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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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Volume 20, Issue 1

Issues

Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

André EikmeierORCID iD: https://orcid.org/0000-0002-0270-6491 / Etienne EmmrichORCID iD: https://orcid.org/0000-0001-9869-0334 / Hans-Christian Kreusler
Published Online: 2018-11-21 | DOI: https://doi.org/10.1515/cmam-2018-0268

Abstract

The initial value problem for an evolution equation of type v+Av+BKv=f is studied, where A:VAVA is a monotone, coercive operator and where B:VBVB induces an inner product. The Banach space VA is not required to be embedded in VB or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

Keywords: Nonlinear Evolution Equation; Monotone Operator; Volterra Operator; Exponentially Decaying Memory; Existence; Uniqueness; Stability; Time Discretisation; Convergence

MSC 2010: 47J35; 45K05; 34K30; 35K90; 35R09; 65J08; 65M12

Dedicated to Professor Rolf D. Grigorieff on the occasion of his 80th birthday

1 Introduction

1.1 Problem Statement and Main Result

We are concerned with initial value problems of the type

v+Av+BKv=fin (0,T),(1.1a)v(0)=v0,(1.1b)where(Kv)(t)=u0+0tk(t-s)v(s)ds,k(z)=λe-λz.(1.1c)

Here, T>0 is the time under consideration, λ>0 is a given parameter, and u0, v0, f are given data of the problem.

The operator A:VAVA is a hemicontinuous, monotone, coercive operator satisfying a certain growth condition, where VA denotes a real, reflexive Banach space. The operator B:VBVB is a linear, symmetric, bounded, strongly positive operator on a real Hilbert space VB. We assume that both VA and VB are densely and continuously embedded in a real Hilbert space H, which is identified with its dual. We emphasise that we do not require VA to be embedded in VB (or vice versa) but only assume that V=VAVB is densely embedded in VA as well as in VB. This yields the scale

VAVB=VVCH=HVCV=VA+VB,C{A,B},(1.2)

of Banach and Hilbert spaces with dense and continuous embeddings. Moreover, we assume that V is separable.

Since the kernel k in (1.1c) is assumed to be of exponential type, (1.1a) can easily be derived from the system

v+Av+Cu=f,(1.3a)(u-u0)+λ(u-u0)=λDv,(1.3b)

with suitable linear operators C, D such that B=CD (and with u0 being replaced by Du0).

Such systems appear, e.g., in the description of viscoelastic fluid flow (see, e.g., [15, 38]) as well as heat flow in materials with memory (see, e.g., [39, 43]).

Other applications arise in, e.g., non-Fickian diffusion models describing diffusion processes of a penetrant through a viscoelastic material (see, e.g., [17, 18, 46]), where, apart from the usual diffusion rate of the penetrant, the change in the internal structure of the viscoelastic material has to be considered. Non-Fickian diffusion also plays a role in, e.g., mathematical biology (see, e.g., [13, 20, 42]).

Note that all the results obtained in this paper are also valid for kernels of the type k(z)=ce-λz (c,λ>0). The proof only differs for the stability with respect to perturbations of the kernel. However, in many applications, kernels of the type given in (1.1c) naturally arise. In such applications, one often deals with a coupled system of the type (1.3), where 1λ can be interpreted as a relaxation or averaged delay time. If λ tends to 0, then the system decouples with u(t)u0. The limit λ results in a first-order equation for v without memory. In contrast to this, in the case where k(z)=ce-λz, the limit λ0 results in an evolution equation for u of second order in time (see [19]), whereas λ results again in a first-order evolution equation for v.

Our goal in this paper is twofold: On the one hand, we wish to prove existence of generalised solutions to (1.1). On the other hand, we wish to prove convergence of a simple time discretisation method, which relies on the implicit Euler or Rothe method combined with a product quadrature. Moreover, we prove stability of the solution against perturbations of the problem data (including perturbations of λ), which also implies uniqueness.

1.2 Literature Overview

Most of the results on evolutionary Volterra integral equations available so far are, indeed, more general with respect to the integral kernel (often dealing with memory of positive type). Our results, however, include the case of different domains of definition of the underlying operators involved without assuming that VA is embedded in VB (or vice versa). Such a situation occurs, e.g., if A is a spatial differential operator of order lower than that of B. This has, to the best knowledge of the authors, not yet been studied. Dealing with memory of exponentially decaying type can thus be seen as a first step within this more general functional analytic framework.

Well-posedness for linear evolutionary integral equations in Banach or Hilbert spaces has been studied in detail by many authors for a long time. We only refer to the standard monographs [30, 44] and the references cited therein. For the semigroup approach, we also refer to [2]. For the finite-dimensional case (including nonlinear problems), one may also consult [9].

Whereas there are many results available for linear problems, there is less known for the nonlinear case. Classical results for nonlinear problems are due to Clément, Crandall, Dafermos, Desch, Gripenberg, Londen, MacCamy, Nohel, and others. One of the equations that is mostly studied is of the type

v+aAv=f,

where a is a suitable kernel function (e.g., completely monotone or of positive type) and A is a suitable nonlinear operator satisfying certain monotonicity assumptions; see, e.g., [14, 33, 27, 28, 29, 50, 51, 52].

For example, MacCamy, and Wong [40] consider a class of nonlinear integro-differential equations with completely monotone kernel in a Hilbert space setting. Gajewski, Gröger, and Zacharias [22] study rather general classes of nonlinear evolution equations with Volterra operators, but are restricted to the Hilbert space case VA=VB. Crandall, Londen, and Nohel [10] consider a doubly nonlinear problem governed by maximal monotone subdifferential operators, where the domain of definition of one of the operators is dense and continuously embedded into the domain of definition of the other operator; see also [11]. An interesting result for a class of doubly nonlinear integro-differential equations governed by an m-dissipative operator can be found in [26]. The nonlocality in time here is of a special form incorporating this m-dissipative operator again. The domain of definition of the principal part of the equations, however, equals the domain of definition of the nonlocality in time. The method of proof relies upon nonlinear semigroup theory. More recent results can be found, e.g., in [1, 25, 6, 23, 24, 48, 53, 54, 3]. The above-cited references do not cover the class of problems we consider here.

The question of numerical approximation, mostly for linear or semilinear problems with memory of positive type, has been dealt with by McLean, Sloan, Thomée, and Wahlbin (see, e.g., [47, 41, 37, 31]). The focus is on Galerkin finite element methods combined with suitable time discretisation methods based on the backward Euler scheme (see, e.g., [8, 7]). Time discretisation methods have also been studied by Calvo, Lubich, and Palencia (see, e.g., [5, 12]) and, in particular, convolution quadrature by Lubich [34, 35, 36] and Ferreira [21]. Indeed, our proof of existence also relies upon the convergence of a numerical scheme that is based on a convolution quadrature.

