AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 797516 10.1155/2012/797516 797516 Research Article Regularity for Variational Evolution Integrodifferential Inequalities Kang Yong Han 1 Jeong Jin-Mun 2 Piskarev Sergey 1 Institute of Liberal Education Catholic University of Daegu, Daegue 712-702 Republic of Korea cataegu.ac.kr 2 Department of Applied Mathematics Pukyong National University, Busan 608-737 Republic of Korea pknu.ac.kr 2012 3 10 2012 2012 08 05 2012 28 06 2012 2012 Copyright © 2012 Yong Han Kang and Jin-Mun Jeong. 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 deal with the regularity for solutions of nonlinear functional integrodifferential equations governed by the variational inequality in a Hilbert space. Moreover, by using the simplest definition of interpolation spaces and the known regularity result, we also prove that the solution mapping from the set of initial and forcing data to the state space of solutions is continuous, which very often arises in application. Finally, an example is also given to illustrate our main result.

1. Introduction

In this paper, we deal with the regularity for solutions of nonlinear functional integrodifferential equations governed by the variational inequality in a Hilbert space H: (VIP)(x(t)+Ax(t),x(t)-z)+ϕ(x(t))-ϕ(z)(0tk(t-s)g(s,x(s))ds+h(t),x(t)-z),a.e.,0<tT,zH,x(0)=x0, where A is a unbounded linear operator associated with a sesquilinear form satisfying Gårding’s inequality and ϕ:H(-,+] is a lower semicontinuous, proper convex function. The nonlinear mapping g is a Lipschitz continuous from ×V into H in the second coordinate, where V is a dense subspace of H.

The background of these problems has emerged vigorously in such applied fields as automatic control theory, network theory, and the dynamic systems.

By using the subdifferential operator ϕ, the control system (VIP) is represented by the following nonlinear functional differential equation on H: (NDE)x(t)+Ax(t)+ϕ(x(t))0tk(t-s)g(s,x(s))ds+h(t),0<tT,x(0)=x0.

In Section 4.3.2 of Barbu  (also see Section  4.3.1 in ) is widely developed the existence of solutions for the case g0. Recently, the regular problem for solutions of the nonlinear functional differential equations with a nonlinear hemicontinuous and coercive operator A was studied in . Some results for solutions of a class of semilinear equations with the nonlinear terms have been dealt with in . As for nontrivial physical examples from the field of visco-elastic materials modeled by integrodifferential equations on Banach spaces, we refer to .

In this paper, we will define ϕϵ:HH(ϵ>0) such that the function ϕϵ is Fréchet differentiable on H and its Frećhet differential ϕϵ is a single valued and Lipschitz continuous on H with Lipschitz constant ϵ-1, where ϕϵ=ϵ-1(I-(I+ϵϕ)-1) as is seen in Corollary  2.2 in [1, Chapter II]. It is also well-known results that limϵ0ϕϵ=ϕ and limϵ0ϕϵ(x)=(ϕ)0(x) for every xD(ϕ), where (ϕ)0 is the minimal segment of ϕ. Now, we introduce the smoothing system corresponding to (NDE) as follows: (SDE 1)x(t)+Ax(t)+ϕϵ(x(t))=0tk(t-s)g(s,x(s))ds+h(t),0<tT,x(0)=x0.

First we recall some regularity results and a variation of constant formula for solutions of the semilinear functional differential equation (in the case g0 in (SDE 1): (1.1)x(t)+Ax(t)+ϕϵ(x(t))=h(t) in a Hilbert space H.

Next, based on the regularity results for (1.1), we intend to establish the regularity for solutions of (NDE). Here, our approach is that results of a class of semilinear equations as (1.1) on L2-regularity remain valid under the above formulation perturbed of nonlinear terms. Here, we note that sine A is not bounded operator H into itself, the Lipschitz continuity of nonlinear terms must be defined on some adjusted spaces (see Section 3). Moreover, using the simplest definition of interpolation spaces and known regularity, we have that the solution mapping from the set of initial and forcing data to the state space of solutions is continuous, which very often arises in application. Finally, an example is also given to illustrate our main result.

