1.1 Problem Statement and Main Result
We are concerned with initial value problems of the type
Here, is the time under consideration, is a given parameter, and , , f are given data of the problem.
The operator is a hemicontinuous, monotone, coercive operator satisfying a certain growth condition, where denotes a real, reflexive Banach space. The operator is a linear, symmetric, bounded, strongly positive operator on a real Hilbert space . We assume that both and are densely and continuously embedded in a real Hilbert space H, which is identified with its dual. We emphasise that we do not require to be embedded in (or vice versa) but only assume that is densely embedded in as well as in . This yields the scale
of Banach and Hilbert spaces with dense and continuous embeddings. Moreover, we assume that V is separable.
with suitable linear operators C, D such that (and with being replaced by ).
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 (). 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 can be interpreted as a relaxation or averaged delay time. If λ tends to 0, then the system decouples with . The limit results in a first-order equation for v without memory. In contrast to this, in the case where , the limit results in an evolution equation for u of second order in time (see ), whereas 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 is embedded in (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 . For the finite-dimensional case (including nonlinear problems), one may also consult .
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
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  consider a class of nonlinear integro-differential equations with completely monotone kernel in a Hilbert space setting. Gajewski, Gröger, and Zacharias  study rather general classes of nonlinear evolution equations with Volterra operators, but are restricted to the Hilbert space case . Crandall, Londen, and Nohel  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 . An interesting result for a class of doubly nonlinear integro-differential equations governed by an m-dissipative operator can be found in . 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 . 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.
For a Banach space X, we denote its norm by , its dual by , equipped with the standard norm , and the duality pairing by . We recall that is reflexive and separable if X is. For a Hilbert space X, we denote the inner product (inducing the norm ) by . The intersection of two Banach spaces X, Y is equipped with the norm , whereas the sum is equipped with the norm
We recall that ; see also [22, pp. 12ff.].
For a real, reflexive, separable Banach space X, the Bochner–Lebesgue spaces () are defined in the usual way and equipped with the standard norm. Denoting by the conjugate of with if , we have that if ; the duality pairing is given by
By () we denote the Banach space of functions whose distributional time derivative is again in ; the space is equipped with the standard norm. Note that if , then u equals almost everywhere a function that is in , i.e., a function that is absolutely continuous on as a function taking values in X. Moreover, is continuously embedded in the Banach space of functions that are continuous on as functions with values in X (see, e.g., [45, Chapter 7] for more details). By we denote the space of functions that are continuous on with respect to the weak topology in X.
The space of infinitely many times differentiable real functions with compact support in is denoted by . The space of on continuously differentiable real functions is denoted by . By c we denote a generic positive constant. We set for whatsoever.
3 Main Assumptions and Preliminary Results
Let be a real, reflexive Banach space and let and H be real Hilbert spaces satisfying (1.2). Moreover, we assume that is separable.
The structural properties we assume for the operators A and B throughout this paper are as follows.
The operator satisfies for all ,
and there exist , , such that for all ,
The linear operator is symmetric and there exist such that for all ,
By we denote the norm induced by B, which is equivalent to . Further, by we denote the space .
We shall remark that the operators A and B can be extended, as usual, to operators acting on functions defined on and taking values in and , respectively. Since the operator 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 and the theorem of Pettis (see, e.g., [16, Theorem 2 on p. 42]), a Bochner measurable function into a Bochner measurable function with (). Because of the growth condition, the operator A then maps into . The linear, symmetric, bounded, strongly positive operator extends, via for a function , to a linear, symmetric, bounded, strongly positive operator mapping into its dual. Indeed, B can also be seen as a linear, bounded operator mapping into for any .
With respect to the operator K defined by (1.1c), we make the following observations.
Let k be as in (1.1c) and let . Then K is an affine-linear, bounded mapping of into itself with
where . Further, K is an affine-linear, bounded mapping of into with
The standard proof is omitted here.
Note that in the assertions above can be replaced by H. An immediate consequence of the lemma above is that as well as 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 , with X being an arbitrary Banach space, we have that for almost all ,
If X is a Hilbert space and if , then testing this relation by immediately implies for all ,
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 , let and (). Let , , and be given approximations of the problem data , , and f, respectively. We look for approximations ().
The numerical scheme we consider combines the implicit Euler method with a convolution or product quadrature for the integral operator K and reads
To be precise, acts on a grid function and, with a slight abuse of notation, the evaluation at n is denoted by . The coefficients () are given by
The idea behind is the approximation
As we deal with kernels k of exponential type given by (1.1c), we can explicitly calculate
which immediately leads to the properties
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).
For better readability, we write instead of 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 from and the data of the problem by solving
Because of the continuous embeddings (1.2), we know that the right-hand side of the foregoing relation is in . The operator is easily shown to be hemicontinuous, coercive, and strictly monotone as a mapping of V into . Here, we make use of the fact that . In particular, we observe that for all ,
The famous theorem of Browder and Minty (see, e.g., [55, Theorem 26.A on p. 557]) now provides existence of a solution ; uniqueness immediately follows from the strict monotonicity.
For proving the a priori estimate, we test (4.1a) by (). With
the coercivity of A, and Young’s inequality, we find for ,
By summing up, we obtain for ,
Altogether, we find for ,
Taking the maximum over all first on the right-hand side and then on the left-hand side leads to
where depends on , , and . This yields
which, inserted in (4.7), proves the assertion. ∎
4.3 Convergence of Approximate Solutions and Existence of a Generalised Solution
Let be a sequence of positive integers such that as . We consider the corresponding sequence of time discrete problems (4.1) with step sizes , starting values with in H as well as with in , and right-hand sides given by
which is well-defined for . As a slight abuse of notation, in general we do not call the dependence of , and of the time instances on .