1.3 Organisation of the Paper

The paper is organised as follows: The general notation is explained in Section 2. In Section 3, we state the main assumptions on the operators A and B and collect some preliminary results on their properties. The main existence result (Theorem 4.2) is provided in Section 4 by showing (weak) convergence of a suitable time discretisation to (1.1). The crucial point here is an integration-by-parts formula given in Lemma 4.3. In Section 5, we show uniqueness and stability with respect to perturbations of the problem data. In particular, we also consider perturbations of the kernel parameter λ.

2 Notation

For a Banach space X, we denote its norm by X, its dual by X, equipped with the standard norm X, and the duality pairing by ,. We recall that X is reflexive and separable if X is. For a Hilbert space X, we denote the inner product (inducing the norm X) by (,)X. The intersection of two Banach spaces X, Y is equipped with the norm XY=X+Y, whereas the sum X+Y is equipped with the norm

gX+Y=inf{max(gXX,gYY):g=gX+gY with gXX,gYY}.

We recall that (XY)=X+Y; see also [22, pp. 12ff.].

For a real, reflexive, separable Banach space X, the Bochner–Lebesgue spaces Lr(0,T;X) (r[1,]) are defined in the usual way and equipped with the standard norm. Denoting by r=rr-1 the conjugate of r(1,) with r= if r=1, we have that (Lr(0,T;X))=Lr(0,T;X) if r[1,); the duality pairing is given by

g,v=0Tg(t),v(t)dt;

see, e.g., [16, Theorem 1 on p. 98, Corollary 13 on p. 76, Theorem 1 on p. 79]. Moreover, Lr(0,T;X) is reflexive if r(1,) (see [16, Corollary 2 on p. 100]) and L1(0,T;X) is separable.

By W1,r(0,T;X) (r[1,]) we denote the Banach space of functions uLr(0,T;X) whose distributional time derivative u is again in Lr(0,T;X); the space is equipped with the standard norm. Note that if uW1,1(0,T;X), then u equals almost everywhere a function that is in 𝒜𝒞([0,T];X), i.e., a function that is absolutely continuous on [0,T] as a function taking values in X. Moreover, W1,1(0,T;X) is continuously embedded in the Banach space 𝒞([0,T];X) of functions that are continuous on [0,T] as functions with values in X (see, e.g., [45, Chapter 7] for more details). By 𝒞w([0,T];X) we denote the space of functions that are continuous on [0,T] with respect to the weak topology in X.

The space of infinitely many times differentiable real functions with compact support in (0,T) is denoted by 𝒞c(0,T). The space of on [0,T] continuously differentiable real functions is denoted by 𝒞1([0,T]). By c we denote a generic positive constant. We set j=10xj=0 for xj whatsoever.

3 Main Assumptions and Preliminary Results

Let VA be a real, reflexive Banach space and let VB and H be real Hilbert spaces satisfying (1.2). Moreover, we assume that V=VAVB is separable.

The structural properties we assume for the operators A and B throughout this paper are as follows.

Assumption A.

The operator A:VAVA satisfies for all u,v,wVA,

  • (i)

    θA(u+θv),w𝒞([0,1]) (hemicontinuity),

and there exist p(2,), μA,βA>0, cA0 such that for all v,wVA,

  • (ii)

    Av-Aw,v-w0 (monotonicity),

  • (iii)

    Av,vμAvVAp-cA (p-coercivity),

  • (iv)

    AvVAβA(1+vVAp-1) ((p-1)-growth).

Assumption B.

The linear operator B:VBVB is symmetric and there exist μB,βB>0 such that for all vVB,

  • (i)

    Bv,vμBvVB2 (strong positivity),

  • (ii)

    BvVBβBvVB (boundedness).

By B=B,1/2 we denote the norm induced by B, which is equivalent to VB. Further, by L2(0,T;B) we denote the space L2(0,T;(VB,B)).

We shall remark that the operators A and B can be extended, as usual, to operators acting on functions defined on [0,T] and taking values in VA and VB, respectively. Since the operator A:VAVA is hemicontinuous and monotone, it is also demicontinuous (see, e.g., [55, Proposition 26.4 on p. 555]). Therefore, the operator A maps, in view of the separability of VA and the theorem of Pettis (see, e.g., [16, Theorem 2 on p. 42]), a Bochner measurable function v:[0,T]VA into a Bochner measurable function Av:[0,T]VA with (Av)(t)=Av(t) (t[0,T]). Because of the growth condition, the operator A then maps Lp(0,T;VA) into (Lp(0,T;VA))=Lp(0,T;VA). The linear, symmetric, bounded, strongly positive operator B:VBVB extends, via (Bu)(t)=Bu(t) for a function u:[0,T]VB, to a linear, symmetric, bounded, strongly positive operator mapping L2(0,T;VB) into its dual. Indeed, B can also be seen as a linear, bounded operator mapping Lr(0,T;VB) into Lr(0,T;VB) for any r[1,].

With respect to the operator K defined by (1.1c), we make the following observations.

Lemma 3.1.

Let k be as in (1.1c) and let u0VB. Then K is an affine-linear, bounded mapping of L2(0,T;VB) into itself with

Kv-u0L2(0,T;VB)kL1(0,T)vL2(0,T;VB),vL2(0,T;VB),

where kL1(0,T)=1-e-λT. Further, K is an affine-linear, bounded mapping of L1(0,T;VB) into AC([0,T];VB) with

Kv-u0𝒞([0,T];VB)λvL1(0,T;VB),vL1(0,T;VB).

The standard proof is omitted here.

Note that in the assertions above VB can be replaced by H. An immediate consequence of the lemma above is that BK:L2(0,T;VB)L2(0,T;VB) as well as BK:L1(0,T;VB)L(0,T;VB) is affine-linear and bounded if Assumption B is satisfied.

We will often make use of the following relation, which indeed is crucial within this work and reflects the exponential type of the memory kernel. For a function vL1(0,T;X), with X being an arbitrary Banach space, we have that for almost all t(0,T),

(Kv)(t)=λ(v(t)-((Kv)(t)-u0)).(3.1)

If X is a Hilbert space and if vL2(0,T;X), then testing this relation by Kv-u0 immediately implies for all t[0,T],

0t((Kv)(s)-u0,v(s))Xds=0t(Kv)(s)-u0X2ds+12λ(Kv)(t)-u0X2,

which shows that the memory term is of positive type.