2. Preliminaries

Let V and H be complex Hilbert spaces forming Gelfand triple VHV* with pivot space H. The norms of V, H and V* are denoted by ||·||, |·|, and ||·||*, respectively. The inner product in H is defined by (·,·). The embeddings (2.1)VHV* are continuous. Then the following inequality easily follows: (2.2)u*|u|u,uV.

Let a(·,·) be a bounded sesquilinear form defined in V×V and satisfying Gårding’s inequality (2.3)Re  a(u,u)ω1u2-ω2|u|2,ω1>0,ω20. Let A be the operator associated with the sesquilinear form a(·,·): (2.4)(Au,v)=a(u,v),u,vV. Then A is a bounded linear operator from V to V* and -A generates an analytic semigroup in both of H and V* as is seen in [9, Theorem 6.1]. The realization for the operator A in H which is the restriction of A to (2.5)D(A)={uV;  AuH} is also denoted by A. From the following inequalities: (2.6)ω1u2Re  a(u,u)+ω2|u|2C|Au|  |u|+ω2|u|2max{C,ω2}uD(A)|u|, where (2.7)uD(A)=(|Au|2+|u|2)1/2 is the graph norm of D(A), it follows that there exists a constant C1>0 such that (2.8)uC1uD(A)1/2|u|1/2. Thus, we have the following sequence: (2.9)D(A)VHV*D(A)*, where each space is dense in the next one and continuous injection.

Lemma 2.1.

With the notations (2.8), (2.9), one has (2.10)(D(A),H)1/2,2=V, where (D(A),H)1/2,2 denotes the real interpolation space between D(A) and H (Section 2.4 of  or ).

The following abstract linear parabolic equation: (LE)x(t)+Ax(t)=h(t),0<tT,x(0)=x0, has a unique solution xL2(0,T;D(A))W1,2(0,T,H) for each T>0 if x0V(D(A),H)1/2,2 and hL2(0,T;H). Moreover, one has (2.11)xL2(0,T;D(A))W1,2(0,T,H)C2(x0(D(A),H)1/2,2+hL2(0,T;H)), where C2 depends on T and M (see [12, Theorem 2.3], ).

In order to substitute H for the intermediate space V considering A as an operator in B(V,V*) instead of B(D(A),H) one proves the following result.

Lemma 2.2.

Let T>0. Then (2.12)H={xV*:0TAetAx*2dt<}. Hence, it implies that H=(V,V*)1/2,2 in the sense of intermediate spaces generated by an analytic semigroup.

Proof.

Put u(t)=etAx for xH. From the result of Theorem  2.3 in  it follows (2.13)uL2(0,T;V)W1,2(0,T;V*), hence (2.14)0TAetAx*2dt=0Tu(t)*2dt<.

Conversely, suppose that xV* and 0T||AetAx||*2dt<. Put u(t)=etAx. Then since A is an isomorphism from V to V* there exists a constant c>0 such that (2.15)0Tu(t)2dtc0TAu(t)*2dt=c0TAetAx*2dt. Thus, we have uL2(0,T;V)W1,2(0,T;V*). By using the definition of real interpolation spaces by trace method, it is known that the embedding L2(0,T;V)W1,2(0,T;V*)C([0,T];H) is continuous. Hence, it follows x=u(0)H.

In view of Lemma 2.2 we can apply (2.11) to (LE) in the space V* as follows.

Proposition 2.3.

Let x0H and hL2(0,T;V*), T>0. Then there exists a unique solution x of (LE) belonging to (2.16)L2(0,T;V)W1,2(0,T;V*)C([0,T];H) and satisfying (2.17)xL2(0,T;V)W1,2(0,T;V*)C2(|x0|+hL2(0,T;V*)), where C2 is a constant depending on T.

Let ϕ:V(-,+] be a lower semicontinuous, proper convex function. Then the subdifferential operator ϕ of ϕ is defined by (2.18)ϕ(x)={x*V*;ϕ(x)ϕ(y)+(x*,x-y),  yV}. First, let us concern with the following perturbation of subdifferential operator: (VE)x(t)+Ax(t)+ϕ(x(t))h(t),0<tT,x(0)=x0.

