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

André Eikmeier
/ Etienne Emmrich
/ 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 ${v}^{\prime }+Av+BKv=f$ is studied, where $A:{V}_{A}\to {V}_{A}^{\prime }$ is a monotone, coercive operator and where $B:{V}_{B}\to {V}_{B}^{\prime }$ induces an inner product. The Banach space ${V}_{A}$ is not required to be embedded in ${V}_{B}$ 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.

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

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

1.1 Problem Statement and Main Result

We are concerned with initial value problems of the type

(1.1a)$v\left(0\right)={v}_{0},$(1.1b)$\text{where} \left(Kv\right)\left(t\right)={u}_{0}+{\int }_{0}^{t}k\left(t-s\right)v\left(s\right)ds,k\left(z\right)=\lambda {e}^{-\lambda z}.$(1.1c)

Here, $T>0$ is the time under consideration, $\lambda >0$ is a given parameter, and ${u}_{0}$, ${v}_{0}$, f are given data of the problem.

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

${V}_{A}\cap {V}_{B}=V\subset {V}_{C}\subset H={H}^{\prime }\subset {V}_{C}^{\prime }\subset {V}^{\prime }={V}_{A}^{\prime }+{V}_{B}^{\prime },C\in \left\{A,B\right\},$(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

${v}^{\prime }+Av+Cu=f,$(1.3a)${\left(u-{u}_{0}\right)}^{\prime }+\lambda \left(u-{u}_{0}\right)=\lambda Dv,$(1.3b)

with suitable linear operators C, D such that $B=CD$ (and with ${u}_{0}$ being replaced by $D{u}_{0}$).

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 $k\left(z\right)=c{e}^{-\lambda z}$ ($c,\lambda >0$). 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 $\frac{1}{\lambda }$ can be interpreted as a relaxation or averaged delay time. If λ tends to 0, then the system decouples with $u\left(t\right)\equiv {u}_{0}$. The limit $\lambda \to \mathrm{\infty }$ results in a first-order equation for v without memory. In contrast to this, in the case where $k\left(z\right)=c{e}^{-\lambda z}$, the limit $\lambda \to 0$ results in an evolution equation for u of second order in time (see [19]), whereas $\lambda \to \mathrm{\infty }$ 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 ${V}_{A}$ is embedded in ${V}_{B}$ (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

${v}^{\prime }+a\ast Av=f,$

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 ${V}_{A}={V}_{B}$. 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 ${\parallel \cdot \parallel }_{X}$, its dual by ${X}^{\prime }$, equipped with the standard norm ${\parallel \cdot \parallel }_{{X}^{\prime }}$, and the duality pairing by $〈\cdot ,\cdot 〉$. We recall that ${X}^{\prime }$ is reflexive and separable if X is. For a Hilbert space X, we denote the inner product (inducing the norm ${\parallel \cdot \parallel }_{X}$) by ${\left(\cdot ,\cdot \right)}_{X}$. The intersection of two Banach spaces X, Y is equipped with the norm ${\parallel \cdot \parallel }_{X\cap Y}={\parallel \cdot \parallel }_{X}+{\parallel \cdot \parallel }_{Y}$, whereas the sum $X+Y$ is equipped with the norm

We recall that ${\left(X\cap Y\right)}^{\prime }={X}^{\prime }+{Y}^{\prime }$; see also [22, pp. 12ff.].

For a real, reflexive, separable Banach space X, the Bochner–Lebesgue spaces ${L}^{r}\left(0,T;X\right)$ ($r\in \left[1,\mathrm{\infty }\right]$) are defined in the usual way and equipped with the standard norm. Denoting by ${r}^{\prime }=\frac{r}{r-1}$ the conjugate of $r\in \left(1,\mathrm{\infty }\right)$ with ${r}^{\prime }=\mathrm{\infty }$ if $r=1$, we have that ${\left({L}^{r}\left(0,T;X\right)\right)}^{\prime }={L}^{{r}^{\prime }}\left(0,T;{X}^{\prime }\right)$ if $r\in \left[1,\mathrm{\infty }\right)$; the duality pairing is given by

$〈g,v〉={\int }_{0}^{T}〈g\left(t\right),v\left(t\right)〉dt;$

see, e.g., [16, Theorem 1 on p. 98, Corollary 13 on p. 76, Theorem 1 on p. 79]. Moreover, ${L}^{r}\left(0,T;X\right)$ is reflexive if $r\in \left(1,\mathrm{\infty }\right)$ (see [16, Corollary 2 on p. 100]) and ${L}^{1}\left(0,T;X\right)$ is separable.

By ${W}^{1,r}\left(0,T;X\right)$ ($r\in \left[1,\mathrm{\infty }\right]$) we denote the Banach space of functions $u\in {L}^{r}\left(0,T;X\right)$ whose distributional time derivative ${u}^{\prime }$ is again in ${L}^{r}\left(0,T;X\right)$; the space is equipped with the standard norm. Note that if $u\in {W}^{1,1}\left(0,T;X\right)$, then u equals almost everywhere a function that is in $\mathcal{𝒜}\mathcal{𝒞}\left(\left[0,T\right];X\right)$, i.e., a function that is absolutely continuous on $\left[0,T\right]$ as a function taking values in X. Moreover, ${W}^{1,1}\left(0,T;X\right)$ is continuously embedded in the Banach space $\mathcal{𝒞}\left(\left[0,T\right];X\right)$ of functions that are continuous on $\left[0,T\right]$ as functions with values in X (see, e.g., [45, Chapter 7] for more details). By ${\mathcal{𝒞}}_{w}\left(\left[0,T\right];X\right)$ we denote the space of functions that are continuous on $\left[0,T\right]$ with respect to the weak topology in X.

The space of infinitely many times differentiable real functions with compact support in $\left(0,T\right)$ is denoted by ${\mathcal{𝒞}}_{c}^{\mathrm{\infty }}\left(0,T\right)$. The space of on $\left[0,T\right]$ continuously differentiable real functions is denoted by ${\mathcal{𝒞}}^{1}\left(\left[0,T\right]\right)$. By c we denote a generic positive constant. We set ${\sum }_{j=1}^{0}{x}_{j}=0$ for ${x}_{j}$ whatsoever.

3 Main Assumptions and Preliminary Results

Let ${V}_{A}$ be a real, reflexive Banach space and let ${V}_{B}$ and H be real Hilbert spaces satisfying (1.2). Moreover, we assume that $V={V}_{A}\cap {V}_{B}$ is separable.

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

Assumption A.

The operator $A:{V}_{A}\to {V}_{A}^{\prime }$ satisfies for all $u,v,w\in {V}_{A}$,

• (i)

$\theta ↦〈A\left(u+\theta v\right),w〉\in \mathcal{𝒞}\left(\left[0,1\right]\right)$ (hemicontinuity),

and there exist $p\in \left(2,\mathrm{\infty }\right)$, ${\mu }_{A},{\beta }_{A}>0$, ${c}_{A}\ge 0$ such that for all $v,w\in {V}_{A}$,

• (ii)

$〈Av-Aw,v-w〉\ge 0$ (monotonicity),

• (iii)

$〈Av,v〉\ge {\mu }_{A}{\parallel v\parallel }_{{V}_{A}}^{p}-{c}_{A}$ (p-coercivity),

• (iv)

${\parallel Av\parallel }_{{V}_{A}^{\prime }}\le {\beta }_{A}\left(1+{\parallel v\parallel }_{{V}_{A}}^{p-1}\right)$ ($\left(p-1\right)$-growth).

Assumption B.

The linear operator $B:{V}_{B}\to {V}_{B}^{\prime }$ is symmetric and there exist ${\mu }_{B},{\beta }_{B}>0$ such that for all $v\in {V}_{B}$,

• (i)

$〈Bv,v〉\ge {\mu }_{B}{\parallel v\parallel }_{{V}_{B}}^{2}$ (strong positivity),

• (ii)

${\parallel Bv\parallel }_{{V}_{B}^{\prime }}\le {\beta }_{B}{\parallel v\parallel }_{{V}_{B}}$ (boundedness).

By ${\parallel \cdot \parallel }_{B}={〈B\cdot ,\cdot 〉}^{1/2}$ we denote the norm induced by B, which is equivalent to ${\parallel \cdot \parallel }_{{V}_{B}}$. Further, by ${L}^{2}\left(0,T;B\right)$ we denote the space ${L}^{2}\left(0,T;\left({V}_{B},{\parallel \cdot \parallel }_{B}\right)\right)$.

We shall remark that the operators A and B can be extended, as usual, to operators acting on functions defined on $\left[0,T\right]$ and taking values in ${V}_{A}$ and ${V}_{B}$, respectively. Since the operator $A:{V}_{A}\to {V}_{A}^{\prime }$ 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 ${V}_{A}^{\prime }$ and the theorem of Pettis (see, e.g., [16, Theorem 2 on p. 42]), a Bochner measurable function $v:\left[0,T\right]\to {V}_{A}$ into a Bochner measurable function $Av:\left[0,T\right]\to {V}_{A}^{\prime }$ with $\left(Av\right)\left(t\right)=Av\left(t\right)$ ($t\in \left[0,T\right]$). Because of the growth condition, the operator A then maps ${L}^{p}\left(0,T;{V}_{A}\right)$ into ${\left({L}^{p}\left(0,T;{V}_{A}\right)\right)}^{\prime }={L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)$. The linear, symmetric, bounded, strongly positive operator $B:{V}_{B}\to {V}_{B}^{\prime }$ extends, via $\left(Bu\right)\left(t\right)=Bu\left(t\right)$ for a function $u:\left[0,T\right]\to {V}_{B}$, to a linear, symmetric, bounded, strongly positive operator mapping ${L}^{2}\left(0,T;{V}_{B}\right)$ into its dual. Indeed, B can also be seen as a linear, bounded operator mapping ${L}^{r}\left(0,T;{V}_{B}\right)$ into ${L}^{r}\left(0,T;{V}_{B}^{\prime }\right)$ for any $r\in \left[1,\mathrm{\infty }\right]$.

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 ${u}_{\mathrm{0}}\mathrm{\in }{V}_{B}$. Then K is an affine-linear, bounded mapping of ${L}^{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}\mathrm{\right)}$ into itself with

${\parallel Kv-{u}_{0}\parallel }_{{L}^{2}\left(0,T;{V}_{B}\right)}\le {\parallel k\parallel }_{{L}^{1}\left(0,T\right)}{\parallel v\parallel }_{{L}^{2}\left(0,T;{V}_{B}\right)},v\in {L}^{2}\left(0,T;{V}_{B}\right),$

where ${\mathrm{\parallel }k\mathrm{\parallel }}_{{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{\right)}}\mathrm{=}\mathrm{1}\mathrm{-}{e}^{\mathrm{-}\lambda \mathit{}T}$. Further, K is an affine-linear, bounded mapping of ${L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}\mathrm{\right)}$ into $\mathcal{A}\mathit{}\mathcal{C}\mathit{}\mathrm{\left(}\mathrm{\left[}\mathrm{0}\mathrm{,}T\mathrm{\right]}\mathrm{;}{V}_{B}\mathrm{\right)}$ with

${\parallel Kv-{u}_{0}\parallel }_{\mathcal{𝒞}\left(\left[0,T\right];{V}_{B}\right)}\le \lambda {\parallel v\parallel }_{{L}^{1}\left(0,T;{V}_{B}\right)},v\in {L}^{1}\left(0,T;{V}_{B}\right).$

The standard proof is omitted here.