4 Main Result: Existence via Time Discretisation

In this section, we show existence of generalised solutions to (1.1) by proving weak or weak* convergence of a sequence of approximate solutions constructed from a suitable time discretisation. We commence by studying the corresponding numerical scheme and its properties.

4.1 Time Discretisation

For N, let τ=TN and tn=nτ (n=0,1,,N). Let v0v0, u0u0, and {fn}n=1Nf be given approximations of the problem data v0, u0, and f, respectively. We look for approximations vnv(tn) (n=1,2,,N).

The numerical scheme we consider combines the implicit Euler method with a convolution or product quadrature for the integral operator K and reads

1τ(vn-vn-1)+Avn+(BKu0τv)n=fn,n=1,2,,N,(4.1a)where(Ku0τv)n:=u0+τj=1nγn-j+1vj.(4.1b)

To be precise, Ku0τ acts on a grid function {vn}n=1N and, with a slight abuse of notation, the evaluation at n is denoted by (Ku0τv)n. The coefficients γi (i=1,2,,N) are given by

γi=01k((i-s^)τ)ds^,(4.1c)

and hence

γn-j+1=01k((n-j+1-s^)τ)ds^=1τtj-1tjk(tn-s)ds.

The idea behind is the approximation

0tk(t-s)v(s)dsj=1ntj-1tjk(tn-s)dsv(tj)for t(tn-1,tn](n=1,2,,N).

As we deal with kernels k of exponential type given by (1.1c), we can explicitly calculate

γi=eλτ-1τe-λti,i=1,2,,N,

which immediately leads to the properties

0<γ1=1-e-λττλ=k(0)as τ0,(4.2)

and

γi+1-γi=-(eλτ-1)γi+1,i=1,2,,N-1.(4.3)

4.2 Existence, Uniqueness, and A Priori Estimates for the Time Discrete Problem

In what follows, we show existence of solutions for the time discrete equation (4.1) and derive suitable a priori estimates.

Theorem 4.1 (Time Discrete Problem).

Let Assumptions A and B be fulfilled and let u0VB, v0H, and {fn}n=1N={f0n}n=1N+{f1n}n=1NVA+H be given. Then there is a unique solution {vn}n=1NV=VAVB to (4.1). Moreover, there holds for n=1,2,,N,

vnH2+j=1nvj-vj-1H2+μAτj=1nvjVAp+TeλT-1(Ku0τv)nVB2c(1+u0VB2+v0H2+τj=1Nf0jVAp+(τj=1Nf1jH)2).(4.4)

Proof.

For better readability, we write Kτ instead of Ku0τ during this proof. We commence with proving existence and uniqueness of a solution step by step. In the n-th step, (4.1a) is equivalent to determine vn from {vj}j=1n-1V and the data of the problem by solving

(1τI+A+τγ1B)vn=fn+1τvn-1-Bu0-τj=1n-1γn-j+1Bvj.

Because of the continuous embeddings (1.2), we know that the right-hand side of the foregoing relation is in V. The operator M:=1τI+A+τγ1B is easily shown to be hemicontinuous, coercive, and strictly monotone as a mapping of V into V. Here, we make use of the fact that γ1>0. In particular, we observe that for all wV,

Mw,w1τwH2+μAwVAp-cA+τγ1μBwVB2cwV2-c.

The famous theorem of Browder and Minty (see, e.g., [55, Theorem 26.A on p. 557]) now provides existence of a solution vnV; uniqueness immediately follows from the strict monotonicity.

For proving the a priori estimate, we test (4.1a) by vnV (n=1,2,,N). With

(a-b)a=12(a2-b2+(a-b)2),a,b,(4.5)

the coercivity of A, and Young’s inequality, we find for n=1,2,,N,

12τ(vnH2-vn-1H2+vn-vn-1H2)+μAvnVAp-cA+(BKτv)n,vnfn,vnf0nVAvnVA+f1nHvnHμA2vnVAp+cf0nVAp+f1nHvnH.

By summing up, we obtain for n=1,2,,N,

vnH2+j=1nvj-vj-1H2+μAτj=1nvnVAp+2τj=1n(BKτv)j,vjc(1+v0H2+τj=1nf0jVAp)+2τj=1nf1jHvjH,

where

2τj=1nf1jHvjH2τj=1nf1jHmaxj=1,,nvjH2(τj=1nf1jH)2+12maxj=1,,nvjH2.

We observe that (4.1b) together with (4.2), (4.3) implies for n=2,3,,N,

1τ((Kτv)n-(Kτv)n-1)=γ1vn+j=1n-1(γn-j+1-γn-j)vj=eλτ-1τ(vn-((Kτv)n-u0)),(4.6)

which is the discrete analogue of the crucial relation (3.1). Note that (eλτ-1)/τλ as τ0. By setting (Kτv)0:=u0, relation (4.6) remains true for n=1.

Resolving (4.6) for vn, together with (4.5) (recall that B:VBVB induces an inner product on VB), gives

2τj=1n(BKτv)j,vj=2τeλτ-1j=1n(BKτv)j,(Kτv)j-(Kτv)j-1+2τj=1n(BKτv)j,(Kτv)j-u0τeλτ-1((Kτv)nB2-(Kτv)0B2)+τj=1n(Kτv)jB2-Tu0B2,

where

τeλτ-1TeλT-1.

Altogether, we find for n=1,2,,N,

vnH2+j=1nvj-vj-1H2+μAτj=1nvnVAp+TeλT-1(Kτv)nB2c(1+u0VB2+v0H2+τj=1Nf0jVAp+(τj=1Nf1jH)2)+12maxj=1,,nvjH2.(4.7)

Taking the maximum over all n{1,,N} first on the right-hand side and then on the left-hand side leads to

maxj=1,,NvjH2C+12maxj=1,,NvjH2,

where C>0 depends on u0, v0, and {fn}. This yields

maxj=1,,NvjH22C,

which, inserted in (4.7), proves the assertion. ∎

4.3 Convergence of Approximate Solutions and Existence of a Generalised Solution

Let {N} be a sequence of positive integers such that N as . We consider the corresponding sequence of time discrete problems (4.1) with step sizes τ=T/N, starting values v0H with v0v0 in H as well as u0VB with u0u0 in VB, and right-hand sides {fn}n=1NVA+H given by

fn=f0n+f1n,f0,1n:=1τtn-1tnf0,1(t)dt,

which is well-defined for f=f0+f1Lp(0,T;VA)+L1(0,T;H). As a slight abuse of notation, in general we do not call the dependence of un,vn,fn, and of the time instances tn on .