Using the regularity for the variational inequality of parabolic type in case where ϕ:V(-,+] is a lower semicontinuous, proper convex function as is seen in [1, Section 4.3] one has the following result on (VE).

Proposition 2.4.

(1) Let hL2(0,T;V*) and x0V satisfying that ϕ(x0)<. Then (VE) has a unique solution: (2.19)xL2(0,T;V)W1,2(0,T;V*)C([0,T];H), which satisfies (2.20)x(t)=(h(t)-Ax(t)-ϕ(x(t)))0,xL2W1,2CC3(1+x0+hL2(0,T;V*)), where C3 is a constant and L2W1,2C=L2(0,T;V)W1,2(0,T;V*)C([0,T];H).

(2) Let A be symmetric and let us assume that there exist gH such that for every ϵ>0 and any yD(ϕ)(2.21)Jϵ(y+ϵg)D(ϕ),ϕ(Jϵ(y+ϵg))ϕ(y). Then for hL2(0,T;H) and x0D(ϕ)¯V,  (VE) has a unique solution: (2.22)xL2(0,T;D(A))W1,2(0,T;H)C([0,T];H), which satisfies (2.23)xL2W1,2CC3(1+x0+hL2(0,T;H)).

Remark 2.5.

When the principal operator A is bounded from H to itself, we assume that ϕ:H(-,+] is a lower semicontinuous, proper convex function and g:[0,T]×HH be a nonlinear mapping satisfying the following: (2.24)|g(t,x1)-g(t,x2)|L|x1-x2|,x1,x2H. Then it is easily seen that the result of (2) of Proposition 2.4. is immediately obtained.

Remark 2.6.

Here, we remark that if V is compactly embedded in H and xL2(0,T;V)) (or the semigroup operator S(t) is compact), the following embedding: (2.25)L2(0,T;V)W1,2(0,T;V*)L2(0,T;H) is compact in view of Theorem 2 of Aubin . Hence, the mapping (x0,f)x is compact from V×L2(0,T;V*) to L2(0,T;H), which is also applicable to optimal control problem.

3. Regularity for Solutions

Assumption (<italic>F</italic>).

Let g:[0,T]×VH be a nonlinear mapping satisfying the following: (3.1)|g(t,x)-g(t,y)|  Lx-y,g(t,0)=0x,yV for a positive constant L.

For xL2(0,T;V) we set (3.2)f(t,x)=0tk(t-s)g(s,x(s))ds, where k belongs to L2(0,T).

Lemma 3.1.

Let xL2(0,T;V), T>0. Then f(·,x)L2(0,T;H). And (3.3)f(·,x)L2(0,T;H)LkL2TxL2(0,T;V). Moreover, if x1,x2L2(0,T;H), then (3.4)f(·,x1)-f(·,x2)L2(0,T;H)LkTx1-x2L2(0,T;V).

The proof is immediately obtained from Assumption (F).

For every ϵ>0, define (3.5)ϕϵ(x)=inf{x-Jϵx*22ϵ+ϕ(Jϵx):xH}, where Jϵ=(I+ϵϕ)-1. Then the function ϕϵ is Frećhet differentiable on H and its Frećhet differential ϕϵ is Lipschitz continuous on H with Lipschitz constant ϵ-1 where ϕϵ=ϵ-1(I-(I+ϵϕ)-1) as is seen in Corollary 2.2 in [1, Chapter II]. It is also well-known results that limϵ0ϕϵ=ϕ and limϵ0ϕϵ(x)=(ϕ)0(x) for every xD(ϕ), where (ϕ)0 is the minimal segment of ϕ.

Now, one introduces the smoothing system corresponding to (NDE) as follows: (SDE 2)x(t)+Ax(t)+ϕϵ(x(t))=f(t,x)+h(t),0<tT,x(0)  =x0. Since -A generates a semigroup S(t) on H, the mild solution of (SDE 2) can be represented by (3.6)xϵ(t)=S(t)x0+0tS(t-s){f(s,xϵ)+h(s)-ϕϵ(xϵ(s))}ds.