Note that in the assertions above ${V}_{B}$ can be replaced by H. An immediate consequence of the lemma above is that $BK:{L}^{2}\left(0,T;{V}_{B}\right)\to {L}^{2}\left(0,T;{V}_{B}^{\prime }\right)$ as well as $BK:{L}^{1}\left(0,T;{V}_{B}\right)\to {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}^{\prime }\right)$ 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 $v\in {L}^{1}\left(0,T;X\right)$, with X being an arbitrary Banach space, we have that for almost all $t\in \left(0,T\right)$,

${\left(Kv\right)}^{\prime }\left(t\right)=\lambda \left(v\left(t\right)-\left(\left(Kv\right)\left(t\right)-{u}_{0}\right)\right).$(3.1)

If X is a Hilbert space and if $v\in {L}^{2}\left(0,T;X\right)$, then testing this relation by $Kv-{u}_{0}$ immediately implies for all $t\in \left[0,T\right]$,

${\int }_{0}^{t}{\left(\left(Kv\right)\left(s\right)-{u}_{0},v\left(s\right)\right)}_{X}ds={\int }_{0}^{t}{\parallel \left(Kv\right)\left(s\right)-{u}_{0}\parallel }_{X}^{2}ds+\frac{1}{2\lambda }{\parallel \left(Kv\right)\left(t\right)-{u}_{0}\parallel }_{X}^{2},$

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 $N\in ℕ$, let $\tau =\frac{T}{N}$ and ${t}_{n}=n\tau$ ($n=0,1,\mathrm{\dots },N$). Let ${v}^{0}\approx {v}_{0}$, ${u}^{0}\approx {u}_{0}$, and ${\left\{{f}^{n}\right\}}_{n=1}^{N}\approx f$ be given approximations of the problem data ${v}_{0}$, ${u}_{0}$, and f, respectively. We look for approximations ${v}^{n}\approx v\left({t}_{n}\right)$ ($n=1,2,\mathrm{\dots },N$).

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

$\frac{1}{\tau }\left({v}^{n}-{v}^{n-1}\right)+A{v}^{n}+{\left(B{K}_{{u}^{0}}^{\tau }v\right)}^{n}={f}^{n},n=1,2,\mathrm{\dots },N,$(4.1a)$\text{where}{\left({K}_{{u}^{0}}^{\tau }v\right)}^{n}:={u}^{0}+\tau \sum _{j=1}^{n}{\gamma }_{n-j+1}{v}^{j}.$(4.1b)

To be precise, ${K}_{{u}^{0}}^{\tau }$ acts on a grid function ${\left\{{v}^{n}\right\}}_{n=1}^{N}$ and, with a slight abuse of notation, the evaluation at n is denoted by ${\left({K}_{{u}^{0}}^{\tau }v\right)}^{n}$. The coefficients ${\gamma }_{i}$ ($i=1,2,\mathrm{\dots },N$) are given by

${\gamma }_{i}={\int }_{0}^{1}k\left(\left(i-\stackrel{^}{s}\right)\tau \right)d\stackrel{^}{s},$(4.1c)

and hence

${\gamma }_{n-j+1}={\int }_{0}^{1}k\left(\left(n-j+1-\stackrel{^}{s}\right)\tau \right)d\stackrel{^}{s}=\frac{1}{\tau }{\int }_{{t}_{j-1}}^{{t}_{j}}k\left({t}_{n}-s\right)ds.$

The idea behind is the approximation

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

${\gamma }_{i}=\frac{{e}^{\lambda \tau }-1}{\tau }{e}^{-\lambda {t}_{i}},i=1,2,\mathrm{\dots },N,$

which immediately leads to the properties

(4.2)

and

${\gamma }_{i+1}-{\gamma }_{i}=-\left({e}^{\lambda \tau }-1\right){\gamma }_{i+1},i=1,2,\mathrm{\dots },N-1.$(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 ${u}^{\mathrm{0}}\mathrm{\in }{V}_{B}$, ${v}^{\mathrm{0}}\mathrm{\in }H$, and ${\mathrm{\left\{}{f}^{n}\mathrm{\right\}}}_{n\mathrm{=}\mathrm{1}}^{N}\mathrm{=}{\mathrm{\left\{}{f}_{\mathrm{0}}^{n}\mathrm{\right\}}}_{n\mathrm{=}\mathrm{1}}^{N}\mathrm{+}{\mathrm{\left\{}{f}_{\mathrm{1}}^{n}\mathrm{\right\}}}_{n\mathrm{=}\mathrm{1}}^{N}\mathrm{\subset }{V}_{A}^{\mathrm{\prime }}\mathrm{+}H$ be given. Then there is a unique solution ${\mathrm{\left\{}{v}^{n}\mathrm{\right\}}}_{n\mathrm{=}\mathrm{1}}^{N}\mathrm{\subset }V\mathrm{=}{V}_{A}\mathrm{\cap }{V}_{B}$ to (4.1). Moreover, there holds for $n\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}N$,

${\parallel {v}^{n}\parallel }_{H}^{2}+\sum _{j=1}^{n}{\parallel {v}^{j}-{v}^{j-1}\parallel }_{H}^{2}+{\mu }_{A}\tau \sum _{j=1}^{n}{\parallel {v}^{j}\parallel }_{{V}_{A}}^{p}+\frac{T}{{e}^{\lambda T}-1}{\parallel {\left({K}_{{u}^{0}}^{\tau }v\right)}^{n}\parallel }_{{V}_{B}}^{2}$$\le c\left(1+{\parallel {u}^{0}\parallel }_{{V}_{B}}^{2}+{\parallel {v}^{0}\parallel }_{H}^{2}+\tau \sum _{j=1}^{N}{\parallel {f}_{0}^{j}\parallel }_{{V}_{A}^{\prime }}^{{p}^{\prime }}+{\left(\tau \sum _{j=1}^{N}{\parallel {f}_{1}^{j}\parallel }_{H}\right)}^{2}\right).$(4.4)

Proof.

For better readability, we write ${K}^{\tau }$ instead of ${K}_{{u}^{0}}^{\tau }$ 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 ${v}^{n}$ from ${\left\{{v}^{j}\right\}}_{j=1}^{n-1}\subset V$ and the data of the problem by solving

$\left(\frac{1}{\tau }I+A+\tau {\gamma }_{1}B\right){v}^{n}={f}^{n}+\frac{1}{\tau }{v}^{n-1}-B{u}^{0}-\tau \sum _{j=1}^{n-1}{\gamma }_{n-j+1}B{v}^{j}.$

Because of the continuous embeddings (1.2), we know that the right-hand side of the foregoing relation is in ${V}^{\prime }$. The operator $M:=\frac{1}{\tau }I+A+\tau {\gamma }_{1}B$ is easily shown to be hemicontinuous, coercive, and strictly monotone as a mapping of V into ${V}^{\prime }$. Here, we make use of the fact that ${\gamma }_{1}>0$. In particular, we observe that for all $w\in V$,

$〈Mw,w〉\ge \frac{1}{\tau }{\parallel w\parallel }_{H}^{2}+{\mu }_{A}{\parallel w\parallel }_{{V}_{A}}^{p}-{c}_{A}+\tau {\gamma }_{1}{\mu }_{B}{\parallel w\parallel }_{{V}_{B}}^{2}\ge c{\parallel w\parallel }_{V}^{2}-c.$

The famous theorem of Browder and Minty (see, e.g., [55, Theorem 26.A on p. 557]) now provides existence of a solution ${v}^{n}\in V$; uniqueness immediately follows from the strict monotonicity.

For proving the a priori estimate, we test (4.1a) by ${v}^{n}\in V$ ($n=1,2,\mathrm{\dots },N$). With

$\left(a-b\right)a=\frac{1}{2}\left({a}^{2}-{b}^{2}+{\left(a-b\right)}^{2}\right),a,b\in ℝ,$(4.5)

the coercivity of A, and Young’s inequality, we find for $n=1,2,\mathrm{\dots },N$,

$\frac{1}{2\tau }\left({\parallel {v}^{n}\parallel }_{H}^{2}-{\parallel {v}^{n-1}\parallel }_{H}^{2}+{\parallel {v}^{n}-{v}^{n-1}\parallel }_{H}^{2}\right)+{\mu }_{A}{\parallel {v}^{n}\parallel }_{{V}_{A}}^{p}-{c}_{A}+〈{\left(B{K}^{\tau }v\right)}^{n},{v}^{n}〉$$\le 〈{f}^{n},{v}^{n}〉$$\le {\parallel {f}_{0}^{n}\parallel }_{{V}_{A}^{\prime }}{\parallel {v}^{n}\parallel }_{{V}_{A}}+{\parallel {f}_{1}^{n}\parallel }_{H}{\parallel {v}^{n}\parallel }_{H}$$\le \frac{{\mu }_{A}}{2}{\parallel {v}^{n}\parallel }_{{V}_{A}}^{p}+c{\parallel {f}_{0}^{n}\parallel }_{{V}_{A}^{\prime }}^{{p}^{\prime }}+{\parallel {f}_{1}^{n}\parallel }_{H}{\parallel {v}^{n}\parallel }_{H}.$

By summing up, we obtain for $n=1,2,\mathrm{\dots },N$,

${\parallel {v}^{n}\parallel }_{H}^{2}+\sum _{j=1}^{n}{\parallel {v}^{j}-{v}^{j-1}\parallel }_{H}^{2}+{\mu }_{A}\tau \sum _{j=1}^{n}{\parallel {v}^{n}\parallel }_{{V}_{A}}^{p}+2\tau \sum _{j=1}^{n}〈{\left(B{K}^{\tau }v\right)}^{j},{v}^{j}〉$$\le c\left(1+{\parallel {v}^{0}\parallel }_{H}^{2}+\tau \sum _{j=1}^{n}{\parallel {f}_{0}^{j}\parallel }_{{V}_{A}^{\prime }}^{{p}^{\prime }}\right)+2\tau \sum _{j=1}^{n}{\parallel {f}_{1}^{j}\parallel }_{H}{\parallel {v}^{j}\parallel }_{H},$

where

$2\tau \sum _{j=1}^{n}{\parallel {f}_{1}^{j}\parallel }_{H}{\parallel {v}^{j}\parallel }_{H}\le 2\tau \sum _{j=1}^{n}{\parallel {f}_{1}^{j}\parallel }_{H}\underset{j=1,\mathrm{\dots },n}{\mathrm{max}}{\parallel {v}^{j}\parallel }_{H}\le 2{\left(\tau \sum _{j=1}^{n}{\parallel {f}_{1}^{j}\parallel }_{H}\right)}^{2}+\frac{1}{2}\underset{j=1,\mathrm{\dots },n}{\mathrm{max}}{\parallel {v}^{j}\parallel }_{H}^{2}.$

We observe that (4.1b) together with (4.2), (4.3) implies for $n=2,3,\mathrm{\dots },N$,

$\frac{1}{\tau }\left({\left({K}^{\tau }v\right)}^{n}-{\left({K}^{\tau }v\right)}^{n-1}\right)={\gamma }_{1}{v}^{n}+\sum _{j=1}^{n-1}\left({\gamma }_{n-j+1}-{\gamma }_{n-j}\right){v}^{j}=\frac{{e}^{\lambda \tau }-1}{\tau }\left({v}^{n}-\left({\left({K}^{\tau }v\right)}^{n}-{u}^{0}\right)\right),$(4.6)

which is the discrete analogue of the crucial relation (3.1). Note that $\left({e}^{\lambda \tau }-1\right)/\tau \to \lambda$ as $\tau \to 0$. By setting ${\left({K}^{\tau }v\right)}^{0}:={u}^{0}$, relation (4.6) remains true for $n=1$.