Let {vn}n=1NV denote the solution to (4.1) with step size τ. We then consider the piecewise constant functions v with v(t)=vn for t(tn-1,tn] (n=1,2,,N) and v(0)=v1. Moreover, let v^ be the piecewise affine-linear interpolation of the points (tn,vn) (n=0,1,,N) and let f be the piecewise constant function with f(t)=fn for t(tn-1,tn] (n=1,2,,N) and f(0)=f1.

Regarding the integral operator K, we define for any integrable function w the piecewise constant function Kw by means of

(Kw)(t)=u0+0tnk(tn-s)w(s)dsif t(tn-1,tn](n=1,,N),

with (Kw)(0):=u0. As an immediate consequence, with (4.1b) and (4.1c) we obtain

(Kv)(t)=(Ku0τv)nif t(tn-1,tn](n=1,,N).(4.8)

We are now able to state the main result.

Theorem 4.2 (Existence via Convergence).

Let Assumptions A and B be fulfilled, and let u0VB, v0H, and fLp(0,T;VA)+L1(0,T;H). Then there exists a solution vLp(0,T;VA)Cw([0,T];H) to problem (1.1) with KvCw([0,T];VB) such that (1.1a) holds in the sense of Lp(0,T;VA)+L1(0,T;H).

By passing to a subsequence if necessary, both the piecewise constant prolongation v and the piecewise affine-linear prolongation v^ of the approximate solutions to (4.1) converge weakly* in L(0,T;H) to v as . Furthermore, again by passing to a subsequence if necessary and as , the piecewise constant prolongation v converges weakly to v in Lp(0,T;VA) and the approximation Kv of the memory term converges weakly* to Kv in L(0,T;VB). The approximation v^ of the time derivative converges to

vLp(0,T;VA)+L1(0,T;H)+L(0,T;VB)

in the sense that v^,wv,w for all wLp(0,T;VA)L(0,T;H)L1(0,T;VB).1

Note that if {τv0VAp} is bounded, then also v^ converges weakly in Lp(0,T;VA) to v as (passing to a subsequence if necessary).

The proof of Theorem 4.2 will be prepared by the following integration-by-parts formula.

Lemma 4.3.

Let u0VB and wLp(0,T;VA)L(0,T;H) such that

KwL(0,T;VB)𝑎𝑛𝑑w+BKwLp(0,T;VA)+L1(0,T;H).

Then for all t[0,T] there holds

0tw(s)+(BKw)(s),w(s)ds=12w(t)H2-12w(0)H2+12λ(Kw)(t)B2-12λu0B2-0t(BKw)(s),u0ds+0t(Kw)(s)B2ds.(4.9)

Proof.

The difficulty in proving (4.9) is that neither

wLp(0,T;VA)+L1(0,T;H)norBKwLp(0,T;VA)+L1(0,T;H)

can be assumed but only the sum w+BKw is known to be in Lp(0,T;VA)+L1(0,T;H). Hence it is not possible to split the sum on the left-hand side and to carry out integration by parts separately for both terms.

As w+BKwLp(0,T;VA)+L1(0,T;H) and BKwL(0,T;VB), it is easy to show that

wLp(0,T;VA)+L1(0,T;VB)L1(0,T;VA+VB),

and thus w𝒜𝒞([0,T];VA+VB). In view of [32, Chapter 3, Lemma 8.1], we find w𝒞w([0,T];H). Because of (3.1), we also see that (Kw)L1(0,T;VA+VB), and thus Kw𝒜𝒞([0,T];VA+VB), which implies Kw𝒞w([0,T];VB). Because of (3.1), we also have (Kw)𝒞w([0,T];H). This justifies evaluating w(t)H, Kw(t)B, and (Kw)(t)H for fixed t.

Further, we remark that Kw-u0L(0,T;V) since wLp(0,T;VA)L(0,T;H). Due to (3.1), we thus have (Kw)Lp(0,T;VA)L(0,T;H). Differentiating (3.1) yields

(Kw)′′(t)=λ(w(t)-(Kw)(t))

for almost all t(0,T). Therefore, we have

w+BKw,w=1λ(Kw)′′+(Kw)+BKw,1λ(Kw)+Kw-u0.

Since 1λ(Kw)′′+(Kw)+BKwLp(0,T;VA)+L1(0,T;H) and Kw-u0L(0,T;V), we are allowed to split the right-hand side into

1λ(Kw)′′+(Kw)+BKw,1λ(Kw)+1λ(Kw)′′+(Kw)+BKw,Kw-u0=1λ(Kw)′′+BKw,1λ(Kw)+1λ(Kw)L2(0,T;H)2+1λ(Kw)′′+(Kw)+BKw,Kw-u0.(4.10)

For the first term on the right-hand side, we apply the result of Strauss [49, Theorem 4.1] providing the integration-by-parts formula

1λ(Kw)′′+BKw,1λ(Kw)=12λ2((Kw)(T)H2-(Kw)(0)H2+λ(Kw)(T)B2-λu0B2).

Since (Kw)Lp(0,T;VA)L(0,T;H), BKwL(0,T;VB), and Kw-u0L(0,T;V), the third term on the right-hand side of (4.10) yields

1λ(Kw)′′+(Kw)+BKw,Kw-u0=1λ(Kw)′′,Kw-u0+(Kw),Kw-u0+BKw,Kw-u0=1λ(Kw-u0)(T),(Kw-u0)(T)-1λ(Kw-u0)(0),(Kw-u0)(0)-1λ(Kw-u0)L2(0,T;H)2   +12(Kw)(T)-u0H2-12(Kw)(0)-u0H2+KwL2(0,T;B)2-BKw,u0.

Altogether, we have

w+BKw,w=12λ2((Kw)(T)H2-(Kw)(0)H2+λ(Kw)(T)B2-λu0B2)+1λ(Kw-u0)(T),(Kw-u0)(T)+12(Kw)(T)-u0H2+KwL2(0,T;B)2-BKw,u0.(4.11)

Due to (3.1), there holds

12w(T)H2=12λ2(Kw)(T)H2+12(Kw)(T)-u0H2+1λ(Kw)(T),(Kw)(T)-u0

as well as

12w(0)H2=12λ2(Kw)(0)H2.

Inserting this into (4.11) yields the desired integration-by-parts formula for t=T. However, everything above remains true for an arbitrary t[0,T], which proves the assertion. ∎

Proof of Theorem 4.2.

