∂t2-∇·∇y1+∇y2-μΔ∂y∂t=Uy+finQ,y=0onΣ,y0,x=y0x,∂y∂t0,x=y1x,inΩ,where U is a bilinear forcing term which is usually referred to as bilinear control variable acting as a multiplier of the displacement term.Bilinear optimal control problems with the state equation being a linear first or second PDEs such as reaction diffusion equation or Kirchhoff plate equation are studied by some authors (see [9–12] and references therein).For future work, we will study bilinear optimal control problem with state equation (2). Then we must be faced with many difficulties because of the quasilinear term in (2). However, more regular solution of (2) corresponding to more regular data than weak one enables us to overcome these difficulties. This is motivation of this paper.As is recognized, existence of regular solution of a quasilinear PDE is also quite model-dependent due to the strong nonlinearity.We briefly summarize this paper as follows. At first, referring to the results in [1], we shall prove the well-posedness of weak solution of (2). Secondly, we shall analyze (2) in higher regularity than in [1] by employing newly constructed energy equality for (2). Finally, we shall prove by exploiting the well-posedness of weak solution of (2) that the regular solution of (2) is continuous with respect to regular data.The most difficult part of the existence proof is to show the strong convergence of nonlinear terms, and the part is completed by using the argument in [3, p. 569]. This is another novelty of the paper.


## 2. Notations and Main Results

IfX is a Banach space, we denote by X′ its topological dual and by ·,·X′,X the duality pairing between X′ and X. We introduce the following abbreviations:(3)Lp=LpΩ,Hk=HkΩ,·p=·Lp, with p≥1. And H0k mean the completions of C0∞Ω in Hk for k≥1. Let DΔ=H2∩H01. If we denote the scalar product on L2 by ·,·2, then the scalar products on H01 and DΔ are given as follows: (4)ψ,ϕH01=∇ψ,∇ϕ2,∀ψ,ϕ∈H01;ψ,ϕDΔ=Δψ,Δϕ2,∀ψ,ϕ∈DΔ. Then obviously,(5)ψH01=∇ψ2,∀ψ∈H01,ϕDΔ=Δϕ2,∀ϕ∈DΔ. The duality pairing between H01 and H-1 is denoted by ϕ,ψ1,-1.It is clear that(6)DΔ↪H01↪L2↪H-1↪DΔ′, and each space is dense in the following one and the injections are continuous.Related to the nonlinear term in (2), we define the function G:Rn→Rn by Gx=x/1+x2,x∈Rn. Then it is easily verified that(7)Gx-Gy≤2x-y,∀x,y∈Rn.The nonlinear operator G∇·:H01→L∞n is defined by(8)G∇ϕx=∇ϕx1+∇ϕx2,a.e.x∈Ω,∀ϕ∈H01.By the definition of G∇· in (8), we have the following useful property on G∇·:(9)G∇ϕ2≤∇ϕ2,G∇ϕ-G∇ψ2≤2∇ϕ-∇ψ2,∀ϕ,ψ∈H01.The solution space W0,T for weak solutions of (2) is defined by(10)W0,T=g∣g∈L20,T;H01,g′∈L20,T;H01,g′′∈L20,T;H-1 endowed with a norm(11)gW0,T=gL20,T;H012+g′L20,T;H012+g′′L20,T;H-121/2, where g′ and g′′ denote the first- and second-order distributive derivatives of g. We remark that W0,T is continuously embedded in C0,T;H01∩C10,T;L2 (cf. Dautray and Lions [3, p. 555]).Definition 1.
A functiony is said to be a weak solution of (2) if y∈W0,T and y satisfies(12)y′′·,ϕ-1,1+G∇y·,∇ϕ2+μ∇y′·,∇ϕ2=U·y·,ϕ2+f·,ϕ-1,1∀ϕ∈H01in  the  sense  ofD′0,T,y0=y0∈H01,y′0=y1∈L2.The following theorem gives the fundamental results on existence, uniqueness, and regularity of weak solutions of (2).Theorem 2.