Resolving (4.6) for ${v}^{n}$, together with (4.5) (recall that $B:{V}_{B}\to {V}_{B}^{\prime }$ induces an inner product on ${V}_{B}$), gives

$2\tau \sum _{j=1}^{n}〈{\left(B{K}^{\tau }v\right)}^{j},{v}^{j}〉=\frac{2\tau }{{e}^{\lambda \tau }-1}\sum _{j=1}^{n}〈{\left(B{K}^{\tau }v\right)}^{j},{\left({K}^{\tau }v\right)}^{j}-{\left({K}^{\tau }v\right)}^{j-1}〉+2\tau \sum _{j=1}^{n}〈{\left(B{K}^{\tau }v\right)}^{j},{\left({K}^{\tau }v\right)}^{j}-{u}^{0}〉$$\ge \frac{\tau }{{e}^{\lambda \tau }-1}\left({\parallel {\left({K}^{\tau }v\right)}^{n}\parallel }_{B}^{2}-{\parallel {\left({K}^{\tau }v\right)}^{0}\parallel }_{B}^{2}\right)+\tau \sum _{j=1}^{n}{\parallel {\left({K}^{\tau }v\right)}^{j}\parallel }_{B}^{2}-T{\parallel {u}^{0}\parallel }_{B}^{2},$

where

$\frac{\tau }{{e}^{\lambda \tau }-1}\ge \frac{T}{{e}^{\lambda T}-1}.$

Altogether, we find for $n=1,2,\mathrm{\dots },N$,

${\parallel {v}^{n}\parallel }_{H}^{2}+\sum _{j=1}^{n}{\parallel {v}^{j}-{v}^{j-1}\parallel }_{H}^{2}+{\mu }_{A}\tau \sum _{j=1}^{n}{\parallel {v}^{n}\parallel }_{{V}_{A}}^{p}+\frac{T}{{e}^{\lambda T}-1}{\parallel {\left({K}^{\tau }v\right)}^{n}\parallel }_{B}^{2}$$\le c\left(1+{\parallel {u}^{0}\parallel }_{{V}_{B}}^{2}+{\parallel {v}^{0}\parallel }_{H}^{2}+\tau \sum _{j=1}^{N}{\parallel {f}_{0}^{j}\parallel }_{{V}_{A}^{\prime }}^{{p}^{\prime }}+{\left(\tau \sum _{j=1}^{N}{\parallel {f}_{1}^{j}\parallel }_{H}\right)}^{2}\right)+\frac{1}{2}\underset{j=1,\mathrm{\dots },n}{\mathrm{max}}{\parallel {v}^{j}\parallel }_{H}^{2}.$(4.7)

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

$\underset{j=1,\mathrm{\dots },N}{\mathrm{max}}{\parallel {v}^{j}\parallel }_{H}^{2}\le C+\frac{1}{2}\underset{j=1,\mathrm{\dots },N}{\mathrm{max}}{\parallel {v}^{j}\parallel }_{H}^{2},$

where $C>0$ depends on ${u}^{0}$, ${v}^{0}$, and $\left\{{f}^{n}\right\}$. This yields

$\underset{j=1,\mathrm{\dots },N}{\mathrm{max}}{\parallel {v}^{j}\parallel }_{H}^{2}\le 2C,$

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

4.3 Convergence of Approximate Solutions and Existence of a Generalised Solution

Let $\left\{{N}_{\mathrm{\ell }}\right\}$ be a sequence of positive integers such that ${N}_{\mathrm{\ell }}\to \mathrm{\infty }$ as $\mathrm{\ell }\to \mathrm{\infty }$. We consider the corresponding sequence of time discrete problems (4.1) with step sizes ${\tau }_{\mathrm{\ell }}=T/{N}_{\mathrm{\ell }}$, starting values ${v}_{\mathrm{\ell }}^{0}\in H$ with ${v}_{\mathrm{\ell }}^{0}\to {v}_{0}$ in H as well as ${u}_{\mathrm{\ell }}^{0}\in {V}_{B}$ with ${u}_{\mathrm{\ell }}^{0}\to {u}_{0}$ in ${V}_{B}$, and right-hand sides ${\left\{{f}^{n}\right\}}_{n=1}^{{N}_{\mathrm{\ell }}}\subset {V}_{A}^{\prime }+H$ given by

${f}^{n}={f}_{0}^{n}+{f}_{1}^{n},{f}_{0,1}^{n}:=\frac{1}{{\tau }_{\mathrm{\ell }}}{\int }_{{t}_{n-1}}^{{t}_{n}}{f}_{0,1}\left(t\right)dt,$

which is well-defined for $f={f}_{0}+{f}_{1}\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$. As a slight abuse of notation, in general we do not call the dependence of ${u}^{n},{v}^{n},{f}^{n}$, and of the time instances ${t}_{n}$ on $\mathrm{\ell }$.

Let ${\left\{{v}^{n}\right\}}_{n=1}^{{N}_{\mathrm{\ell }}}\subset V$ denote the solution to (4.1) with step size ${\tau }_{\mathrm{\ell }}$. We then consider the piecewise constant functions ${v}_{\mathrm{\ell }}$ with ${v}_{\mathrm{\ell }}\left(t\right)={v}^{n}$ for $t\in \left({t}_{n-1},{t}_{n}\right]$ ($n=1,2,\mathrm{\dots },{N}_{\mathrm{\ell }}$) and ${v}_{\mathrm{\ell }}\left(0\right)={v}^{1}$. Moreover, let ${\stackrel{^}{v}}_{\mathrm{\ell }}$ be the piecewise affine-linear interpolation of the points $\left({t}_{n},{v}^{n}\right)$ ($n=0,1,\mathrm{\dots },{N}_{\mathrm{\ell }}$) and let ${f}_{\mathrm{\ell }}$ be the piecewise constant function with ${f}_{\mathrm{\ell }}\left(t\right)={f}^{n}$ for $t\in \left({t}_{n-1},{t}_{n}\right]$ ($n=1,2,\mathrm{\dots },{N}_{\mathrm{\ell }}$) and ${f}_{\mathrm{\ell }}\left(0\right)={f}^{1}$.

Regarding the integral operator K, we define for any integrable function w the piecewise constant function ${K}_{\mathrm{\ell }}w$ by means of

with $\left({K}_{\mathrm{\ell }}w\right)\left(0\right):={u}_{\mathrm{\ell }}^{0}$. As an immediate consequence, with (4.1b) and (4.1c) we obtain

(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 ${u}_{\mathrm{0}}\mathrm{\in }{V}_{B}$, ${v}_{\mathrm{0}}\mathrm{\in }H$, and $f\mathrm{\in }{L}^{{p}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{+}{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$. Then there exists a solution $v\mathrm{\in }{L}^{p}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}\mathrm{\right)}\mathrm{\cap }{\mathcal{C}}_{w}\mathit{}\mathrm{\left(}\mathrm{\left[}\mathrm{0}\mathrm{,}T\mathrm{\right]}\mathrm{;}H\mathrm{\right)}$ to problem (1.1) with $K\mathit{}v\mathrm{\in }{\mathcal{C}}_{w}\mathit{}\mathrm{\left(}\mathrm{\left[}\mathrm{0}\mathrm{,}T\mathrm{\right]}\mathrm{;}{V}_{B}\mathrm{\right)}$ such that (1.1a) holds in the sense of ${L}^{{p}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{+}{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$.

By passing to a subsequence if necessary, both the piecewise constant prolongation ${v}_{\mathrm{\ell }}$ and the piecewise affine-linear prolongation ${\stackrel{\mathrm{^}}{v}}_{\mathrm{\ell }}$ of the approximate solutions to (4.1) converge weakly* in ${L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$ to v as $\mathrm{\ell }\mathrm{\to }\mathrm{\infty }$. Furthermore, again by passing to a subsequence if necessary and as $\mathrm{\ell }\mathrm{\to }\mathrm{\infty }$, the piecewise constant prolongation ${v}_{\mathrm{\ell }}$ converges weakly to v in ${L}^{p}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}\mathrm{\right)}$ and the approximation ${K}_{\mathrm{\ell }}\mathit{}{v}_{\mathrm{\ell }}$ of the memory term converges weakly* to Kv in ${L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}\mathrm{\right)}$. The approximation ${\stackrel{\mathrm{^}}{v}}_{\mathrm{\ell }}^{\mathrm{\prime }}$ of the time derivative converges to

${v}^{\prime }\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)+{L}^{\mathrm{\infty }}\left(0,T;{V}_{B}^{\prime }\right)$

in the sense that $\mathrm{〈}{\stackrel{\mathrm{^}}{v}}_{\mathrm{\ell }}^{\mathrm{\prime }}\mathrm{,}w\mathrm{〉}\mathrm{\to }\mathrm{〈}{v}^{\mathrm{\prime }}\mathrm{,}w\mathrm{〉}$ for all $w\mathrm{\in }{L}^{p}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}\mathrm{\right)}$.1

Note that if $\left\{{\tau }_{\mathrm{\ell }}{\parallel {v}_{\mathrm{\ell }}^{0}\parallel }_{{V}_{A}}^{p}\right\}$ is bounded, then also ${\stackrel{^}{v}}_{\mathrm{\ell }}$ converges weakly in ${L}^{p}\left(0,T;{V}_{A}\right)$ to v as $\mathrm{\ell }\to \mathrm{\infty }$ (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 ${u}_{\mathrm{0}}\mathrm{\in }{V}_{B}$ and $w\mathrm{\in }{L}^{p}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$ such that

$Kw\in {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }{w}^{\prime }+BKw\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right).$

Then for all $t\mathrm{\in }\mathrm{\left[}\mathrm{0}\mathrm{,}T\mathrm{\right]}$ there holds

${\int }_{0}^{t}〈{w}^{\prime }\left(s\right)+\left(BKw\right)\left(s\right),w\left(s\right)〉ds$$=\frac{1}{2}{\parallel w\left(t\right)\parallel }_{H}^{2}-\frac{1}{2}{\parallel w\left(0\right)\parallel }_{H}^{2}+\frac{1}{2\lambda }{\parallel \left(Kw\right)\left(t\right)\parallel }_{B}^{2}-\frac{1}{2\lambda }{\parallel {u}_{0}\parallel }_{B}^{2}-{\int }_{0}^{t}〈\left(BKw\right)\left(s\right),{u}_{0}〉ds+{\int }_{0}^{t}{\parallel \left(Kw\right)\left(s\right)\parallel }_{B}^{2}ds.$(4.9)

Proof.

The difficulty in proving (4.9) is that neither

${w}^{\prime }\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)\mathit{ }\text{nor}\mathit{ }BKw\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$

can be assumed but only the sum ${w}^{\prime }+BKw$ is known to be in ${L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$. 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 ${w}^{\prime }+BKw\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$ and $BKw\in {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}^{\prime }\right)$, it is easy to show that

${w}^{\prime }\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;{V}_{B}^{\prime }\right)\subset {L}^{1}\left(0,T;{V}_{A}^{\prime }+{V}_{B}^{\prime }\right),$

and thus $w\in \mathcal{𝒜}\mathcal{𝒞}\left(\left[0,T\right];{V}_{A}^{\prime }+{V}_{B}^{\prime }\right)$. In view of [32, Chapter 3, Lemma 8.1], we find $w\in {\mathcal{𝒞}}_{w}\left(\left[0,T\right];H\right)$. Because of (3.1), we also see that ${\left(Kw\right)}^{\prime }\in {L}^{1}\left(0,T;{V}_{A}+{V}_{B}\right)$, and thus $Kw\in \mathcal{𝒜}\mathcal{𝒞}\left(\left[0,T\right];{V}_{A}+{V}_{B}\right)$, which implies $Kw\in {\mathcal{𝒞}}_{w}\left(\left[0,T\right];{V}_{B}\right)$. Because of (3.1), we also have ${\left(Kw\right)}^{\prime }\in {\mathcal{𝒞}}_{w}\left(\left[0,T\right];H\right)$. This justifies evaluating ${\parallel w\left(t\right)\parallel }_{H}$, ${\parallel Kw\left(t\right)\parallel }_{B}$, and ${\parallel {\left(Kw\right)}^{\prime }\left(t\right)\parallel }_{H}$ for fixed t.