For simplicity, we write τ and N instead of τ and N, respectively, during this proof. Since, by assumption, v0v0 in H, u0u0 in VB, and since ff in Lp(0,T;VA)+L1(0,T;H), the right-hand side of the a priori estimate (4.4) stays bounded as . Since

vL(0,T;H)=maxn=1,2,,NvnH,v^L(0,T;H)=maxn=0,1,,NvnH,vLp(0,T;VA)=(τn=1NvnVAp)1/p,

the a priori estimate (4.4) implies that the sequence {v} is bounded in L(0,T;H)Lp(0,T;VA), the sequence {v^} is bounded in L(0,T;H), and – in view of (4.8) – the sequence {Kv} is bounded in L(0,T;VB). Due to the growth condition on A, we also conclude that the sequence {Av} is bounded in Lp(0,T;VA).

Since L(0,T;H) and L(0,T;VB) are the duals of the separable normed spaces L1(0,T;H) and L1(0,T;VB), respectively, and since Lp(0,T;VA) and Lp(0,T;VA) are reflexive Banach spaces, there exists a common subsequence, again denoted by , and elements vL(0,T;H)Lp(0,T;VA), v^L(0,T;H), uL(0,T;VB), and aLp(0,T;VA) such that, as

v*vin L(0,T;H),vvin Lp(0,T;VA),v^*v^in L(0,T;H),Kv*uin L(0,T;VB),Avain Lp(0,T;VA);

see, e.g., [4, Corollaire III.26, Théorème III.27]. Since B:VBVB is linear and bounded, we also find that BKv*Bu in L(0,T;VB).

A short calculation shows that

v-v^L2(0,T;H)2=τ3j=1Nvj-vj-1H2,

which tends to 0 as because of the a priori estimate (4.4). Hence v=v^.

Let us show that u=Kv, and thereby KvL(0,T;VB). We already know that Kv*u in L(0,T;VB), and thus Kvu in L2(0,T;H). The integral operator K^ defined by K^w:=Kw-u0 can easily be shown to be a linear, bounded and thus weakly-weakly continuous mapping of L2(0,T;H) into itself (see also Lemma 3.1). Therefore, in view of v*v in L(0,T;H), and thus vv in L2(0,T;H), we find that Kv-Kv=K^v-K^v0 in L2(0,T;H). Because of

u-Kv=u-Kv+Kv-Kv+Kv-Kv,

it thus remains to prove that Kv-Kv converges at least weakly in L2(0,T;H) to 0 in order to prove u=Kv. For wL2(0,T;H) and t(tn-1,tn] (n=1,2,,N), we have

(Kw)(t)-(Kw)(t)=u0-u0+0tn(k(tn-s)-k(t-s))w(s)ds+ttnk(t-s)w(s)ds.

Since |k(t-s)|=λe-λ(t-s)λeλτ for s(t,tn)(tn-1,tn) as well as

|k(tn-s)-k(t-s)|λ(eλτ-1)cτfor s(0,tn),

the Cauchy–Schwarz inequality yields

Kw-KwL2(0,T;H)Tu0-u0H+cτwL2(0,T;H).

Since u0u0 in VB and v is bounded in L2(0,T;H), we immediately get Kv-Kv0 in L2(0,T;H) as (and thus τ0), which is the last step to prove u=KvL(0,T;VB).

With the help of the piecewise constant and piecewise linear interpolation of the corresponding grid functions, we can now rewrite the numerical scheme (4.1a) as

v^+Av+BKv=f.(4.12)

For any wV and ϕ𝒞c(0,T), we thus obtain

-0Tv^(t),wϕ(t)dt=0Tf(t)-Av(t)-(BKv)(t),wϕ(t)dt.

Since ff in Lp(0,T;VA)+L1(0,T;H), and employing the (weak and weak*) convergence just shown, we can pass to the limit in the foregoing equation to come up with

-0Tv(t),wϕ(t)dt=0Tf(t)-a(t)-(BKv)(t),wϕ(t)dt.

This shows that v is differentiable in the weak sense with

v=f-a-BKvLp(0,T;VA)+L(0,T;VB)+L1(0,T;H).(4.13)

Let us summarise the regularity properties of the limits we have proven so far: As

vLp(0,T;VA)L(0,T;H)andvLp(0,T;VA)+L(0,T;VB)+L1(0,T;H)L1(0,T;V),

we conclude that v𝒜𝒞([0,T];V). This implies v𝒞w([0,T];H) (see [32, Chapter 3, Lemma 8.1]). Likewise, from

KvL(0,T;VB)and(Kv)=λv-λKv+λu0Lp(0,T;VA)+L(0,T;VB)L1(0,T;VA+VB)

(see (3.1)), we deduce that Kv𝒜𝒞([0,T];VA+VB), and thus Kv𝒞w([0,T];VB). We remark that, in addition, one can easily show that Kv-u0W1,p(0,T;VA).

By construction, we have v^(0)=v0v0 in H and v^(T)=vN. In view of the a priori estimate (4.4), the sequence {v^(T)} is bounded in H, and hence possesses a subsequence, again denoted by , that converges weakly in H to an element vTH. We shall show that vT=v(T).

For any ϕ𝒞1([0,T]), wV, we test (4.13) with the function tϕ(t)w𝒞1([0,T];V), which allows us to integrate by parts. Together with (4.12), we find

(v(T),w)ϕ(T)-(v(0),w)ϕ(0)=0Tv(t),wϕ(t)dt+0Tv(t),wϕ(t)dt=0Tf(t)-f(t)+Av(t)-a(t)+(BKv)(t)-(BKv)(t),wϕ(t)dt+0Tv(t),wϕ(t)dt+0Tv^(t),wϕ(t)dt=0Tf(t)-f(t)+Av(t)-a(t)+(BKv)(t)-(BKv)(t),wϕ(t)dt+0Tv(t)-v^(t),wϕ(t)dt+(v^(T),w)ϕ(T)-(v^(0),w)ϕ(0).(4.14)

The first two terms on the right-hand side tend to zero as . This yields

(v(T),w)ϕ(T)-(v(0),w)ϕ(0)=(vT,w)ϕ(T)-(v0,w)ϕ(0)for all wV.

Choosing ϕ such that ϕ(T)=0 and ϕ(0)=0, respectively, shows that v(0)=v0 and v(T)=vT, respectively.

By definition, we have (Kv)(0)=u0u0 in VB and (Kv)(T)=(Ku0τv)N. Again, the a priori estimate yields the boundedness of {(Kv)(T)} in VB, and thus the existence of a weakly convergent subsequence, again denoted by , and an element uTVB such that (Kv)(T)uT in VB.