One will use a fixed point theorem and a step and step method to get the global solution for (NDE). Then one needs the following hypothesis.

Assumption (<italic>A</italic>).

( ϕ ) 0 is uniformly bounded, that is, (3.7)|(ϕ)0x|M1,xV.

Lemma 3.2.

For given ϵ, λ>0, let xϵ and xλ be the solutions of (SDE 2) corresponding to ϵ and λ, respectively. Then there exists a constant C independent of ϵ and λ such that (3.8)xϵ-xλC([0,T];H)L2(0,T;V)C(ϵ+λ),0<T.

Proof.

From (SDE 2) we have (3.9)xϵ(t)-xλ(t)+A(xϵ(t)-xλ(t))+ϕϵ(xϵ(t))-ϕλ(xλ(t))=f(t,xϵ)-f(t,xλ), and hence, from (2.3) and multiplying by xϵ(t)-xλ(t), it follows that (3.10)12ddt|xϵ(t)-xλ(t)|2+ω1xϵ(t)-xλ(t)2+(ϕϵ(xϵ(t))-ϕλ(xλ(t)),xϵ(t)-xλ(t))(f(t,xϵ)-f(t,xλ),xϵ(t)-xλ(t))+ω2|xϵ(t)-xλ(t)|2. Here, we note (3.11)|f(t,xϵ)-f(t,xλ)|LkL2xϵ(·)-xλ(·)L2(0,t;V)0Txϵ(·)-xλ(·)L2(0,t;V)2dt=T0Txϵ(t)-xλ(t)2dt. Thus, we have (3.12)(f(t,xϵ)-f(t,xλ),xϵ(t)-xλ(t))|f(t,xϵ)-f(t,xλ)|·|xϵ(t)-xλ(t)|ω12T(LkL2)2|f(t,xϵ)-f(t,xλ)|2+T(LkL2)22ω1|xϵ(t)-xλ(t)|2ω12Txϵ(·)-xλ(·)L2(0,t;H)2+T(LkL2)22ω1|xϵ(t)-xλ(t)|2. Therefore, by using the monotonicity of ϕ and integrating (3.10) over [0,T] it holds (3.13)12|xϵ(t)-xλ(t)|2+ω120Txϵ(t)-xλ(t)2dt0T(ϕϵ(xϵ(t))-ϕλ(xλ(t)),λϕλ(xλ(t))-ϵϕϵ(xϵ(t)))dt+{T(L||k||L2)22ω1+ω2}0T|xϵ(t)-xλ(t)|2dt. Here, we used that (3.14)ϕϵ(xϵ(t))=ϵ-1(xϵ(t)-(I+ϵϕ)-1xϵ(t)). Since |ϕϵ(x)||(ϕ)0x| for every xD(ϕ) it follows from Assumption (A) and using Gronwall’s inequality that (3.15)xϵ-xλC([0,T];H)L2(0,T;V)C(ϵ+λ),0<T.

Let xL1(0,T;V). Then it is well known that (3.16)limh0h-10hx(t+s)-x(t)ds=0 for almost all point of t(0,T).

Definition 3.3.

The point t which permits (3.16) to hold is called the Lebesgue point of x.

We establish the following results on the solvability of (NDE).

Theorem 3.4.

Let Assumptions (F) and (A) be satisfied. Then for every (x0,h)V×L2(0,T;V*), (NDE) has a unique solution: (3.17)xL2(0,T;V)W1,2(0,T;V*)C([0,T];H), and there exists a constant C4 depending on T such that (3.18)xL2W1,2CC4(1+x0+hL2(0,T;V*)).

Proof.