Further, we remark that $Kw-{u}_{0}\in {L}^{\mathrm{\infty }}\left(0,T;V\right)$ since $w\in {L}^{p}\left(0,T;{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(0,T;H\right)$. Due to (3.1), we thus have ${\left(Kw\right)}^{\prime }\in {L}^{p}\left(0,T;{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(0,T;H\right)$. Differentiating (3.1) yields

${\left(Kw\right)}^{\prime \prime }\left(t\right)=\lambda \left({w}^{\prime }\left(t\right)-{\left(Kw\right)}^{\prime }\left(t\right)\right)$

for almost all $t\in \left(0,T\right)$. Therefore, we have

$〈{w}^{\prime }+BKw,w〉=〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+{\left(Kw\right)}^{\prime }+BKw,\frac{1}{\lambda }{\left(Kw\right)}^{\prime }+Kw-{u}_{0}〉.$

Since $\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+{\left(Kw\right)}^{\prime }+BKw\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$ and $Kw-{u}_{0}\in {L}^{\mathrm{\infty }}\left(0,T;V\right)$, we are allowed to split the right-hand side into

$〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+{\left(Kw\right)}^{\prime }+BKw,\frac{1}{\lambda }{\left(Kw\right)}^{\prime }〉+〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+{\left(Kw\right)}^{\prime }+BKw,Kw-{u}_{0}〉$$=〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+BKw,\frac{1}{\lambda }{\left(Kw\right)}^{\prime }〉+\frac{1}{\lambda }{\parallel {\left(Kw\right)}^{\prime }\parallel }_{{L}^{2}\left(0,T;H\right)}^{2}+〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+{\left(Kw\right)}^{\prime }+BKw,Kw-{u}_{0}〉.$(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

$〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+BKw,\frac{1}{\lambda }{\left(Kw\right)}^{\prime }〉=\frac{1}{2{\lambda }^{2}}\left({\parallel {\left(Kw\right)}^{\prime }\left(T\right)\parallel }_{H}^{2}-{\parallel {\left(Kw\right)}^{\prime }\left(0\right)\parallel }_{H}^{2}+\lambda {\parallel \left(Kw\right)\left(T\right)\parallel }_{B}^{2}-\lambda {\parallel {u}_{0}\parallel }_{B}^{2}\right).$

Since ${\left(Kw\right)}^{\prime }\in {L}^{p}\left(0,T;{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(0,T;H\right)$, $BKw\in {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}^{\prime }\right)$, and $Kw-{u}_{0}\in {L}^{\mathrm{\infty }}\left(0,T;V\right)$, the third term on the right-hand side of (4.10) yields

$〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime }+{\left(Kw\right)}^{\prime }+BKw,Kw-{u}_{0}〉$$=〈\frac{1}{\lambda }{\left(Kw\right)}^{\prime \prime },Kw-{u}_{0}〉+〈{\left(Kw\right)}^{\prime },Kw-{u}_{0}〉+〈BKw,Kw-{u}_{0}〉$$=\frac{1}{\lambda }〈{\left(Kw-{u}_{0}\right)}^{\prime }\left(T\right),\left(Kw-{u}_{0}\right)\left(T\right)〉-\frac{1}{\lambda }〈{\left(Kw-{u}_{0}\right)}^{\prime }\left(0\right),\left(Kw-{u}_{0}\right)\left(0\right)〉-\frac{1}{\lambda }{\parallel {\left(Kw-{u}_{0}\right)}^{\prime }\parallel }_{{L}^{2}\left(0,T;H\right)}^{2}$$+\frac{1}{2}{\parallel \left(Kw\right)\left(T\right)-{u}_{0}\parallel }_{H}^{2}-\frac{1}{2}{\parallel \left(Kw\right)\left(0\right)-{u}_{0}\parallel }_{H}^{2}+{\parallel Kw\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}-〈BKw,{u}_{0}〉.$

Altogether, we have

$〈{w}^{\prime }+BKw,w〉=\frac{1}{2{\lambda }^{2}}\left({\parallel {\left(Kw\right)}^{\prime }\left(T\right)\parallel }_{H}^{2}-{\parallel {\left(Kw\right)}^{\prime }\left(0\right)\parallel }_{H}^{2}+\lambda {\parallel \left(Kw\right)\left(T\right)\parallel }_{B}^{2}-\lambda {\parallel {u}_{0}\parallel }_{B}^{2}\right)$$+\frac{1}{\lambda }〈{\left(Kw-{u}_{0}\right)}^{\prime }\left(T\right),\left(Kw-{u}_{0}\right)\left(T\right)〉+\frac{1}{2}{\parallel \left(Kw\right)\left(T\right)-{u}_{0}\parallel }_{H}^{2}$$+{\parallel Kw\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}-〈BKw,{u}_{0}〉.$(4.11)

Due to (3.1), there holds

$\frac{1}{2}{\parallel w\left(T\right)\parallel }_{H}^{2}=\frac{1}{2{\lambda }^{2}}{\parallel {\left(Kw\right)}^{\prime }\left(T\right)\parallel }_{H}^{2}+\frac{1}{2}{\parallel \left(Kw\right)\left(T\right)-{u}_{0}\parallel }_{H}^{2}+\frac{1}{\lambda }〈{\left(Kw\right)}^{\prime }\left(T\right),\left(Kw\right)\left(T\right)-{u}_{0}〉$

as well as

$\frac{1}{2}{\parallel w\left(0\right)\parallel }_{H}^{2}=\frac{1}{2{\lambda }^{2}}{\parallel {\left(Kw\right)}^{\prime }\left(0\right)\parallel }_{H}^{2}.$

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

Proof of Theorem 4.2.

For simplicity, we write τ and N instead of ${\tau }_{\mathrm{\ell }}$ and ${N}_{\mathrm{\ell }}$, respectively, during this proof. Since, by assumption, ${v}_{\mathrm{\ell }}^{0}\to {v}_{0}$ in H, ${u}_{\mathrm{\ell }}^{0}\to {u}_{0}$ in ${V}_{B}$, and since ${f}_{\mathrm{\ell }}\to f$ in ${L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$, the right-hand side of the a priori estimate (4.4) stays bounded as $\mathrm{\ell }\to \mathrm{\infty }$. Since

${\parallel {v}_{\mathrm{\ell }}\parallel }_{{L}^{\mathrm{\infty }}\left(0,T;H\right)}=\underset{n=1,2,\mathrm{\dots },N}{\mathrm{max}}{\parallel {v}^{n}\parallel }_{H},{\parallel {\stackrel{^}{v}}_{\mathrm{\ell }}\parallel }_{{L}^{\mathrm{\infty }}\left(0,T;H\right)}=\underset{n=0,1,\mathrm{\dots },N}{\mathrm{max}}{\parallel {v}^{n}\parallel }_{H},$${\parallel {v}_{\mathrm{\ell }}\parallel }_{{L}^{p}\left(0,T;{V}_{A}\right)}={\left(\tau \sum _{n=1}^{N}{\parallel {v}^{n}\parallel }_{{V}_{A}}^{p}\right)}^{1/p},$

the a priori estimate (4.4) implies that the sequence $\left\{{v}_{\mathrm{\ell }}\right\}$ is bounded in ${L}^{\mathrm{\infty }}\left(0,T;H\right)\cap {L}^{p}\left(0,T;{V}_{A}\right)$, the sequence $\left\{{\stackrel{^}{v}}_{\mathrm{\ell }}\right\}$ is bounded in ${L}^{\mathrm{\infty }}\left(0,T;H\right)$, and – in view of (4.8) – the sequence $\left\{{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right\}$ is bounded in ${L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)$. Due to the growth condition on A, we also conclude that the sequence $\left\{A{v}_{\mathrm{\ell }}\right\}$ is bounded in ${L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)$.

Since ${L}^{\mathrm{\infty }}\left(0,T;H\right)$ and ${L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)$ are the duals of the separable normed spaces ${L}^{1}\left(0,T;H\right)$ and ${L}^{1}\left(0,T;{V}_{B}^{\prime }\right)$, respectively, and since ${L}^{p}\left(0,T;{V}_{A}\right)$ and ${L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)$ are reflexive Banach spaces, there exists a common subsequence, again denoted by $\mathrm{\ell }$, and elements $v\in {L}^{\mathrm{\infty }}\left(0,T;H\right)\cap {L}^{p}\left(0,T;{V}_{A}\right)$, $\stackrel{^}{v}\in {L}^{\mathrm{\infty }}\left(0,T;H\right)$, $u\in {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)$, and $a\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)$ such that, as $\mathrm{\ell }\to \mathrm{\infty }$

see, e.g., [4, Corollaire III.26, Théorème III.27]. Since $B:{V}_{B}\to {V}_{B}^{\prime }$ is linear and bounded, we also find that $B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\stackrel{*}{⇀}Bu$ in ${L}^{\mathrm{\infty }}\left(0,T;{V}_{B}^{\prime }\right)$.

A short calculation shows that

${\parallel {v}_{\mathrm{\ell }}-{\stackrel{^}{v}}_{\mathrm{\ell }}\parallel }_{{L}^{2}\left(0,T;H\right)}^{2}=\frac{\tau }{3}\sum _{j=1}^{N}{\parallel {v}^{j}-{v}^{j-1}\parallel }_{H}^{2},$

which tends to 0 as $\mathrm{\ell }\to \mathrm{\infty }$ because of the a priori estimate (4.4). Hence $v=\stackrel{^}{v}$.

Let us show that $u=Kv$, and thereby $Kv\in {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)$. We already know that ${K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\stackrel{*}{⇀}u$ in ${L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)$, and thus ${K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}⇀u$ in ${L}^{2}\left(0,T;H\right)$. The integral operator $\stackrel{^}{K}$ defined by $\stackrel{^}{K}w:=Kw-{u}_{0}$ can easily be shown to be a linear, bounded and thus weakly-weakly continuous mapping of ${L}^{2}\left(0,T;H\right)$ into itself (see also Lemma 3.1). Therefore, in view of ${v}_{\mathrm{\ell }}\stackrel{*}{⇀}v$ in ${L}^{\mathrm{\infty }}\left(0,T;H\right)$, and thus ${v}_{\mathrm{\ell }}⇀v$ in ${L}^{2}\left(0,T;H\right)$, we find that $K{v}_{\mathrm{\ell }}-Kv=\stackrel{^}{K}{v}_{\mathrm{\ell }}-\stackrel{^}{K}v⇀0$ in ${L}^{2}\left(0,T;H\right)$. Because of

$u-Kv=u-{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}+{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}-K{v}_{\mathrm{\ell }}+K{v}_{\mathrm{\ell }}-Kv,$

it thus remains to prove that ${K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}-K{v}_{\mathrm{\ell }}$ converges at least weakly in ${L}^{2}\left(0,T;H\right)$ to 0 in order to prove $u=Kv$. For $w\in {L}^{2}\left(0,T;H\right)$ and $t\in \left({t}_{n-1},{t}_{n}\right]$ ($n=1,2,\mathrm{\dots },N$), we have

$\left({K}_{\mathrm{\ell }}w\right)\left(t\right)-\left(Kw\right)\left(t\right)={u}_{\mathrm{\ell }}^{0}-{u}_{0}+{\int }_{0}^{{t}_{n}}\left(k\left({t}_{n}-s\right)-k\left(t-s\right)\right)w\left(s\right)ds+{\int }_{t}^{{t}_{n}}k\left(t-s\right)w\left(s\right)ds.$

Since $|k\left(t-s\right)|=\lambda {e}^{-\lambda \left(t-s\right)}\le \lambda {e}^{\lambda \tau }$ for $s\in \left(t,{t}_{n}\right)\subset \left({t}_{n-1},{t}_{n}\right)$ as well as

the Cauchy–Schwarz inequality yields

${\parallel {K}_{\mathrm{\ell }}w-Kw\parallel }_{{L}^{2}\left(0,T;H\right)}\le \sqrt{T}{\parallel {u}_{\mathrm{\ell }}^{0}-{u}_{0}\parallel }_{H}+c\tau {\parallel w\parallel }_{{L}^{2}\left(0,T;H\right)}.$

Since ${u}_{\mathrm{\ell }}^{0}\to {u}_{0}$ in ${V}_{B}$ and ${v}_{\mathrm{\ell }}$ is bounded in ${L}^{2}\left(0,T;H\right)$, we immediately get ${K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}-K{v}_{\mathrm{\ell }}\to 0$ in ${L}^{2}\left(0,T;H\right)$ as $\mathrm{\ell }\to \mathrm{\infty }$ (and thus $\tau \to 0$), which is the last step to prove $u=Kv\in {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)$.