We shall show that uT=(Kv)(T). Analogously to (4.14), choosing ϕ(t)=tT, t[0,T], we have for any wV,

((Kv)(T),w)=0T(Kv)(t),wtTdt+0T(Kv)(t),w1Tdt.

Using (3.1), we obtain

((Kv)(T),w)=λ0Tv(t)-((Kv)(t)-u0),wtTdt+0T(Kv)(t),w1Tdt=lim(λ0Tv(t)-((Kv)(t)-u0),wtTdt+0T(Kv)(t),w1Tdt).(4.15)

For the first integral, (4.6) yields

0Tv(t)-((Kv)(t)-u0),wtTdt=j=1Ntj-1tjvj-((Ku0τv)j-u0),wtTdt=j=1Ntj-1tj1eλτ-1(Ku0τv)j-(Ku0τv)j-1,wtTdt.

For better readability, we write Kτ instead of Ku0τ for the rest of this proof. Splitting up the sum and shifting the index in the second sum yields

j=1Ntj-1tj(Kτv)j-(Kτv)j-1,wtTdt=(Kτv)N,wtN-1tNtTdt-u0,w0τtTdt+j=1N-1(Kτv)j,w(tj-1tjtTdt-tjtj+1tTdt)=(2T-τ)τ2T(Kτv)N,w-τ22Tu0,w-τ2Tj=1N(Kτv)j,w+τ2T(Kτv)N,w=((2T-τ)τ2T+τ2T)(Kv)(T),w-τ22Tu0,w-τT0T(Kv)(t),wdt.

Inserting this into (4.15) and using that {u0} is bounded in VB, (Kv)(T)uT in VB, Kv*Kv in L(0,T;VB) as well as λτ(eλτ-1)-11 as (and thus τ0), we end up with

((Kv)(T),w)=lim(λτeλτ-1((2T-τ2T+τT)(Kv)(T),w-τ2Tu0,w-1T0T(Kv)(t),wdt)+0T(Kv)(t),w1Tdt)=uT,w,

which proves uT=(Kv)(T).

It remains to show that a equals Av. We recall that vL(0,T;V). Hence, for arbitrary wLp(0,T;VA) the monotonicity of A implies

Av,v=Av-Aw,v-w+Aw,v-w+Av,wAw,v-w+Av,w.

The right-hand side converges to

Aw,v-w+a,w.

This already implies

lim infAv,vAw,v-w+a,w.

Below we will show

lim supAv,va,v,(4.16)

hence altogether proving

Aw-a,v-w0.

Taking w=v±rz for any zLp(0,T;VA), r>0, passing to the limit r0, and Lebesgue’s theorem on dominated convergence together with the hemicontinuity and the growth condition of A:VAVA yield

Av-a,z=0for all zLp(0,T;VA),

which proves the assertion.

To finish the proof of existence, it remains to show (4.16). We start with

Av,v=f,v-v^,v-BKv,v.(4.17)

Since ff in Lp(0,T;VA)+L1(0,T;H), the first term on the right-hand side tends to f,v as . Using (4.5), we estimate the second term as

v^,v=j=1N(vj-vj-1,vj)12vNH2-12v0H2.

As v0v0=v(0) and vNvT=v(T) in H, the weak lower semicontinuity of the norm provides

lim infv^,v12v(T)H2-12v(0)H2.

Regarding the last term on the right-hand side of equation (4.17), note that only vv in Lp(0,T;VA) and BKvBKv in L2(0,T;VB). Therefore, a finer examination of that term is necessary. To start with, a short calculation (see (4.6)) shows that

vn=e-λττγ1((Kτv)n-(Kτv)n-1)+1-e-λττγ1(Kτv)n-1-e-λττγ1u0.

We conclude with (4.5) that

BKv,v=τj=1N(BKτv)j,vj=e-λτγ1j=1N(BKτv)j,(Kτv)j-(Kτv)j-1+1-e-λτγ1j=1N(BKτv)j,(Kτv)j-1-e-λτγ1j=1N(BKτv)j,u0e-λτ2γ1((Kτv)NB2-(Kτv)0B2)+1-e-λττγ1KvL2(0,T;B)2-1-e-λττγ10T(BKv)(t),u0dt.

Since (Kτv)0=u0u0=(Kv)(0) in VB and (Kτv)N=(Kv)(T)uT=(Kv)(T) in VB as well as KvKv in L2(0,T;VB), the weak lower semicontinuity of the norms involved shows, by passing to the limit and employing γ1λ (see (4.2)),

lim infBKv,v12λ(Kv)(T)B2-12λ(Kv)(0)B2+KvL2(0,T;B)2-0T(BKv)(t),u0dt.

Altogether, we find

lim supAv,vf,v-12v(T)H2+12v(0)H2-12λ(Kv)(T)B2+12λ(Kv)(0)B2-KvL2(0,T;B)2+0T(BKv)(t),u0dt.

We replace f by v+a+BKv and remind that v𝒞w([0,T];H) as well as Kv𝒞w([0,T];VB). Thus we are allowed to apply Lemma 4.3, which yields (4.16). ∎

5 Uniqueness and Continuous Dependence on the Problem Data

Using again the integration-by-parts formula of Lemma 4.3, we are able to prove a stability and uniqueness result. Further, we are able to prove stability with respect to perturbations of the kernel parameter λ. In order to do so, we again have to prove an integration-by-parts formula using a different approach that requires less structure than the one used in the proof of Lemma 4.3.

Theorem 5.1 (Stability).

Suppose Assumptions A and B are fulfilled and let u0,u^0VB, v0,v^0H, and f,f^Lp(0,T;VA)+L1(0,T;H).

  • (i)

    Let f-f^L1(0,T;H) and let v and v^ be solutions to ( 1.1 ) with data (u0,v0,f) and (u^0,v^0,f^) , respectively. Then the stability estimate

    v(t)-v^(t)H2+1λ(Ku0v)(t)-(Ku^0v^)(t)B2+0t(Ku0v)(s)-(Ku^0v^)(s)B2dsc(v0-v^0H2+u0-u^0VB2+f-f^L1(0,T;H)2)

    holds for all t[0,T] , where the index in the notations Ku0 and Ku^0 denotes the dependence of the integral operator on u0 and u^0 , respectively.

  • (ii)

    Let f-f^Lp(0,T;VA) . Assume in addition that A is uniformly monotone in the sense that there is μ>0 such that

    Av-Aw,v-wμv-wVAp

    for all v,wVA . Let v and v^ be solutions to ( 1.1 ) with data (u0,v0,f) and (u^0,v^0,f^) , respectively. Then the stability estimate

    v(t)-v^(t)H2+μ0tv(s)-v^(s)VAp𝑑s+1λ(Ku0v)(t)-(Ku^0v^)(t)B2+0t(Ku0v)(s)-(Ku^0v^)(s)B2dsc(v0-v^0H2+u0-u^0VB2+f-f^Lp(0,T;VA)2)

    holds for all t[0,T].