Assume thaty0,y1,f∈H01×L2×L20,T;H-1, and U∈L∞Q. Then problem (2) has a unique weak solution y∈W0,T. Moreover, the solution mapping p=y0,y1,f,U→yp of P≡H01×L2×L20,T;H-1×L∞Q into W0,T is locally Lipschitz continuous.Indeed, letp1=y01,y11,f1,U1∈P and p2=y02,y12,f2,U2∈P. We prove Theorem 2 by showing the inequality(13)yp1-yp2W0,T≤C∇y01-y0222+y11-y1222+f1-f2L20,T;H-12+U1-U2L∞Q21/2≡Cp1-p2P,where C>0 is a constant depending on data.Next we introduce the solution spaceS0,T for strong solutions of (2) defined by(14)S0,T=g∣g∈L20,T;DΔ,g′∈L20,T;DΔ,g′′∈L2Q endowed with a norm(15)gS0,T=gL20,T;DΔ2+g′L20,T;DΔ2+g′′L2Q21/2, where g′ and g′′ denote the first- and second-order distributive derivatives of g. We remark also from Dautray and Lions [3, p. 555] that S0,T is continuously embedded in C0,T;DΔ∩C10,T;H01.Definition 3.
A functiony is said to be a strong solution of (2) if y∈S0,T and y satisfies(16)y′′t-∇·G∇yt-μΔy′t=Utyt+ft,a.e.t∈0,T,y0=y0∈DΔ,y′0=y1∈H01.The next theorem gives a well-posedness result for strong solutions of (2).Theorem 4.
Assume thaty0,y1,f∈DΔ×H01×L2Q, and U∈L∞Q. Then (2) has a unique strong solution y∈S0,T and it satisfies(17)yS0,T≤Cy0DΔ2+y1H012+fL2Q2+UL∞Q21/2,where C is a constant depending on data.Now we give the result on the continuous dependence of strong solutions of (2) on p=y0,y1,f,U. Let F be a product space defined by(18)F=DΔ×H01×L2Q×L∞Qendowed with a norm(19)pF=y0DΔ2+y1H012+fL2Q2+UL∞Q21/2. For each p=y0,y1,f,U∈F we have a strong solution y=yp∈S0,T of (2) by Theorem 4. Thus, we can define the solution mapping p=y0,y1,f,U→yp of F into S0,T.Theorem 5.
The nonlinear solution mappingp=y0,y1,f,U→yp of F into S0,T of (2) is continuous.Throughout this paper, we will useC as a generic constant and omit writing the integral variables in any definite integrals without confusion.


## 3. Proof of Main Results

Proof of Theorem2.
SinceUi∈L∞Qi=1,2, by the results in [1], we can deduce that the weak solutions ypi of (2) corresponding to pii=1,2 exist in W0,T such that(20)ypiW0,T≤C∇y0i22+y1i22+fiL20,T;H-121/2≤CpiPi=1,2.We denote y1-y2≡yp1-yp2 by ψ. Then, we can get from (2) that ψ satisfies the following equation in weak sense:(21)ψ′′-μΔψ′=∇·G∇y1-∇·G∇y2+U1ψ+Uy2+finQ,ψ=0onΣ,ψ0=y01-y02,ψ′0=y11-y12inΩ,where U=U1-U2 and f=f1-f2.
We multiply (21) by ψ′+ψ to have(22)12ddtψ′t22+μ∇ψt22+μ∇ψ′t22=-G∇y1t-G∇y2t,∇ψ′t+ψt2+U1tψt+Uty2t,ψ′t+ψt2-ψ′′t,ψt-1,1+ft,ψ′t-ψt-1,1.