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

${\stackrel{^}{v}}_{\mathrm{\ell }}^{\prime }+A{v}_{\mathrm{\ell }}+B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}={f}_{\mathrm{\ell }}.$(4.12)

For any $w\in V$ and $\varphi \in {\mathcal{𝒞}}_{c}^{\mathrm{\infty }}\left(0,T\right)$, we thus obtain

$-{\int }_{0}^{T}〈{\stackrel{^}{v}}_{\mathrm{\ell }}\left(t\right),w〉{\varphi }^{\prime }\left(t\right)dt={\int }_{0}^{T}〈{f}_{\mathrm{\ell }}\left(t\right)-A{v}_{\mathrm{\ell }}\left(t\right)-\left(B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right),w〉\varphi \left(t\right)dt.$

Since ${f}_{\mathrm{\ell }}\to f$ in ${L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$, and employing the (weak and weak*) convergence just shown, we can pass to the limit in the foregoing equation to come up with

$-{\int }_{0}^{T}〈v\left(t\right),w〉{\varphi }^{\prime }\left(t\right)dt={\int }_{0}^{T}〈f\left(t\right)-a\left(t\right)-\left(BKv\right)\left(t\right),w〉\varphi \left(t\right)dt.$

This shows that v is differentiable in the weak sense with

${v}^{\prime }=f-a-BKv\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{\mathrm{\infty }}\left(0,T;{V}_{B}^{\prime }\right)+{L}^{1}\left(0,T;H\right).$(4.13)

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

$v\in {L}^{p}\left(0,T;{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(0,T;H\right)\mathit{ }\text{and}\mathit{ }{v}^{\prime }\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{\mathrm{\infty }}\left(0,T;{V}_{B}^{\prime }\right)+{L}^{1}\left(0,T;H\right)\subset {L}^{1}\left(0,T;{V}^{\prime }\right),$

we conclude that $v\in \mathcal{𝒜}\mathcal{𝒞}\left(\left[0,T\right];{V}^{\prime }\right)$. This implies $v\in {\mathcal{𝒞}}_{w}\left(\left[0,T\right];H\right)$ (see [32, Chapter 3, Lemma 8.1]). Likewise, from

$Kv\in {L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)\mathit{ }\text{and}\mathit{ }{\left(Kv\right)}^{\prime }=\lambda v-\lambda Kv+\lambda {u}_{0}\in {L}^{p}\left(0,T;{V}_{A}\right)+{L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)\subset {L}^{1}\left(0,T;{V}_{A}+{V}_{B}\right)$

(see (3.1)), we deduce that $Kv\in \mathcal{𝒜}\mathcal{𝒞}\left(\left[0,T\right];{V}_{A}+{V}_{B}\right)$, and thus $Kv\in {\mathcal{𝒞}}_{w}\left(\left[0,T\right];{V}_{B}\right)$. We remark that, in addition, one can easily show that $Kv-{u}_{0}\in {W}^{1,p}\left(0,T;{V}_{A}\right)$.

By construction, we have ${\stackrel{^}{v}}_{\mathrm{\ell }}\left(0\right)={v}_{\mathrm{\ell }}^{0}\to {v}_{0}$ in H and ${\stackrel{^}{v}}_{\mathrm{\ell }}\left(T\right)={v}^{N}$. In view of the a priori estimate (4.4), the sequence $\left\{{\stackrel{^}{v}}_{\mathrm{\ell }}\left(T\right)\right\}$ is bounded in H, and hence possesses a subsequence, again denoted by $\mathrm{\ell }$, that converges weakly in H to an element ${v}_{T}\in H$. We shall show that ${v}_{T}=v\left(T\right)$.

For any $\varphi \in {\mathcal{𝒞}}^{1}\left(\left[0,T\right]\right)$, $w\in V$, we test (4.13) with the function $t↦\varphi \left(t\right)w\in {\mathcal{𝒞}}^{1}\left(\left[0,T\right];V\right)$, which allows us to integrate by parts. Together with (4.12), we find

$\left(v\left(T\right),w\right)\varphi \left(T\right)-\left(v\left(0\right),w\right)\varphi \left(0\right)={\int }_{0}^{T}〈{v}^{\prime }\left(t\right),w〉\varphi \left(t\right)dt+{\int }_{0}^{T}〈v\left(t\right),w〉{\varphi }^{\prime }\left(t\right)dt$$={\int }_{0}^{T}〈f\left(t\right)-{f}_{\mathrm{\ell }}\left(t\right)+A{v}_{\mathrm{\ell }}\left(t\right)-a\left(t\right)+\left(B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right)-\left(BKv\right)\left(t\right),w〉\varphi \left(t\right)dt$$+{\int }_{0}^{T}〈v\left(t\right),w〉{\varphi }^{\prime }\left(t\right)dt+{\int }_{0}^{T}〈{\stackrel{^}{v}}_{\mathrm{\ell }}^{\prime }\left(t\right),w〉\varphi \left(t\right)dt$$={\int }_{0}^{T}〈f\left(t\right)-{f}_{\mathrm{\ell }}\left(t\right)+A{v}_{\mathrm{\ell }}\left(t\right)-a\left(t\right)+\left(B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right)-\left(BKv\right)\left(t\right),w〉\varphi \left(t\right)dt$$+{\int }_{0}^{T}〈v\left(t\right)-{\stackrel{^}{v}}_{\mathrm{\ell }}\left(t\right),w〉{\varphi }^{\prime }\left(t\right)dt+\left({\stackrel{^}{v}}_{\mathrm{\ell }}\left(T\right),w\right)\varphi \left(T\right)-\left({\stackrel{^}{v}}_{\mathrm{\ell }}\left(0\right),w\right)\varphi \left(0\right).$(4.14)

The first two terms on the right-hand side tend to zero as $\mathrm{\ell }\to \mathrm{\infty }$. This yields

Choosing ϕ such that $\varphi \left(T\right)=0$ and $\varphi \left(0\right)=0$, respectively, shows that $v\left(0\right)={v}_{0}$ and $v\left(T\right)={v}_{T}$, respectively.

By definition, we have $\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(0\right)={u}_{\mathrm{\ell }}^{0}\to {u}_{0}$ in ${V}_{B}$ and $\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(T\right)={\left({K}_{{u}_{\mathrm{\ell }}^{0}}^{\tau }v\right)}^{N}$. Again, the a priori estimate yields the boundedness of $\left\{\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(T\right)\right\}$ in ${V}_{B}$, and thus the existence of a weakly convergent subsequence, again denoted by $\mathrm{\ell }$, and an element ${u}_{T}\in {V}_{B}$ such that $\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(T\right)⇀{u}_{T}$ in ${V}_{B}$.

We shall show that ${u}_{T}=\left(Kv\right)\left(T\right)$. Analogously to (4.14), choosing $\varphi \left(t\right)=\frac{t}{T}$, $t\in \left[0,T\right]$, we have for any $w\in V$,

$\left(\left(Kv\right)\left(T\right),w\right)={\int }_{0}^{T}〈{\left(Kv\right)}^{\prime }\left(t\right),w〉\frac{t}{T}dt+{\int }_{0}^{T}〈\left(Kv\right)\left(t\right),w〉\frac{1}{T}dt.$

Using (3.1), we obtain

$\left(\left(Kv\right)\left(T\right),w\right)=\lambda {\int }_{0}^{T}〈v\left(t\right)-\left(\left(Kv\right)\left(t\right)-{u}_{0}\right),w〉\frac{t}{T}dt+{\int }_{0}^{T}〈\left(Kv\right)\left(t\right),w〉\frac{1}{T}dt$$=\underset{\mathrm{\ell }\to \mathrm{\infty }}{lim}\left(\lambda {\int }_{0}^{T}〈{v}_{\mathrm{\ell }}\left(t\right)-\left(\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right)-{u}_{\mathrm{\ell }}^{0}\right),w〉\frac{t}{T}dt+{\int }_{0}^{T}〈\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right),w〉\frac{1}{T}dt\right).$(4.15)

For the first integral, (4.6) yields

${\int }_{0}^{T}〈{v}_{\mathrm{\ell }}\left(t\right)-\left(\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right)-{u}_{\mathrm{\ell }}^{0}\right),w〉\frac{t}{T}dt=\sum _{j=1}^{N}{\int }_{{t}_{j-1}}^{{t}_{j}}〈{v}^{j}-\left({\left({K}_{{u}_{\mathrm{\ell }}^{0}}^{\tau }v\right)}^{j}-{u}_{\mathrm{\ell }}^{0}\right),w〉\frac{t}{T}dt$$=\sum _{j=1}^{N}{\int }_{{t}_{j-1}}^{{t}_{j}}\frac{1}{{e}^{\lambda \tau }-1}〈{\left({K}_{{u}_{\mathrm{\ell }}^{0}}^{\tau }v\right)}^{j}-{\left({K}_{{u}_{\mathrm{\ell }}^{0}}^{\tau }v\right)}^{j-1},w〉\frac{t}{T}dt.$

For better readability, we write ${K}^{\tau }$ instead of ${K}_{{u}_{\mathrm{\ell }}^{0}}^{\tau }$ for the rest of this proof. Splitting up the sum and shifting the index in the second sum yields

$\sum _{j=1}^{N}{\int }_{{t}_{j-1}}^{{t}_{j}}〈{\left({K}^{\tau }v\right)}^{j}-{\left({K}^{\tau }v\right)}^{j-1},w〉\frac{t}{T}dt$$=〈{\left({K}^{\tau }v\right)}^{N},w〉{\int }_{{t}_{N-1}}^{{t}_{N}}\frac{t}{T}dt-〈{u}_{\mathrm{\ell }}^{0},w〉{\int }_{0}^{\tau }\frac{t}{T}dt+\sum _{j=1}^{N-1}〈{\left({K}^{\tau }v\right)}^{j},w〉\left({\int }_{{t}_{j-1}}^{{t}_{j}}\frac{t}{T}dt-{\int }_{{t}_{j}}^{{t}_{j+1}}\frac{t}{T}dt\right)$$=\frac{\left(2T-\tau \right)\tau }{2T}〈{\left({K}^{\tau }v\right)}^{N},w〉-\frac{{\tau }^{2}}{2T}〈{u}_{\mathrm{\ell }}^{0},w〉-\frac{{\tau }^{2}}{T}\sum _{j=1}^{N}〈{\left({K}^{\tau }v\right)}^{j},w〉+\frac{{\tau }^{2}}{T}〈{\left({K}^{\tau }v\right)}^{N},w〉$$=\left(\frac{\left(2T-\tau \right)\tau }{2T}+\frac{{\tau }^{2}}{T}\right)〈\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(T\right),w〉-\frac{{\tau }^{2}}{2T}〈{u}_{\mathrm{\ell }}^{0},w〉-\frac{\tau }{T}{\int }_{0}^{T}〈\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right),w〉dt.$