Let us fix T0>0 such that (3.19)C1C2(ϵ-1+T0  LkL2)(T02)1/2<1. Let yL2(0,T0;V). Then f(·,y(·))L2(0,T0;H) from Assumption (F). Set (3.20)(Fx)(t)=f(t,x(t))-ϕϵ(x(t)),0tT0. Then from Lemma 3.1 it follows that (3.21)|(Fx1)(t)-(Fx2)(t)|(ϵ-1+T0LkL2)x1(t)-x2(t). For i=1,2, we consider the following equation: (3.22)xi(t)+Axi(t)=(Fyi)(t)+h(t),0<tT0,xi(0)=x0. Then (3.23)ddt(x1(t)-x2(t))+A(x1(t)-x2(t))=(Fy1)(t)-(Fy2)(t),t>0,x1(0)-x2(0)=0. From (2.11) it follows that (3.24)x1-x2L2(0,T0;D(A0))W1,2(0,T0;H)C2Fy1-Fy2L2(0,T0;H). Using the Hölder inequality we also obtain that (3.25)x1-x2L2(0,T0;H)={0T0|x1(t)-x2(t)|2dt}1/2={0T0|0t(x˙1(τ)-x˙2(τ))dτ|2dt}1/2{0T0t0t|x˙1(τ)-x˙2(τ)|2dτdt}1/2T02x1-x2W1,2(0,T0;H). Therefore, in terms of (2.8) and (3.25) we have (3.26)x1-x2L2(0,T0;V)C1x1-x2L2(0,T0;D(A0))1/2x1-x2L2(0,T0;H)1/2C1x1-x2L2(0,T0;D(A0))1/2(T02)1/2x1-x2W1,2(0,T0;H)1/2C1(T02)1/2x1-x2L2(0,T0;D(A0))W1,2(0,T0;H)C1C2(T02)1/2Fy1-Fy2L2(0,T0:H)C1C2(ϵ-1+T0LkL2)(T02)1/2y1-y2L2(0,T0;V). So by virtue of the condition (3.19) the contraction principle gives that (SDE 2) has a unique solution in [0,T0]. Thus, letting λ0 in Lemma 3.1 we can see that there exists a constant C independent of ϵ such that (3.27)xϵ-xC([0,T0];H)L2(0,T0;V)Cϵ,0<T0, and hence, limϵ0xϵ(t)=x(t) exists in H. From Assumption (F) and (3.27) it follows that (3.28)f(·,xϵ)f(·,x),strongly  in  L2(0,T0;H),AxnAx,strongly  in  L2(0,T0;V*). Since ϕϵ(xϵ) is uniformly bounded by Assumption (A), from (3.27), (3.28) we have that (3.29)ddtxϵddtx,weakly  in  L2(0,T0;V*), therefore (3.30)ϕϵ(xϵ)f(·,x)+h-x-Ax,weakly  in  L2(0,T0;V*). Since (I+ϵϕ)-1xϵx    strongly and ϕ is demiclosed, we have that (3.31)f(·,x)+h-x'-Axϕ(x)inL2(0,T0;V*). Thus we have proved that x(t) satisfies a.e. on (0,T0) the equation  (NDE).