Proof.

Let v and v^ be solutions for (u0,v0,f) and (u^0,v^0,f^), respectively. The difference of both equations (1.1a) then reads

(v-v^)+Av-Av^+B(Ku0v-Ku^0v^)=f-f^.(5.1)

Lemma 4.3 implies for all t[0,T],

0t(v-v^)(s)+B((Ku0v)(s)-(Ku^0v^)(s)),(v-v^)(s)ds=12v(t)-v^(t)H2-12v0-v^0H2+12λ(Ku0v)(t)-(Ku^0v^)(t)B2   -12λu0-u^0B2-0tB((Ku0v)(s)-(Ku^0v^)(s)),u0-u^0ds+0t(Ku0v)(s)-(Ku^0v^)(s)B2ds.

We test in equation (5.1) with v-v^. To prove (i), the monotonicity of A and Young’s inequality imply

12v(t)-v^(t)H2+12λ(Ku0v)(t)-(Ku^0v^)(t)B2+0t(Ku0v)(s)-(Ku^0v^)(s)B2ds12v0-v^0H2+12λu0-u^0B2+0tB((Ku0v)(s)-(Ku^0v^)(s)),u0-u^0ds+0t(f-f^)(s),(v-v^)(s)dsc(v0-v^0H2+u0-u^0B2+f-f^L1(0,T;H)2)+120t(Ku0v)(s)-(Ku^0v^)(s)B2ds+14v-v^L(0,T;H)2.

An argument analogous to the one used in the proof of Theorem 4.1 yields the first statement. The second one follows analogously from the uniform monotonicity of A and

0t(f-f^)(s),(v-v^)(s)dsf-f^Lp(0,T;VA)0t(v-v^)(s)VApds,

which proves the theorem. ∎

As usually, the stability estimates directly provide a uniqueness result, which we formulate in the following corollary.

Corollary 5.2 (Uniqueness).

Under the assumptions of Theorem 4.2, the solution to problem (1.1) is unique. Furthermore, the whole sequences {v} and {v^} of piecewise constant and of piecewise affine-linear prolongations of solutions to the discrete problem (4.1) converge to the solution in the sense stated in Theorem 4.2.

Proof.

We consider u0=u^0VB, v0=v^0H and f=f^Lp(0,T;VA)+L1(0,T;H) with corresponding solutions v and v^. As f-f^=0L1(0,T;H), the first assumption of Theorem 5.1 is fulfilled, which shows

v(t)-v^(t)H2+1λ(Kv)(t)-(Kv^)(t)B2+0t(Kv)(s)-(Kv^)(s)B2ds0

for all t[0,T]. This proves the uniqueness. Convergence of the whole sequences then follows as usual by contradiction. ∎

Finally, we aim to derive Lipschitz-type dependence on the kernel parameter 1λ, that is, Lipschitz dependence on the average relaxation-time.

Theorem 5.3 (Perturbation of λ).

Let the assumptions of Theorem 4.2 be fulfilled and let v0V. For λ>0, let Kλ denote the operator related to the kernel kλ(t)=λe-λt and let vλLp(0,T;VA)L(0,T;H) denote the corresponding solution to problem (1.1). Moreover, assume that vλL2(0,T;VB). Then for all μ>0 and corresponding solutions vμLp(0,T;VA)L(0,T;H) there holds

(vλ-vμ)(t)H2+0t(Kλvλ-Kμvμ)(s)B2dsλ22(1+λ2T2)|1λ-1μ|2vλL2(0,T;B)2

for almost all t.

Proof.

Taking the difference of the equations (1.1a) for vλ and vμ, respectively, and testing with vλ-vμ leads to

0t(vλ-vμ)(s)+B((Kλvλ)(s)-(Kμvμ)(s)),(vλ-vμ)(s)ds=-0tAvλ(s)-Avμ(s),(vλ-vμ)(s)ds0.(5.2)

To deal with the integral on the left-hand side, we need to prove an integration-by-parts formula similar to (4.9). Unfortunately, due to the fact that we have two different parameters λ and μ instead of one, we are not able to apply the same method as in the proof of Lemma 4.3. Therefore, we use another method of proof using the centered Steklov average.2

In order to prove the integration-by-parts formula on (0,t), we first have to extend the functions considered to the interval (-η,0) for a fixed η>0. We then prove the formula on an arbitrary interval (α,β) with

-η+h0<α<β<T-h0.

At the end, instead of a fixed α, we consider a sequence {αk} with αk<0 and αk0 as k. If we considered a sequence of positive αk, we would not be able to identify the limit properly.

We fix h0>0. The centered Steklov average for a function zLp(-η,T;X), with X being an arbitrary Banach space, is defined by

(Shz)(t):=12ht-ht+hz(s)ds,

where 0<h<h0 and t[-η+h0,T-h0]. It is well known (see [16, Theorem 9, p. 49]) that (Shz)(t)z(t) in X for almost all t(-η+h0,T-h0). In addition, it is easy to show that ShzLp(-η+h0,T-h0;X) as well as, by using Fubini’s theorem,

ShzLp(-η+h0,T-h0;X)zLp(-η,T;X),

and thus Shzz in Lp(-η+h0,T-h0;X).

First, note that both vλ and vμ are in 𝒞w([0,T];H), and take the same value v0 at t=0. Hence, the difference vλ-vμ fulfills the initial condition (vλ-vμ)(0)=0. We assume both vλ and vμ to be extended by v0V for t<0, and thus the difference to be extended by 0 for t<0. We conclude that vλ-vμLp(-η,T;VA)𝒞w([-η,T];H) and similarly Kλvλ-Kμvμ𝒞w([-η,T];VB). Relation (3.1) still holds for both vλ and vμ for almost all t(-η,0).

As

vλ-vμLp(-η,T;VA)𝒞w([-η,T];H),

it is easy to show that Sh(vλ-vμ) is bounded in Lp(-η+h0,T-h0;VA)L(-η+h0,T-h0;H). Therefore, there exists a subsequence, again denoted by h, such that Sh(vλ-vμ)*vλ-vμ in

Lp(-η+h0,T-h0;VA)L(-η+h0,T-h0;H).