Inserting this into (4.15) and using that $\left\{{u}_{\mathrm{\ell }}^{0}\right\}$ is bounded in ${V}_{B}$, $\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(T\right)⇀{u}_{T}$ in ${V}_{B}$, ${K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\stackrel{*}{⇀}Kv$ in ${L}^{\mathrm{\infty }}\left(0,T;{V}_{B}\right)$ as well as $\lambda \tau {\left({e}^{\lambda \tau }-1\right)}^{-1}\to 1$ as $\mathrm{\ell }\to \mathrm{\infty }$ (and thus $\tau \to 0$), we end up with

$\left(\left(Kv\right)\left(T\right),w\right)=\underset{\mathrm{\ell }\to \mathrm{\infty }}{lim}\left(\frac{\lambda \tau }{{e}^{\lambda \tau }-1}\left(\left(\frac{2T-\tau }{2T}+\frac{\tau }{T}\right)〈\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(T\right),w〉-\frac{\tau }{2T}〈{u}_{\mathrm{\ell }}^{0},w〉-\frac{1}{T}{\int }_{0}^{T}〈\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right),w〉\mathrm{d}t\right)$$+{\int }_{0}^{T}〈\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right),w〉\frac{1}{T}\mathrm{d}t\right)$$=〈{u}_{T},w〉,$

which proves ${u}_{T}=\left(Kv\right)\left(T\right)$.

It remains to show that a equals Av. We recall that ${v}_{\mathrm{\ell }}\in {L}^{\mathrm{\infty }}\left(0,T;V\right)$. Hence, for arbitrary $w\in {L}^{p}\left(0,T;{V}_{A}\right)$ the monotonicity of A implies

$〈A{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉=〈A{v}_{\mathrm{\ell }}-Aw,{v}_{\mathrm{\ell }}-w〉+〈Aw,{v}_{\mathrm{\ell }}-w〉+〈A{v}_{\mathrm{\ell }},w〉\ge 〈Aw,{v}_{\mathrm{\ell }}-w〉+〈A{v}_{\mathrm{\ell }},w〉.$

The right-hand side converges to

$〈Aw,v-w〉+〈a,w〉.$

$\underset{\mathrm{\ell }\to \mathrm{\infty }}{lim inf}〈A{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉\ge 〈Aw,v-w〉+〈a,w〉.$

Below we will show

$\underset{\mathrm{\ell }\to \mathrm{\infty }}{lim sup}〈A{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉\le 〈a,v〉,$(4.16)

hence altogether proving

$〈Aw-a,v-w〉\le 0.$

Taking $w=v±rz$ for any $z\in {L}^{p}\left(0,T;{V}_{A}\right)$, $r>0$, passing to the limit $r\to 0$, and Lebesgue’s theorem on dominated convergence together with the hemicontinuity and the growth condition of $A:{V}_{A}\to {V}_{A}^{\prime }$ yield

which proves the assertion.

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

$〈A{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉=〈{f}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉-〈{\stackrel{^}{v}}_{\mathrm{\ell }}^{\prime },{v}_{\mathrm{\ell }}〉-〈B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉.$(4.17)

Since ${f}_{\mathrm{\ell }}\to f$ in ${L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$, the first term on the right-hand side tends to $〈f,v〉$ as $\mathrm{\ell }\to \mathrm{\infty }$. Using (4.5), we estimate the second term as

$〈{\stackrel{^}{v}}_{\mathrm{\ell }}^{\prime },{v}_{\mathrm{\ell }}〉=\sum _{j=1}^{N}\left({v}^{j}-{v}^{j-1},{v}^{j}\right)\ge \frac{1}{2}{\parallel {v}^{N}\parallel }_{H}^{2}-\frac{1}{2}{\parallel {v}_{\mathrm{\ell }}^{0}\parallel }_{H}^{2}.$

As ${v}_{\mathrm{\ell }}^{0}\to {v}_{0}=v\left(0\right)$ and ${v}^{N}⇀{v}_{T}=v\left(T\right)$ in H, the weak lower semicontinuity of the norm provides

$\underset{\mathrm{\ell }\to \mathrm{\infty }}{lim inf}〈{\stackrel{^}{v}}_{\mathrm{\ell }}^{\prime },{v}_{\mathrm{\ell }}〉\ge \frac{1}{2}{\parallel v\left(T\right)\parallel }_{H}^{2}-\frac{1}{2}{\parallel v\left(0\right)\parallel }_{H}^{2}.$

Regarding the last term on the right-hand side of equation (4.17), note that only ${v}_{\mathrm{\ell }}⇀v$ in ${L}^{p}\left(0,T;{V}_{A}\right)$ and $B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}⇀BKv$ in ${L}^{2}\left(0,T;{V}_{B}^{\prime }\right)$. Therefore, a finer examination of that term is necessary. To start with, a short calculation (see (4.6)) shows that

${v}^{n}=\frac{{e}^{-\lambda \tau }}{\tau {\gamma }_{1}}\left({\left({K}^{\tau }v\right)}^{n}-{\left({K}^{\tau }v\right)}^{n-1}\right)+\frac{1-{e}^{-\lambda \tau }}{\tau {\gamma }_{1}}{\left({K}^{\tau }v\right)}^{n}-\frac{1-{e}^{-\lambda \tau }}{\tau {\gamma }_{1}}{u}_{\mathrm{\ell }}^{0}.$

We conclude with (4.5) that

$〈B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉=\tau \sum _{j=1}^{N}〈{\left(B{K}^{\tau }v\right)}^{j},{v}^{j}〉$$=\frac{{e}^{-\lambda \tau }}{{\gamma }_{1}}\sum _{j=1}^{N}〈{\left(B{K}^{\tau }v\right)}^{j},{\left({K}^{\tau }v\right)}^{j}-{\left({K}^{\tau }v\right)}^{j-1}〉+\frac{1-{e}^{-\lambda \tau }}{{\gamma }_{1}}\sum _{j=1}^{N}〈{\left(B{K}^{\tau }v\right)}^{j},{\left({K}^{\tau }v\right)}^{j}〉$$-\frac{1-{e}^{-\lambda \tau }}{{\gamma }_{1}}\sum _{j=1}^{N}〈{\left(B{K}^{\tau }v\right)}^{j},{u}_{\mathrm{\ell }}^{0}〉$$\ge \frac{{e}^{-\lambda \tau }}{2{\gamma }_{1}}\left({\parallel {\left({K}^{\tau }v\right)}^{N}\parallel }_{B}^{2}-{\parallel {\left({K}^{\tau }v\right)}^{0}\parallel }_{B}^{2}\right)+\frac{1-{e}^{-\lambda \tau }}{\tau {\gamma }_{1}}{\parallel {K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}-\frac{1-{e}^{-\lambda \tau }}{\tau {\gamma }_{1}}{\int }_{0}^{T}〈\left(B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(t\right),{u}_{\mathrm{\ell }}^{0}〉dt.$

Since ${\left({K}^{\tau }v\right)}^{0}={u}_{\mathrm{\ell }}^{0}\to {u}_{0}=\left(Kv\right)\left(0\right)$ in ${V}_{B}$ and ${\left({K}^{\tau }v\right)}^{N}=\left({K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}\right)\left(T\right)⇀{u}_{T}=\left(Kv\right)\left(T\right)$ in ${V}_{B}$ as well as ${K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }}⇀Kv$ in ${L}^{2}\left(0,T;{V}_{B}\right)$, the weak lower semicontinuity of the norms involved shows, by passing to the limit and employing ${\gamma }_{1}\to \lambda$ (see (4.2)),

$\underset{\mathrm{\ell }\to \mathrm{\infty }}{lim inf}〈B{K}_{\mathrm{\ell }}{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉\ge \frac{1}{2\lambda }{\parallel \left(Kv\right)\left(T\right)\parallel }_{B}^{2}-\frac{1}{2\lambda }{\parallel \left(Kv\right)\left(0\right)\parallel }_{B}^{2}+{\parallel Kv\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}-{\int }_{0}^{T}〈\left(BKv\right)\left(t\right),{u}_{0}〉dt.$

Altogether, we find

$\underset{\mathrm{\ell }\to \mathrm{\infty }}{lim sup}〈A{v}_{\mathrm{\ell }},{v}_{\mathrm{\ell }}〉\le 〈f,v〉-\frac{1}{2}{\parallel v\left(T\right)\parallel }_{H}^{2}+\frac{1}{2}{\parallel v\left(0\right)\parallel }_{H}^{2}-\frac{1}{2\lambda }{\parallel \left(Kv\right)\left(T\right)\parallel }_{B}^{2}+\frac{1}{2\lambda }{\parallel \left(Kv\right)\left(0\right)\parallel }_{B}^{2}$$-{\parallel Kv\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}+{\int }_{0}^{T}〈\left(BKv\right)\left(t\right),{u}_{0}〉dt.$

We replace f by ${v}^{\prime }+a+BKv$ and remind that $v\in {\mathcal{𝒞}}_{w}\left(\left[0,T\right];H\right)$ as well as $Kv\in {\mathcal{𝒞}}_{w}\left(\left[0,T\right];{V}_{B}\right)$. 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 ${u}_{\mathrm{0}}\mathrm{,}{\stackrel{\mathrm{^}}{u}}_{\mathrm{0}}\mathrm{\in }{V}_{B}$, ${v}_{\mathrm{0}}\mathrm{,}{\stackrel{\mathrm{^}}{v}}_{\mathrm{0}}\mathrm{\in }H$, and $f\mathrm{,}\stackrel{\mathrm{^}}{f}\mathrm{\in }{L}^{{p}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{+}{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$.

• (i)

Let $f-\stackrel{^}{f}\in {L}^{1}\left(0,T;H\right)$ and let v and $\stackrel{^}{v}$ be solutions to ( 1.1 ) with data $\left({u}_{0},{v}_{0},f\right)$ and $\left({\stackrel{^}{u}}_{0},{\stackrel{^}{v}}_{0},\stackrel{^}{f}\right)$ , respectively. Then the stability estimate

${\parallel v\left(t\right)-\stackrel{^}{v}\left(t\right)\parallel }_{H}^{2}+\frac{1}{\lambda }{\parallel \left({K}_{{u}_{0}}v\right)\left(t\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(t\right)\parallel }_{B}^{2}+{\int }_{0}^{t}{\parallel \left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\parallel }_{B}^{2}ds$$\le c\left({\parallel {v}_{0}-{\stackrel{^}{v}}_{0}\parallel }_{H}^{2}+{\parallel {u}_{0}-{\stackrel{^}{u}}_{0}\parallel }_{{V}_{B}}^{2}+{\parallel f-\stackrel{^}{f}\parallel }_{{L}^{1}\left(0,T;H\right)}^{2}\right)$

holds for all $t\in \left[0,T\right]$ , where the index in the notations ${K}_{{u}_{0}}$ and ${K}_{{\stackrel{^}{u}}_{0}}$ denotes the dependence of the integral operator on ${u}_{0}$ and ${\stackrel{^}{u}}_{0}$ , respectively.