By integrating (22) over 0,t, we obtain(23)ψ′t22+2μ∫0t∇ψ′22ds+μ∇ψt22=ψ′022+μ∇ψ022-2∫0tG∇y1-G∇y2,∇ψ′+ψ2ds+2∫0tU1ψ+Uy2,ψ′+ψ2ds-2ψ′t,ψt2+2ψ′0,ψ02+2∫0tψ′22ds+2∫0tf,ψ′+ψ-1,1ds.Let ϵ>0 be an arbitrary real number. Then, by (9), (20), and the Schwartz inequality we can obtain the following:(24)2∫0tG∇y1-G∇y2,∇ψ′+ψ2ds≤4∫0t∇ψ2∇ψ′2ds+4∫0t∇ψ22ds≤4ϵ+4∫0t∇ψ22ds+ϵ∫0t∇ψ′22ds;2∫0tU1ψ+Uy2,ψ′+ψ2ds≤2∫0tU1ψ2+Uy22ψ′2+ψ2ds≤2U1L∞Q∫0tψ2ψ′2+ψ22ds+∫0tUy222ds+2∫0tψ22+ψ′22ds≤C∫0t∇ψ22+ψ′22ds+UL∞Q2y2L2Q2≤C∫0t∇ψ22+ψ′22ds+UL∞Q2y2W0,T2≤C∫0t∇ψ22+ψ′22ds+UL∞Q2p2P2;2∫0tf,ψ′+ψ-1,1ds≤2∫0tfH-1∇ψ′2+∇ψ2ds≤1ϵ+1fL20,T;H-12+ϵ∫0t∇ψ′22ds+∫0t∇ψ22ds;2ψ′t,ψt2≤2ψ′t2ψ0+∫0tψ′ds2≤ϵψ′t22+1ϵψ0+∫0tψ′ds22≤ϵψ′t22+2ϵψ022+2Tϵ∫0tψ′22ds.We also note that 2ψ′0,ψ02≤C∇ψ022+ψ′022. Therefore, from (23) and (24), we can obtain the following inequality:(25)1-ϵψ′t22+μ∇ψt22+2μ-2ϵ∫0t∇ψ′22ds≤C1+1+Tϵ+p2P2∇ψ022+ψ′022+fL20,T;H-12+UL∞Q2+∫0tψ′22+∇ψ22ds.If we choose ϵ=min⁡1/2,μ/2, then by Bellman-Gronwall’s inequality it follows that(26)ψ′t22+∇ψt22+∫0t∇ψ′22ds≤CT,p2∇ψ022+ψ′022+fL20,T;H-12+UL∞Q2=CT,p2p1-p2P2.By (21) and (26) we have(27)ψ′′L20,T;H-1≤C∇ψ′L2Q+G∇y1-G∇y2L2Q+ψL2Q+UL∞Qy2L2Q+fL20,T;H-1≤C∇ψ′L2Q+∇ψL2Q+UL∞Qy2W0,T+fL20,T;H-1≤C1T,p2p1-p2P+UL∞Qp2P+fL20,T;H-1≤C2T,p2p1-p2P.Finally, by combining (26) and (27) we obtain (13).
This completes the proof.Lemma 6.
LetX,Y, and Z be Banach spaces such that the imbeddings X⊂Y⊂Z are continuous and the imbedding X⊂Y is compact. Then a bounded set of W1,∞0,T;X,Z=g∣g∈L∞0,T;X,g′∈L∞0,T;Z is relatively compact in C0,T;Y.Proof.
See Simon [13].Proof of Theorem4.
We divide the proof into three steps.
Step
1 (approximate solutions and a priori estimates). We construct approximate solutions of (2) by a Faedo–Galerkin’s procedure. Since DΔ is separable, there exists a complete orthonormal system wmm=1∞ in L2 such that wmm=1∞ is free and total in DΔ. For each m∈N we can define an approximate solution of (2) by(28)ymt=∑j=1mgjmtwj, where ymt satisfies (2). Then (2) can be written as m vector differential equations(29)E~d2dt2g→m+G~∇ymg→m+μΔ~ddtg→m=U~g→m+f→mwith initial values (30)g→m0=y0,w12,y0,w22,…,y0,wm2t,ddtg→m0=y1,w12,y1,w22,…,y1,wm2t. Notations of (29) can be explained as follows: (31)g→m=g1m,…,gmmt,E~=wi,wj2=δij:i=1,…,m,j=1,…,m≡Im,G~∇ym=11+∇ym2∇wi,∇wj2:i=1,…,m,j=1,…,m,μΔ~=μ∇wi,∇wj2:i=1,…,m,j=1,…,m,U~=Uwi,wj2:i=1,…,m,j=1,…,m,f→m=f,w12,…,f,wm2t, where ⋯t denotes the transpose of ⋯. Since G∇·:H01→L∞n is Lipschitz continuous and U∈L∞Q, we can deduce by Carathéodory type existence theorem that the nonlinear vector differential equation (29) admits a unique solution g1m,g2m,…,gmmt on 0,T. Hence, we can construct the approximate solution ymt of (2). Next we shall derive a priori estimates of ymt.