Let y be the solution of (3.32)y(t)+Ay(t)+ϕ(y(t))0,0<tT0,y(0)=x0, then, it implies (3.33)ddt(x(t)-y(t))+A(x(t)-y(t))+ϕ(x(t))-ϕ(y(t))f(t,x)+h(t). Noting that ||·|||·|||·||, by multiplying by x(t)-y(t) and using the monotonicity of ϕ and (2.3), we obtain (3.34)12ddt|x(t)-y(t)|2+ω1x(t)-y(t)2ω2|x(t)-y(t)|2+|f(t,x)+h(t)|·x(t)-y(t). Since (3.35)|f(t,x)+h(t)|·x(t)-y(t)12ω1|f(t,x)+h(t))|2+ω12x(t)-y(t)2 for every c>0 and by integrating on (3.34) over (0,t) we have (3.36)|x(t)-y(t)|2+ω10tx(s)-y(s)2ds1ω1f(·,x)+hL2(0,T0;V*)+2ω20t|x(s)-y(s)|2ds and by Gronwall’s inequality: (3.37)|x(t)-y(t)|2+ω10tx(s)-y(s)2dsω1-1e2ω2T0f(·,x)+hL2(0,T0;V*)2. Let us fix T0>T1>0 so that T1 is a Lebesgue point of x, ϕ(x(T1))<, and (3.38)ω1-1e2ω2T1T1LkL2<ω1. Put (3.39)N=ω1-2eω2T1, then from Assumption (F) it follows (3.40)x-yL2(0,T1;V)Nf(·,x)+hL2(0,T1;V*)x-yL2(0,T1;V)NT1  LkL2xL2(0,T1;V)+NhL2(0,T1:V*) and hence, from (2.17) in Proposition 2.3, we have that (3.41)xL2(0,T1;V)11-NT1LkL2(yL2(0,T1;V)+NhL2(0,T1:V*))11-NT1LkL2{C2(1+x0)+NhL2(0,T1:V*)}C4(1+x0+hL2(0,T1:V*)) for some positive constant C4. Since the condition (3.38) is independent of initial values, noting the Assumption (A), the solution of (NDE) can be extended to the internal [0,nT1] for natural number n, that is, for the initial x(nT1) in the interval [nT1,(n+1)T1], as analogous estimate (3.41) holds for the solution in [0,(n+1)T1]. The norm estimate of x in W1,2(0,T;H) can be obtained by acting on both side of (NDE) by x'(t) and by using (3.42)ddtϕ(x(t))=(g(t),ddtx(t)),a.e.,0<t, for all g(t)ϕ(x(t)). Furthermore, the estimate (3.18) is immediately obtained from (3.41).

Theorem 3.5.

Let Assumptions (F) and (A) be satisfied and (x0,h)V×L2(0,T;V*), then the solution x of (NDE) belongs to xL2(0,T;V)W1,2(0,T;V*) and the mapping: (3.43)V×L2(0,T;V*)(x0,h)xL2(0,T;V)C([0,T];H) is continuous.

Proof.

If (x0,h)V×L2(0,T;V*) then x belongs to L2(0,T;V)W1,2(0,T;V*) form Theorem 3.4. Let (x0i,hi)V×L2(0,T;V*) and xi be the solution of (NDE) with (x0i,hi) in place of (x0,h) for i=1,2. Multiplying on (NDE) by x1(t)-x2(t), we have (3.44)12ddt|x1(t)-x2(t)|2+ω1x1(t)-x2(t)2ω2|x1(t)-x2(t)|2+|f(t,x1)-f(t,x2)|x1(t)-x2(t)+h1(t)-h2(t)*x1(t)-x2(t). Let us fix T1>T2>0 so that T2 is a Lebesgue point of x, ϕ(x(T2)<, and (3.45)ω1-ω1-1e2ω2T2T2  LKL2>0. Since (3.46)h1(t)-h2(t))*x1(t)-x2(t)1ω1h1(t)-h2(t)*2+ω14x1(t)-x2(t)2, by integrating on (3.44) over [0,T2] where T2<T and as is seen in (3.37), it follows (3.47)x1-x2C([0,T2];H)2+ω12x1-x2L2(0,T2;V)2x01-x022+1ω1f(t,x1)-f(t,x2)L2(0,T2;H)2+2ω1h1-h2L2(0,T2;V*)x01-x022+ω1-1T2  LKL2x1-x2L2(0,T2;V)2+2ω1h1-h2L2(0,T2;V*). Putting that (3.48)N1min  [1,{ω12-ω1-1T2LKL2}]1/2,N2max{1,2ω1}, we have (3.49)x1-x2L2C2N21-N1(x01-x02+h1-h2). Suppose (x0n,hn)(x0,h) in V×L2(0,T;V*), and let xn and x be the solutions (SDE 2) with (x0n,hn) and (x0,h), respectively. Then, by virtue of (3.44) and (3.49), we see that xnx in L2(0,T2,V)W1,2(0,T2,V*)C([0,T2];H). This implies that xn(T2)x(T2) in H. Therefore the same argument shows that xnx in (3.50)L2(T2,min{2T2,T};V)C([T2,min{2T2,T}];H). Repeating this process, we conclude that xnx in L2(0,T;V)W1,2(0,T2,V*)C([0,T2];H).