• (ii)

Let $f-\stackrel{^}{f}\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)$ . Assume in addition that A is uniformly monotone in the sense that there is $\mu >0$ such that

$〈Av-Aw,v-w〉\ge \mu {\parallel v-w\parallel }_{{V}_{A}}^{p}$

for all $v,w\in {V}_{A}$ . Let v and $\stackrel{^}{v}$ be solutions to ( 1.1 ) with data $\left({u}_{0},{v}_{0},f\right)$ and $\left({\stackrel{^}{u}}_{0},{\stackrel{^}{v}}_{0},\stackrel{^}{f}\right)$ , respectively. Then the stability estimate

${\parallel v\left(t\right)-\stackrel{^}{v}\left(t\right)\parallel }_{H}^{2}+\mu {\int }_{0}^{t}{\parallel v\left(s\right)-\stackrel{^}{v}\left(s\right)\parallel }_{{V}_{A}}^{p}𝑑s+\frac{1}{\lambda }{\parallel \left({K}_{{u}_{0}}v\right)\left(t\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(t\right)\parallel }_{B}^{2}+{\int }_{0}^{t}{\parallel \left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\parallel }_{B}^{2}ds$$\le c\left({\parallel {v}_{0}-{\stackrel{^}{v}}_{0}\parallel }_{H}^{2}+{\parallel {u}_{0}-{\stackrel{^}{u}}_{0}\parallel }_{{V}_{B}}^{2}+{\parallel f-\stackrel{^}{f}\parallel }_{{L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)}^{2}\right)$

holds for all $t\in \left[0,T\right]$.

Proof.

Let v and $\stackrel{^}{v}$ be solutions for $\left({u}_{0},{v}_{0},f\right)$ and $\left({\stackrel{^}{u}}_{0},{\stackrel{^}{v}}_{0},\stackrel{^}{f}\right)$, respectively. The difference of both equations (1.1a) then reads

${\left(v-\stackrel{^}{v}\right)}^{\prime }+Av-A\stackrel{^}{v}+B\left({K}_{{u}_{0}}v-{K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)=f-\stackrel{^}{f}.$(5.1)

Lemma 4.3 implies for all $t\in \left[0,T\right]$,

${\int }_{0}^{t}〈{\left(v-\stackrel{^}{v}\right)}^{\prime }\left(s\right)+B\left(\left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\right),\left(v-\stackrel{^}{v}\right)\left(s\right)〉ds$$=\frac{1}{2}{\parallel v\left(t\right)-\stackrel{^}{v}\left(t\right)\parallel }_{H}^{2}-\frac{1}{2}{\parallel {v}_{0}-{\stackrel{^}{v}}_{0}\parallel }_{H}^{2}+\frac{1}{2\lambda }{\parallel \left({K}_{{u}_{0}}v\right)\left(t\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(t\right)\parallel }_{B}^{2}$$-\frac{1}{2\lambda }{\parallel {u}_{0}-{\stackrel{^}{u}}_{0}\parallel }_{B}^{2}-{\int }_{0}^{t}〈B\left(\left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\right),{u}_{0}-{\stackrel{^}{u}}_{0}〉ds+{\int }_{0}^{t}{\parallel \left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\parallel }_{B}^{2}ds.$

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

$\frac{1}{2}{\parallel v\left(t\right)-\stackrel{^}{v}\left(t\right)\parallel }_{H}^{2}+\frac{1}{2\lambda }{\parallel \left({K}_{{u}_{0}}v\right)\left(t\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(t\right)\parallel }_{B}^{2}+{\int }_{0}^{t}{\parallel \left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\parallel }_{B}^{2}ds$$\le \frac{1}{2}{\parallel {v}_{0}-{\stackrel{^}{v}}_{0}\parallel }_{H}^{2}+\frac{1}{2\lambda }{\parallel {u}_{0}-{\stackrel{^}{u}}_{0}\parallel }_{B}^{2}+{\int }_{0}^{t}〈B\left(\left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\right),{u}_{0}-{\stackrel{^}{u}}_{0}〉ds+{\int }_{0}^{t}〈\left(f-\stackrel{^}{f}\right)\left(s\right),\left(v-\stackrel{^}{v}\right)\left(s\right)〉ds$$\le c\left({\parallel {v}_{0}-{\stackrel{^}{v}}_{0}\parallel }_{H}^{2}+{\parallel {u}_{0}-{\stackrel{^}{u}}_{0}\parallel }_{B}^{2}+{\parallel f-\stackrel{^}{f}\parallel }_{{L}^{1}\left(0,T;H\right)}^{2}\right)+\frac{1}{2}{\int }_{0}^{t}{\parallel \left({K}_{{u}_{0}}v\right)\left(s\right)-\left({K}_{{\stackrel{^}{u}}_{0}}\stackrel{^}{v}\right)\left(s\right)\parallel }_{B}^{2}ds+\frac{1}{4}{\parallel v-\stackrel{^}{v}\parallel }_{{L}^{\mathrm{\infty }}\left(0,T;H\right)}^{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

${\int }_{0}^{t}〈\left(f-\stackrel{^}{f}\right)\left(s\right),\left(v-\stackrel{^}{v}\right)\left(s\right)〉ds\le {\parallel f-\stackrel{^}{f}\parallel }_{{L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)}{\int }_{0}^{t}{\parallel \left(v-\stackrel{^}{v}\right)\left(s\right)\parallel }_{{V}_{A}}^{p}ds,$

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 $\mathrm{\left\{}{v}_{\mathrm{\ell }}\mathrm{\right\}}$ and $\mathrm{\left\{}{\stackrel{\mathrm{^}}{v}}_{\mathrm{\ell }}\mathrm{\right\}}$ 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 ${u}_{0}={\stackrel{^}{u}}_{0}\in {V}_{B}$, ${v}_{0}={\stackrel{^}{v}}_{0}\in H$ and $f=\stackrel{^}{f}\in {L}^{{p}^{\prime }}\left(0,T;{V}_{A}^{\prime }\right)+{L}^{1}\left(0,T;H\right)$ with corresponding solutions v and $\stackrel{^}{v}$. As $f-\stackrel{^}{f}=0\in {L}^{1}\left(0,T;H\right)$, the first assumption of Theorem 5.1 is fulfilled, which shows

${\parallel v\left(t\right)-\stackrel{^}{v}\left(t\right)\parallel }_{H}^{2}+\frac{1}{\lambda }{\parallel \left(Kv\right)\left(t\right)-\left(K\stackrel{^}{v}\right)\left(t\right)\parallel }_{B}^{2}+{\int }_{0}^{t}{\parallel \left(Kv\right)\left(s\right)-\left(K\stackrel{^}{v}\right)\left(s\right)\parallel }_{B}^{2}ds\le 0$

for all $t\in \left[0,T\right]$. 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 $\frac{1}{\lambda }$, 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 ${v}_{\mathrm{0}}\mathrm{\in }V$. For $\lambda \mathrm{>}\mathrm{0}$, let ${K}_{\lambda }$ denote the operator related to the kernel ${k}_{\lambda }\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{=}\lambda \mathit{}{e}^{\mathrm{-}\lambda \mathit{}t}$ and let ${v}_{\lambda }\mathrm{\in }{L}^{p}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$ denote the corresponding solution to problem (1.1). Moreover, assume that ${v}_{\lambda }\mathrm{\in }{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}\mathrm{\right)}$. Then for all $\mu \mathrm{>}\mathrm{0}$ and corresponding solutions ${v}_{\mu }\mathrm{\in }{L}^{p}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$ there holds

${\parallel \left({v}_{\lambda }-{v}_{\mu }\right)\left(t\right)\parallel }_{H}^{2}+{\int }_{0}^{t}{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(s\right)\parallel }_{B}^{2}ds\le \frac{{\lambda }^{2}}{2}\left(1+{\lambda }^{2}{T}^{2}\right){|\frac{1}{\lambda }-\frac{1}{\mu }|}^{2}{\parallel {v}_{\lambda }\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}$

for almost all t.

Proof.

Taking the difference of the equations (1.1a) for ${v}_{\lambda }$ and ${v}_{\mu }$, respectively, and testing with ${v}_{\lambda }-{v}_{\mu }$ leads to

${\int }_{0}^{t}〈{\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }\left(s\right)+B\left(\left({K}_{\lambda }{v}_{\lambda }\right)\left(s\right)-\left({K}_{\mu }{v}_{\mu }\right)\left(s\right)\right),\left({v}_{\lambda }-{v}_{\mu }\right)\left(s\right)〉ds$$=-{\int }_{0}^{t}〈A{v}_{\lambda }\left(s\right)-A{v}_{\mu }\left(s\right),\left({v}_{\lambda }-{v}_{\mu }\right)\left(s\right)〉ds\le 0.$(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 $\left(0,t\right)$, we first have to extend the functions considered to the interval $\left(-\eta ,0\right)$ for a fixed $\eta >0$. We then prove the formula on an arbitrary interval $\left(\alpha ,\beta \right)$ with

$-\eta +{h}_{0}<\alpha <\beta

At the end, instead of a fixed α, we consider a sequence $\left\{{\alpha }_{k}\right\}$ with ${\alpha }_{k}<0$ and ${\alpha }_{k}\to 0$ as $k\to \mathrm{\infty }$. If we considered a sequence of positive ${\alpha }_{k}$, we would not be able to identify the limit properly.

We fix ${h}_{0}>0$. The centered Steklov average for a function $z\in {L}^{p}\left(-\eta ,T;X\right)$, with X being an arbitrary Banach space, is defined by

$\left({S}_{h}z\right)\left(t\right):=\frac{1}{2h}{\int }_{t-h}^{t+h}z\left(s\right)ds,$

where $0 and $t\in \left[-\eta +{h}_{0},T-{h}_{0}\right]$. It is well known (see [16, Theorem 9, p. 49]) that $\left({S}_{h}z\right)\left(t\right)\to z\left(t\right)$ in X for almost all $t\in \left(-\eta +{h}_{0},T-{h}_{0}\right)$. In addition, it is easy to show that ${S}_{h}z\in {L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};X\right)$ as well as, by using Fubini’s theorem,

${\parallel {S}_{h}z\parallel }_{{L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};X\right)}\le {\parallel z\parallel }_{{L}^{p}\left(-\eta ,T;X\right)},$

and thus ${S}_{h}z\to z$ in ${L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};X\right)$.

First, note that both ${v}_{\lambda }$ and ${v}_{\mu }$ are in ${\mathcal{𝒞}}_{w}\left(\left[0,T\right];H\right)$, and take the same value ${v}_{0}$ at $t=0$. Hence, the difference ${v}_{\lambda }-{v}_{\mu }$ fulfills the initial condition $\left({v}_{\lambda }-{v}_{\mu }\right)\left(0\right)=0$. We assume both ${v}_{\lambda }$ and ${v}_{\mu }$ to be extended by ${v}_{0}\in V$ for $t<0$, and thus the difference to be extended by 0 for $t<0$. We conclude that ${v}_{\lambda }-{v}_{\mu }\in {L}^{p}\left(-\eta ,T;{V}_{A}\right)\cap {\mathcal{𝒞}}_{w}\left(\left[-\eta ,T\right];H\right)$ and similarly ${K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\in {\mathcal{𝒞}}_{w}\left(\left[-\eta ,T\right];{V}_{B}\right)$. Relation (3.1) still holds for both ${v}_{\lambda }$ and ${v}_{\mu }$ for almost all $t\in \left(-\eta ,0\right)$.