By analogy with (22), we take L2 product of the equations for approximate solutions ymt with -Δym′t-Δymt to have(32)12ddt∇ym′t22+μΔymt22+μΔym′t22=-∇·G∇ymt,Δym′t+Δymt2-Utymt,Δym′t+Δymt2+ym′′t,Δymt2-ft,Δym′t+Δymt2.By integrating (32) over 0,t, we obtain(33)∇ym′t22+2μ∫0tΔym′22ds+μΔymt22=∇y1m22+μΔy0m22-2∫0t∇·G∇ym,Δym′+Δym2ds-2∫0tUym,Δym′+Δym2ds+2ym′t,Δymt2-2y1m,Δy0m2+2∫0t∇ym′22ds-2∫0tf,Δym′+Δym2ds.Here we note from the elliptic regularity theory that(34)∇·G∇ϕ2=Δϕ1+∇ϕ2-∑i,j=1nϕxiϕxjϕxixj1+∇ϕ23/22≤Δϕ2+∑i,j=1nϕxixj2≤CΔϕ2∀ϕ∈DΔ.Thus, by (34) we can deduce for ϵ>0(35)2∫0t∇·G∇ym,Δym′+Δym2ds≤2∫0t∇·G∇ym2Δym′+Δym2ds≤C∫0tΔym2Δym′2+Δym2ds≤ϵ∫0tΔym′22ds+C∫0tΔym22ds;2ym′t,Δymt2≤2∇ym′t2∇y0m+∫0tym′ds2≤ϵ∇ym′t22+1ϵ∇y0m+∫0tym′ds22≤ϵ∇ym′t22+2ϵ∇y0m22+2Tϵ∫0t∇ym′22ds.For other estimations of the terms in the RHS of (33) other than (35), we can follow the analogous process in the proof of Theorem 2 to get(36)∇ym′t22+∫0tΔym′22ds+Δymt22≤C∇y1m22+Δy0m22+fL2Q2+∫0t∇ym′22+Δym22ds.Therefore, it is shown by using Bellman-Gronwall’s inequality that(37)∇ym′t22+Δymt22+∫0tΔym′22ds≤CΔy0m22+∇y1m22+fL2Q2,∀t∈0,T.And also from Theorem 2, (2), (34), and (37), we have(38)ym′′L2Q≤∇·G∇ymL2Q+μΔym′L2Q+UymL2Q+fL2Q≤CΔymL2Q+Δym′L2Q+UL∞QymL2Q+fL2Q≤CΔymL2Q+Δym′L2Q+UL∞QymW0,T+fL2Q≤C1+pmPΔy0m2+∇y1m2+fL2Q+UL∞Q,where pm=y0m,y1m,f,U.
Step
2 (passage to the limits). Equations (37) and (38) imply that(39)ymis  bounded  inS0,T∩W1,∞0,T;DΔ,H01.And the nonlinear term ∇·G∇ym is bounded in L∞0,T;L2. Hence, by the extraction theorem of Rellich, we can extract a subsequence ymk of ym and find y∈S0,T∩W1,∞0,T;DΔ,H01 and G∈L∞0,T;L2 such that(40)(i)ymk⟶yweakly-star  inL∞0,T;DΔand  weakly  inS0,T,(ii)ymk′⟶y′weakly-star  inL∞0,T;H01,(iii)∇·G∇ymk⟶Gweakly-star  inL∞0,T;L2,as k→∞. Since DΔ↪H01 is compact, we can apply Lemma 6 with X=DΔ,Y=Z=H01 and Aubin-Lions-Temam’s compact imbedding theorem (cf. Temam [14, p. 274]) to (39) to verify that ym and ym′ are precompact in C0,T;H01 and L20,T;H01, respectively. Hence, we can find a subsequence ymk⊂ym, if necessary, such that(41)ymk⟶ystrongly  inC0,T;H01,(42)ymk′⟶y′strongly  inL20,T;H01as k→∞.