4. Example

Let Ω be bounded domain in n with smooth boundary Ω. We define the following spaces: (4.1)H1(Ω)={u:u,uxiL2(Ω),i=1,2,,n},H2(Ω)={u:u,xxi,2uxixjL2(Ω),i,j=1,2,,n},H01(Ω)={u:uH1(Ω),u|Ω=0}=the  closure  of  C0(Ω)  in  H1(Ω), where /xiu and 2/xixju are the derivative of u in the distribution sense. The norm of H01(Ω) is defined by (4.2)u={Ωi=1n(u(x)xi)2dx}1/2. Hence H01(Ω) is a Hilbert space. Let H-1(Ω) = H01(Ω)* be a dual space of H01(Ω). For any lH-1(Ω) and vH01(Ω), the notation (l,v) denotes the value l at v. In what follows, we consider the regularity for given equations in the spaces: (4.3)V=H01(Ω)={uH1(Ω);  u=0  on  Ω},H=L2(Ω),V*=H-1(Ω) as introduced in Section 2. We deal with the Dirichlet condition’s case as follows.

Assume that aij=aji are continuous and bounded on Ω¯ and {aij(x)} is positive definite uniformly in Ω, that is, there exists a positive number δ such that (4.4)i,j=1naij(x)ξiξjδ|ξ|2,ξΩ-. Let (4.5)biL(Ω),cL(Ω),βi=j=1naijxj+bi. For each u,vH01(Ω), let us consider the following sesquilinear form: (4.6)a(u,v)=Ω{i,j=1naijuxiv¯xj+j=1nβiuxiv¯+cuv¯}dx. Since {aij} is real symmetric, by (4.4) the inequality: (4.7)i,j=1naij(x)ξiξ-jδ|ξ|2 holds for all complex vectors ξ=(ξ1,,ξn). By hypothesis, there exists a constant K such that |βi(x)|K and c(x)K hold a.e., hence (4.8)Re  a(u,u)Ωδi=1n|uxi|2dx-KΩi=1n|uxi||u|dx-KΩ|u|2dxδΩi=1n|uxi|2dx-KΩi=1n(ϵ2|uxi|2+12ϵ|u|2)dx-KΩ|u|2dx=(δ-ϵ2K)i=1nΩ|uxi|2dx-(nK2ϵ+K)Ω|u|2dx. By choosing ϵ=δK-1, we have (4.9)Rea(u,u)δ2i=1nΩ|uxi|2dx-(nK22δ+K)Ω|u|2dx=δ2u12-(nK22δ+K+δ2)u2. By virtue of Lax-Milgram theorem, we know that for any vV there exists fV* such that (4.10)a(u,v)=(f,v). Therefore, we know that the associated operator A:VV* defined by (4.11)(Au,v)=-a(u,v),u,vV is bounded and satisfies conditions (2.3) in Section 2.

Let g:[0,T]×VH be a nonlinear mapping defined by (4.12)g(t,u(t,x))=0ti=1nxiσi(s,u(s,x))ds.

We assume the following.

Assumption (<italic>F1</italic>).

The partial derivatives σi(s,ξ), /t  σi(s,ξ) and /ξjσi(s,ξ), exist and continuous for i=1,2, j=1,2,,n, and σi(s,ξ) satisfies an uniform Lipschitz condition with respect to ξ, that is, there exists a constant L>0 such that (4.13)|σi(s,ξ)-σi(s,ξ^)|L|ξ-ξ^|, where |·| denotes the norm of L2(Ω).

Lemma 4.1.

If Assumption (F1) is satisfied, then the mapping tg(t,·) is continuously differentiable on [0,T] and ug(·,u) is Lipschitz continuous on V.

Proof.

Put (4.14)g1(s,u)=i=1nxiσi(s,u), then we have g1(s,u)H-1(Ω). For each wH01(Ω), we satisfy the following that (4.15)(g1(s,u),w)=-i=1n(σi(s,u),xiw). The nonlinear term is given by (4.16)g(t,u)=0tg1(s,u)ds. For any wH01(Ω), if u and u^ belong to H01(Ω), by Assumption (F1) we obtain (4.17)|(g(t,u)-g(t,u^)),w|LTu-u^w.