As

${v}_{\lambda }-{v}_{\mu }\in {L}^{p}\left(-\eta ,T;{V}_{A}\right)\cap {\mathcal{𝒞}}_{w}\left(\left[-\eta ,T\right];H\right),$

it is easy to show that ${S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)$ is bounded in ${L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(-\eta +{h}_{0},T-{h}_{0};H\right)$. Therefore, there exists a subsequence, again denoted by h, such that ${S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\stackrel{*}{⇀}{v}_{\lambda }-{v}_{\mu }$ in

${L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(-\eta +{h}_{0},T-{h}_{0};H\right).$

In view of (3.1), we have

$\left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(t\right)=\frac{1}{2h}\left(\left(\frac{1}{\lambda }{K}_{\lambda }{v}_{\lambda }-\frac{1}{\mu }{K}_{\mu }{v}_{\mu }\right)\left(t+h\right)-\left(\frac{1}{\lambda }{K}_{\lambda }{v}_{\lambda }-\frac{1}{\mu }{K}_{\mu }{v}_{\mu }\right)\left(t-h\right)\right)+\left({S}_{h}{K}_{\lambda }{v}_{\lambda }\right)\left(t\right)-\left({S}_{h}{K}_{\mu }{v}_{\mu }\right)\left(t\right)$

such that ${K}_{\lambda }{v}_{\lambda },{K}_{\mu }{v}_{\mu }\in {\mathcal{𝒞}}_{w}\left(\left[-\eta ,T\right];{V}_{B}\right)$ provide ${S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\in {\mathcal{𝒞}}_{w}\left(\left[-\eta +{h}_{0},T-{h}_{0}\right];{V}_{B}\right)$. Thus, in contrast to the difference ${v}_{\lambda }-{v}_{\mu }$ itself, the Steklov average ${S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)$ is an element of

${L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};{V}_{A}\right)\cap {\mathcal{𝒞}}_{w}\left(\left[-\eta +{h}_{0},T-{h}_{0}\right];{V}_{B}\right),$

which provides the regularity needed.

For all $\alpha ,\beta$ with $-\eta +{h}_{0}<\alpha <\beta we find

${\int }_{\alpha }^{\beta }〈{\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }\left(t\right)+B\left(\left({K}_{\lambda }{v}_{\lambda }\right)\left(t\right)-\left({K}_{\mu }{v}_{\mu }\right)\left(t\right)\right),\left({v}_{\lambda }-{v}_{\mu }\right)\left(t\right)〉dt$$={\int }_{\alpha }^{\beta }〈{\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }\left(t\right)+B\left(\left({K}_{\lambda }{v}_{\lambda }\right)\left(t\right)-\left({K}_{\mu }{v}_{\mu }\right)\left(t\right)\right),\left({v}_{\lambda }-{v}_{\mu }\right)\left(t\right)-\left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(t\right)〉dt$$+{\int }_{\alpha }^{\beta }〈{\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }\left(t\right)+B\left(\left({K}_{\lambda }{v}_{\lambda }\right)\left(t\right)-\left({K}_{\mu }{v}_{\mu }\right)\left(t\right)\right)$$-\left({S}_{h}\left({\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }+B\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\right)\left(t\right),\left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(t\right)〉\mathrm{d}t$$+{\int }_{\alpha }^{\beta }〈\left({S}_{h}{\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }\right)\left(t\right)+B\left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right),\left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(t\right)〉dt.$(5.3)

The first term on the right-hand side tends to zero as $h\to 0$ since

${\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }+B\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)=-\left(A{v}_{\lambda }-A{v}_{\mu }\right)\in {L}^{{p}^{\prime }}\left(-\eta +{h}_{0},T-{h}_{0};{V}_{A}^{\prime }\right)+{L}^{1}\left(-\eta +{h}_{0},T-{h}_{0};H\right)$

and

${S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\stackrel{*}{⇀}{v}_{\lambda }-{v}_{\mu }$

in ${L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(-\eta +{h}_{0},T-{h}_{0};H\right)$ as $h\to 0$, the second one since $\left\{{S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right\}$ is bounded in ${L}^{p}\left(-\eta +{h}_{0},T-{h}_{0};{V}_{A}\right)\cap {L}^{\mathrm{\infty }}\left(-\eta +{h}_{0},T-{h}_{0};H\right)$, and

${S}_{h}\left({\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }+B\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\to {\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }+B\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)$

in ${L}^{{p}^{\prime }}\left(-\eta +{h}_{0},T-{h}_{0};{V}_{A}^{\prime }\right)+{L}^{1}\left(-\eta +{h}_{0},T-{h}_{0};H\right)$ as $h\to 0$. The latter follows from the linearity of ${S}_{h}$.

Due to the regularity properties of ${S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)$, we are allowed to split the last integrand in (5.3). We use

${v}_{\lambda }-{v}_{\mu }=\frac{1}{\lambda }{\left({K}_{\lambda }{v}_{\lambda }\right)}^{\prime }-\frac{1}{\mu }{\left({K}_{\mu }{v}_{\mu }\right)}^{\prime }+{K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }$

as well as ${S}_{h}\left({z}^{\prime }\right)={\left({S}_{h}z\right)}^{\prime }$ for any integrable function z on $\left(\alpha ,\beta \right)\subset \left(-\eta +{h}_{0},T-{h}_{0}\right)$. We rewrite the third term as

${\int }_{\alpha }^{\beta }〈\left({S}_{h}{\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }\right)\left(t\right)+B\left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right),\left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(t\right)〉dt$$={\int }_{\alpha }^{\beta }〈\left({S}_{h}{\left({v}_{\lambda }-{v}_{\mu }\right)}^{\prime }\right)\left(t\right),\left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(t\right)〉dt+{\int }_{\alpha }^{\beta }〈B\left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right),\left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right)〉dt$$+{\int }_{\alpha }^{\beta }〈B\left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right),\frac{1}{\lambda }\left({S}_{h}{\left({K}_{\lambda }{v}_{\lambda }\right)}^{\prime }\right)\left(t\right)-\frac{1}{\mu }\left({S}_{h}{\left({K}_{\mu }{v}_{\mu }\right)}^{\prime }\right)\left(t\right)〉dt$$=\frac{1}{2}{\parallel \left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(\beta \right)\parallel }_{H}^{2}-\frac{1}{2}{\parallel \left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(\alpha \right)\parallel }_{H}^{2}+{\int }_{\alpha }^{\beta }{\parallel \left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right)\parallel }_{B}^{2}dt$$+\frac{1}{\mu }{\int }_{\alpha }^{\beta }〈B\left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right),\left({S}_{h}{\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)}^{\prime }\right)\left(t\right)〉dt$$+\left(\frac{1}{\lambda }-\frac{1}{\mu }\right){\int }_{\alpha }^{\beta }〈B\left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right),\left({S}_{h}{\left({K}_{\lambda }{v}_{\lambda }\right)}^{\prime }\right)\left(t\right)〉dt$$\ge \frac{1}{2}{\parallel \left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(\beta \right)\parallel }_{H}^{2}-\frac{1}{2}{\parallel \left({S}_{h}\left({v}_{\lambda }-{v}_{\mu }\right)\right)\left(\alpha \right)\parallel }_{H}^{2}+{\int }_{\alpha }^{\beta }{\parallel \left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right)\parallel }_{B}^{2}dt$$+\frac{1}{2\mu }{\parallel \left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(\beta \right)\parallel }_{B}^{2}-\frac{1}{2\mu }{\parallel \left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(\alpha \right)\parallel }_{B}^{2}$$-|\frac{1}{\lambda }-\frac{1}{\mu }|{\int }_{\alpha }^{\beta }{\parallel \left({S}_{h}\left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\right)\left(t\right)\parallel }_{B}{\parallel \left({S}_{h}{\left({K}_{\lambda }{v}_{\lambda }\right)}^{\prime }\right)\left(t\right)\parallel }_{B}dt.$

Taking the limit $h\to \mathrm{\infty }$ and employing (5.2) results in

$\frac{1}{2}{\parallel \left({v}_{\lambda }-{v}_{\mu }\right)\left(\beta \right)\parallel }_{H}^{2}+{\int }_{\alpha }^{\beta }{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(t\right)\parallel }_{B}^{2}dt+\frac{1}{2\mu }{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(\beta \right)\parallel }_{B}^{2}$$\le \frac{1}{2}{\parallel \left({v}_{\lambda }-{v}_{\mu }\right)\left(\alpha \right)\parallel }_{H}^{2}+\frac{1}{2\mu }{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(\alpha \right)\parallel }_{B}^{2}$$+|\frac{1}{\lambda }-\frac{1}{\mu }|{\int }_{\alpha }^{\beta }{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(t\right)\parallel }_{B}{\parallel {\left({K}_{\lambda }{v}_{\lambda }\right)}^{\prime }\left(t\right)\parallel }_{B}dt$(5.4)

for almost all $\alpha ,\beta$, namely for those that are Lebesgue points of ${v}_{\lambda }-{v}_{\mu }$ and ${K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }$. Finally, we consider a sequence $\left\{{\alpha }_{k}\right\}$ with ${\alpha }_{k}<0$ and ${\alpha }_{k}\to 0$ as $k\to \mathrm{\infty }$. We remark that ${v}_{\lambda }-{v}_{\mu }\in {\mathcal{𝒞}}_{w}\left(\left[-\eta ,T\right];H\right)$ and ${K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\in {\mathcal{𝒞}}_{w}\left(\left[-\eta ,T\right];{V}_{B}\right)$. Due to the choice of the extension for $t<0$, equation (5.4) shows for $\beta =t$ that

$\frac{1}{2}{\parallel \left({v}_{\lambda }-{v}_{\mu }\right)\left(t\right)\parallel }_{H}^{2}+{\int }_{0}^{t}{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(s\right)\parallel }_{B}^{2}ds+\frac{1}{2\mu }{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(t\right)\parallel }_{B}^{2}$$\le |\frac{1}{\lambda }-\frac{1}{\mu }|{\int }_{0}^{t}{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(s\right)\parallel }_{B}{\parallel {\left({K}_{\lambda }{v}_{\lambda }\right)}^{\prime }\left(s\right)\parallel }_{B}ds$

for almost all $t\in \left(0,T\right)$, and thus

$\frac{1}{2}{\parallel \left({v}_{\lambda }-{v}_{\mu }\right)\left(t\right)\parallel }_{H}^{2}+\frac{1}{2}{\int }_{0}^{t}{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(s\right)\parallel }_{B}^{2}ds+\frac{1}{2\mu }{\parallel \left({K}_{\lambda }{v}_{\lambda }-{K}_{\mu }{v}_{\mu }\right)\left(t\right)\parallel }_{B}^{2}$$\le \frac{1}{2}{|\frac{1}{\lambda }-\frac{1}{\mu }|}^{2}{\parallel {\left({K}_{\lambda }{v}_{\lambda }\right)}^{\prime }\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}$$\le \frac{{\lambda }^{2}}{2}{|\frac{1}{\lambda }-\frac{1}{\mu }|}^{2}\left({\parallel {v}_{\lambda }\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}+{\parallel {K}_{\lambda }{v}_{\lambda }-{u}_{0}\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}\right).$

It is easy to show that ${\parallel {K}_{\lambda }{v}_{\lambda }-{u}_{0}\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}\le {\lambda }^{2}{T}^{2}{\parallel {v}_{\lambda }\parallel }_{{L}^{2}\left(0,T;B\right)}^{2}$, which proves the assertion. ∎

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Footnotes

• 1

Here, we slightly abuse the notation of the duality pairing because ${L}^{{p}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{+}{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}\mathrm{+}{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}^{\mathrm{\prime }}\mathrm{\right)}$ is not the dual space of ${L}^{p}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}\mathrm{\right)}$. However, we can consider the sum of the duality pairings between ${L}^{{p}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{A}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{+}{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}H\mathrm{\right)}$ and ${L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{V}_{B}\mathrm{\right)}$ 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 $\left(\alpha ,\beta \right)$ for almost all $\alpha ,\beta \in \left(0,T\right)$.

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,

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