By the standard argument of Dautray and Lions [3, pp. 564–566], it can be verified that the limit y of ymk is a strong solution of the linear problem(43)y′′-G-μΔy′=Uy+finQ,y=0onΣ,y0,x=y0,y′0,x=y1,inΩ.
Step
3 (strong convergence of approximate solutions). In order to prove that y is a strong solution of (2), it is sufficient to prove G=∇·G∇y. For this, we shall show ymt→yt strongly in DΔ for all t∈0,T. To prove the strong convergence, we use the modified arguments in Dautray and Lions [3, pp. 579–581] and the classical compact imbedding theorem.
First as in (33), we take L2 product equation (43) with -Δy′-Δy and integrate it over 0,t to have(44)∇y′t22+2μ∫0tΔy′22ds+μΔyt22=∇y122+μΔy022-2∫0tG,Δy′+Δy2ds-2∫0tUy,Δy′+Δy2ds+2y′t,Δyt2-2y1,Δy02+2∫0t∇y′22ds-2∫0tf,Δy′+Δy2ds.By making use of the following trivial equalities: (45)ϕ22+φ22=ϕ-φ22+2ϕ,φ2,∀ϕ,φ∈L2,ϕ1,φ12=ϕ1-ϕ2,φ1-φ22+ϕ1,φ22+ϕ2,φ1-φ22,∀ϕi,φi∈L2i=1,2, we add (33) to (44) and denote ym-y by ψm to get(46)∇ψm′t22+2μ∫0tΔψm′22ds+μΔψmt22=Φm0+2∑i=13Φmit-2∫0t∇·G∇ym-∇·G∇y,Δψm′+Δψm2ds,where (47)Φm0=∇y1m22+∇y122+μ∇y0m22+∇y022-2y1m,Δy0m2+y1,Δy02,Φm1t=-∇ym′t,∇y′t2-μΔymt,Δyt2-2μ∫0tΔym′,Δy′2ds,Φm2t=-∫0tUym,Δym′+Δym2ds-∫0tUy,Δy′+Δy2ds+∫0t∇ym′22ds+∫0t∇y′22ds-∫0tf,Δym′+Δym2ds-∫0tf,Δy′+Δy2dsds+ym′t,Δymt2+y′t,Δyt2,Φm3t=-∫0t∇·G∇ym,Δy′+Δy2ds-∫0tG,Δy′+Δy2ds-∫0t∇·G∇y,Δψm′+Δψm2ds. For simplicity we set(48)Φmt=Φm0+2∑i=13Φmit.It is verified by direct computations that(49)∇·G∇ym-G∇y=Km+Rm,where(50)Km=Δym-Δy1+∇ym2-∑i,j=1nymxiymxjymxixj-yxixj1+∇ym23/2,(51)Rm=Δy11+∇ym2-11+∇y2+∑i,j=1nyxixjyxiyxj1+∇y23/2-ymxiymxj1+∇ym23/2.By (49)–(51), (46) can be rewritten by(52)∇ψm′t22+2μ∫0tΔψm′22ds+μΔψmt22=Φmt-2∫0tRm,Δψm′+Δψm2ds-2∫0tKm,Δψm′+Δψm2ds.The term Km can be estimated as(53)Km2≤Δψm2+∑i=1nψmxixj2≤CΔψm2.Then by routine calculations in (52) together with (53), we can derive the following inequality:(54)∇ψm′t22+Δψmt22+∫0tΔψm′22ds≤CΦmt+∫0tRm22ds+C∫0tΔψm22ds.By applying the extended Bellman-Gronwall’s inequality to (54), we deduce(55)∇ψm′t22+Δψmt22+∫0tΔψm′22ds≤CΦmt+RmL2Q+C∫0tΦm+RmL2Qds.