We set (4.18)f(t,u)=0tk(t-s)0si=1nxiσi(τ,u(τ,x))dτds, where k belongs to L2(0,T). Let ϕ:H01(Ω)(-,+] be a lower semicontinuous, proper convex function. Now in virtue of Lemma 4.1, we can apply the results of Theorem 3.4 as follows.

Theorem 4.2.

Let Assumption (F1) be satisfied. Then for any u0H01(Ω) and hL2(0,T;H-1(Ω)), the following nonlinear problem: (4.19)(u(t)+Au(t),u(t)-z)+ϕ(u(t))-ϕ(z)(f(t,u)+h(t),u(t)-z),a.e.,0<tT,zL2(Ω),u(0)=u0 has a unique solution: (4.20)uL2(0,T;H01(Ω))W1,2(0,T;H-1(Ω))C([0,T];L2(Ω)).

Furthermore, the following energy inequality holds: there exists a constant CT depending on T such that (4.21)uL2W1,2CT(1+u0+hL2(0,T;H-1(Ω))).

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0026609).

Barbu V. Nonlinear Semigroups and Differential Equations in Banach Spaces 1976 The Netherlands Nordhoff Leiden 352 0390843 ZBL0343.49010 Barbu V. Analysis and Control of Nonlinear Infinite Dimensional Systems 1993 190 Boston, Mass, USA Academic Press x+476 Mathematics in Science and Engineering 1195128 ZBL0840.01033 Jeong J.-M. Jeong D.-H. Park J.-Y. Nonlinear variational evolution inequalities in Hilbert spaces International Journal of Mathematics and Mathematical Sciences 2000 23 1 11 20 10.1155/S0161171200001630 1741320 ZBL0993.34060 Ahmed N. U. Xiang X. Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations Nonlinear Analysis. Series A 1994 22 1 81 89 10.1016/0362-546X(94)90007-8 1256172 ZBL0806.34051 Jeong J.-M. Park J.-Y. Nonlinear variational inequalities of semilinear parabolic type Journal of Inequalities and Applications 2001 6 2 227 245 10.1155/S1025583401000133 896837 1835527 ZBL1032.35116 Jeong J. M. Kwun Y. C. Park J. Y. Approximate controllability for semilinear retarded functional-differential equations Journal of Dynamical and Control Systems 1999 5 3 329 346 10.1023/A:1021714500075 1706797 ZBL0962.93013 Kobayashi Y. Matsumoto T. Tanaka N. Semigroups of locally Lipschitz operators associated with semilinear evolution equations Journal of Mathematical Analysis and Applications 2007 330 2 1042 1067 10.1016/j.jmaa.2006.08.028 2308426 ZBL1123.34044 Ahmed N. U. Optimal control of infinite-dimensional systems governed by integrodifferential equations Differential Equations, Dynamical Systems, and Control Science 1994 152 New York, NY, USA Dekker 383 402 Lecture Notes in Pure and Applied Mathematics 1243213 Tanabe H. Equations of Evolution 1979 6 London, UK Pitman xii+260 Monographs and Studies in Mathematics 533824 Lions J. L. Magenes E. Non-Homogeneous Boundary Value Problmes and Applications 1972 Berlin, Germany Springer Triebel H. Interpolation Theory, Function Spaces, Differential Operators 1978 18 Amsterdam, The Netherlands North-Holland 528 North-Holland Mathematical Library 503903 Di Blasio G. Kunisch K. Sinestrari E. L 2 -regularity for parabolic partial integro-differential equations with delay in the highest-order derivatives Journal of Mathematical Analysis and Applications 1984 102 1 38 57 10.1016/0022-247X(84)90200-2 751340 Lions J. L. Magenes E. Problemes Aux Limites Non Homogenes Et Applications 1968 3 Paris, France Dunod Aubin J. P. Un Théorème de Compacité Comptes Rendus de l'Académie des Sciences 1963 256 5042 5044 0152860 ZBL0195.13002