In view of (3.1), we have

(Sh(vλ-vμ))(t)=12h((1λKλvλ-1μKμvμ)(t+h)-(1λKλvλ-1μKμvμ)(t-h))+(ShKλvλ)(t)-(ShKμvμ)(t)

such that Kλvλ,Kμvμ𝒞w([-η,T];VB) provide Sh(vλ-vμ)𝒞w([-η+h0,T-h0];VB). Thus, in contrast to the difference vλ-vμ itself, the Steklov average Sh(vλ-vμ) is an element of

Lp(-η+h0,T-h0;VA)𝒞w([-η+h0,T-h0];VB),

which provides the regularity needed.

For all α,β with -η+h0<α<β<T-h0 we find

αβ(vλ-vμ)(t)+B((Kλvλ)(t)-(Kμvμ)(t)),(vλ-vμ)(t)dt=αβ(vλ-vμ)(t)+B((Kλvλ)(t)-(Kμvμ)(t)),(vλ-vμ)(t)-(Sh(vλ-vμ))(t)dt   +αβ(vλ-vμ)(t)+B((Kλvλ)(t)-(Kμvμ)(t))   -(Sh((vλ-vμ)+B(Kλvλ-Kμvμ)))(t),(Sh(vλ-vμ))(t)dt   +αβ(Sh(vλ-vμ))(t)+B(Sh(Kλvλ-Kμvμ))(t),(Sh(vλ-vμ))(t)dt.(5.3)

The first term on the right-hand side tends to zero as h0 since

(vλ-vμ)+B(Kλvλ-Kμvμ)=-(Avλ-Avμ)Lp(-η+h0,T-h0;VA)+L1(-η+h0,T-h0;H)

and

Sh(vλ-vμ)*vλ-vμ

in Lp(-η+h0,T-h0;VA)L(-η+h0,T-h0;H) as h0, the second one since {Sh(vλ-vμ)} is bounded in Lp(-η+h0,T-h0;VA)L(-η+h0,T-h0;H), and

Sh((vλ-vμ)+B(Kλvλ-Kμvμ))(vλ-vμ)+B(Kλvλ-Kμvμ)

in Lp(-η+h0,T-h0;VA)+L1(-η+h0,T-h0;H) as h0. The latter follows from the linearity of Sh.

Due to the regularity properties of Sh(vλ-vμ), we are allowed to split the last integrand in (5.3). We use

vλ-vμ=1λ(Kλvλ)-1μ(Kμvμ)+Kλvλ-Kμvμ

as well as Sh(z)=(Shz) for any integrable function z on (α,β)(-η+h0,T-h0). We rewrite the third term as

αβ(Sh(vλ-vμ))(t)+B(Sh(Kλvλ-Kμvμ))(t),(Sh(vλ-vμ))(t)dt=αβ(Sh(vλ-vμ))(t),(Sh(vλ-vμ))(t)dt+αβB(Sh(Kλvλ-Kμvμ))(t),(Sh(Kλvλ-Kμvμ))(t)dt   +αβB(Sh(Kλvλ-Kμvμ))(t),1λ(Sh(Kλvλ))(t)-1μ(Sh(Kμvμ))(t)dt=12(Sh(vλ-vμ))(β)H2-12(Sh(vλ-vμ))(α)H2+αβ(Sh(Kλvλ-Kμvμ))(t)B2dt   +1μαβB(Sh(Kλvλ-Kμvμ))(t),(Sh(Kλvλ-Kμvμ))(t)dt   +(1λ-1μ)αβB(Sh(Kλvλ-Kμvμ))(t),(Sh(Kλvλ))(t)dt12(Sh(vλ-vμ))(β)H2-12(Sh(vλ-vμ))(α)H2+αβ(Sh(Kλvλ-Kμvμ))(t)B2dt   +12μ(Sh(Kλvλ-Kμvμ))(β)B2-12μ(Sh(Kλvλ-Kμvμ))(α)B2   -|1λ-1μ|αβ(Sh(Kλvλ-Kμvμ))(t)B(Sh(Kλvλ))(t)Bdt.

Taking the limit h and employing (5.2) results in

12(vλ-vμ)(β)H2+αβ(Kλvλ-Kμvμ)(t)B2dt+12μ(Kλvλ-Kμvμ)(β)B212(vλ-vμ)(α)H2+12μ(Kλvλ-Kμvμ)(α)B2   +|1λ-1μ|αβ(Kλvλ-Kμvμ)(t)B(Kλvλ)(t)Bdt(5.4)

for almost all α,β, namely for those that are Lebesgue points of vλ-vμ and Kλvλ-Kμvμ. Finally, we consider a sequence {αk} with αk<0 and αk0 as k. We remark that vλ-vμ𝒞w([-η,T];H) and Kλvλ-Kμvμ𝒞w([-η,T];VB). Due to the choice of the extension for t<0, equation (5.4) shows for β=t that

12(vλ-vμ)(t)H2+0t(Kλvλ-Kμvμ)(s)B2ds+12μ(Kλvλ-Kμvμ)(t)B2|1λ-1μ|0t(Kλvλ-Kμvμ)(s)B(Kλvλ)(s)Bds

for almost all t(0,T), and thus

12(vλ-vμ)(t)H2+120t(Kλvλ-Kμvμ)(s)B2ds+12μ(Kλvλ-Kμvμ)(t)B212|1λ-1μ|2(Kλvλ)L2(0,T;B)2λ22|1λ-1μ|2(vλL2(0,T;B)2+Kλvλ-u0L2(0,T;B)2).

It is easy to show that Kλvλ-u0L2(0,T;B)2λ2T2vλL2(0,T;B)2, which proves the assertion. ∎

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Footnotes

  • 1

    Here, we slightly abuse the notation of the duality pairing because Lp(0,T;VA)+L1(0,T;H)+L(0,T;VB) is not the dual space of Lp(0,T;VA)L(0,T;H)L1(0,T;VB). However, we can consider the sum of the duality pairings between Lp(0,T;VA)+L1(0,T;H) and L1(0,T;VB) and their respective dual spaces. 

  • 2

    We could have also used this method to prove Lemma 4.3, but it only provides the integration-by-parts formula to hold on (α,β) for almost all α,β(0,T). 

About the article

Received: 2018-06-20

Accepted: 2018-10-20

Published Online: 2018-11-21

Published in Print: 2020-01-01


This work has been supported by the Deutsche Forschungsgemeinschaft through Collaborative Research Center 910 “Control of self-organizing nonlinear systems: Theoretical methods and concepts of application”.


Citation Information: Computational Methods in Applied Mathematics, Volume 20, Issue 1, Pages 89–108, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0268.

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