By virtue of the strong convergence of the initial values and (40)–(42), we can extract a subsequence ymk of ym such that(56)Φmk0⟶2∇y122+2μΔy022-4y1,Δy02,Φmk1t⟶-∇y′t22-μΔyt22-2μ∫0tΔy′22ds,Φmk2t⟶-2∫0tUy,Δy′+Δy2ds+2∫0t∇y′22ds-2∫0tf,Δy′+Δy2ds+2y′t,Δyt2,Φmk3t⟶-2∫0tG,Δy′+Δy2dsas k→∞. Therefore, in view of (44) we can deduce by the sum of limits in (56) that(57)Φmkt⟶0ask⟶∞.Also from (41), we can easily verify that there exists a subsequence ymk of ym such that(58)ymkxi⟶yxia.e.x∈Ω,i=1,…,nfor all t∈0,T. Then it follows from (58) that(59)11+∇ymkt,x2⟶11+∇yt,x2a.e.x∈Ω,ymkxit,xymkxjt,x1+∇ymkt,x23/2⟶yxit,xyxjt,x1+∇yt,x23/2a.e.x∈Ωfor all t∈0,T. Hence, we have(60)limk→∞⁡Rmkt,x=0a.e.t,x∈Q.Since(61)Rmt,x≤2Δyt,x+2∑i,j=1nyxixjt,xa.e.t,x∈Q,we see from (60), (61), and the Lebesgue dominated convergence theorem that(62)Rmk⟶0strongly  inL2Qask⟶∞.By applying (57) and (62) to (55) with m=mk, we have(63)ψmk,ψmk′⟶0,0strongly  inC0,T;DΔ×C0,T;H01∩L20,T;DΔas k→∞. From (49), (53), (62), and (63), it follows that(64)∇·G∇ymk⟶∇·G∇ystrongly  inL2Qas k→∞. Thus, we readily have(65)G=∇·G∇y.Therefore, we have proved the existence of a strong solution of (2).
By similar estimations as in (37) and (38), we can show (17).
The uniqueness of strong solutions is evident from the uniqueness of weak solutions.Proof of Theorem5.
Letyp+δp and yp be the strong solutions of (2) corresponding to p+δp=y0+δy0,y1+δy1,f+δf,U+δU∈F and p=y0,y1,f,U∈F, respectively. Set ψδ=yp+δp-yp. Then ψδ satisfies(66)ψδ′′-μΔψδ′=∇·G∇yp+δp-∇·G∇yp+Uψδ+δUyp+δp+δfinQ,ψδ=0onΣ,ψδ0,x=δy0,ψδ′0,x=δy1,inΩin the weak sense. Here we can repeat the same estimations in the proof of Theorem 4 mainly by performing L2Q-product of (66) with -Δψδ′-Δψδ to obtain(67)∇ψδ′t22+Δψδt22+∫0tΔψδ′22ds≤C∫0tRδ22ds+Δδy022+∇δy122+δUyp+δp+δfL2Q2≤CRδL2Q2+Δδy022+∇δy122+δUL∞Q2yp+δpL2Q2+δfL2Q2≤CRδL2Q2+Δδy022+∇δy122+δUL∞Q2p+δpP2+δfL2Q2≤C1+p+δpP2RδL2Q2+δpF2,where Rδ is given by (51) in which ym and y are replaced by yp+δp and yp, respectively. By Theorem 2, we know that(68)ψδW0,T≤CδpP. Equation (68) immediately implies(69)ψδ⟶0strongly  inC0,T;H01asδp⟶0inP.Whence from (58)–(62) we know that(70)Rδ⟶0strongly  inL2Qasδp⟶0inP.Finally by (66), (67), and (70), we can conclude that(71)ψδ⟶0strongly  inS0,Tasδp⟶0inF.This completes the proof.



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*Source: 4529847-2017-06-11.xml*