Let denote the solution to (4.1) with step size . We then consider the piecewise constant functions with for () and . Moreover, let be the piecewise affine-linear interpolation of the points () and let be the piecewise constant function with for () and .
Regarding the integral operator K, we define for any integrable function w the piecewise constant function by means of
We are now able to state the main result.
Theorem 4.2 (Existence via Convergence).
By passing to a subsequence if necessary, both the piecewise constant prolongation and the piecewise affine-linear prolongation of the approximate solutions to (4.1) converge weakly* in to v as . Furthermore, again by passing to a subsequence if necessary and as , the piecewise constant prolongation converges weakly to v in and the approximation of the memory term converges weakly* to Kv in . The approximation of the time derivative converges to
in the sense that for all .1
Note that if is bounded, then also converges weakly in 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.
Let and such that
Then for all there holds
The difficulty in proving (4.9) is that neither
can be assumed but only the sum is known to be in . 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 and , it is easy to show that
and thus . In view of [32, Chapter 3, Lemma 8.1], we find . Because of (3.1), we also see that , and thus , which implies . Because of (3.1), we also have . This justifies evaluating , , and for fixed t.
for almost all . Therefore, we have
Since and , we are allowed to split the right-hand side into
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
Since , , and , the third term on the right-hand side of (4.10) yields
Altogether, we have
Due to (3.1), there holds
as well as
Inserting this into (4.11) yields the desired integration-by-parts formula for . However, everything above remains true for an arbitrary , which proves the assertion. ∎
Proof of Theorem 4.2.
For simplicity, we write τ and N instead of and , respectively, during this proof. Since, by assumption, in H, in , and since in , the right-hand side of the a priori estimate (4.4) stays bounded as . Since
the a priori estimate (4.4) implies that the sequence is bounded in , the sequence is bounded in , and – in view of (4.8) – the sequence is bounded in . Due to the growth condition on A, we also conclude that the sequence is bounded in .
Since and are the duals of the separable normed spaces and , respectively, and since and are reflexive Banach spaces, there exists a common subsequence, again denoted by , and elements , , , and such that, as
see, e.g., [4, Corollaire III.26, Théorème III.27]. Since is linear and bounded, we also find that in .
A short calculation shows that
which tends to 0 as because of the a priori estimate (4.4). Hence .
Let us show that , and thereby . We already know that in , and thus in . The integral operator defined by can easily be shown to be a linear, bounded and thus weakly-weakly continuous mapping of into itself (see also Lemma 3.1). Therefore, in view of in , and thus in , we find that in . Because of
it thus remains to prove that converges at least weakly in to 0 in order to prove . For and (), we have
Since for as well as
the Cauchy–Schwarz inequality yields
Since in and is bounded in , we immediately get in as (and thus ), which is the last step to prove .
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
For any and , we thus obtain
Since in , and employing the (weak and weak*) convergence just shown, we can pass to the limit in the foregoing equation to come up with
This shows that v is differentiable in the weak sense with
Let us summarise the regularity properties of the limits we have proven so far: As
we conclude that . This implies (see [32, Chapter 3, Lemma 8.1]). Likewise, from
(see (3.1)), we deduce that , and thus . We remark that, in addition, one can easily show that .
By construction, we have in H and . In view of the a priori estimate (4.4), the sequence is bounded in H, and hence possesses a subsequence, again denoted by , that converges weakly in H to an element . We shall show that .
The first two terms on the right-hand side tend to zero as . This yields
Choosing ϕ such that and , respectively, shows that and , respectively.
By definition, we have in and . Again, the a priori estimate yields the boundedness of in , and thus the existence of a weakly convergent subsequence, again denoted by , and an element such that in .
We shall show that . Analogously to (4.14), choosing , , we have for any ,
Using (3.1), we obtain
For the first integral, (4.6) yields
For better readability, we write instead of for the rest of this proof. Splitting up the sum and shifting the index in the second sum yields
Inserting this into (4.15) and using that is bounded in , in , in as well as as (and thus ), we end up with
which proves .
It remains to show that a equals Av. We recall that . Hence, for arbitrary the monotonicity of A implies
The right-hand side converges to
This already implies
Below we will show
hence altogether proving
Taking for any , , passing to the limit , and Lebesgue’s theorem on dominated convergence together with the hemicontinuity and the growth condition of yield
which proves the assertion.
To finish the proof of existence, it remains to show (4.16). We start with
Since in , the first term on the right-hand side tends to as . Using (4.5), we estimate the second term as
As and in H, the weak lower semicontinuity of the norm provides
Regarding the last term on the right-hand side of equation (4.17), note that only in and in . Therefore, a finer examination of that term is necessary. To start with, a short calculation (see (4.6)) shows that
We conclude with (4.5) that
Since in and in as well as in , the weak lower semicontinuity of the norms involved shows, by passing to the limit and employing (see (4.2)),
Altogether, we find
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).
Let and let v and be solutions to ( 1.1 ) with data and , respectively. Then the stability estimate
holds for all , where the index in the notations and denotes the dependence of the integral operator on and , respectively.
Let . Assume in addition that A is uniformly monotone in the sense that there is such that
for all . Let v and be solutions to ( 1.1 ) with data and , respectively. Then the stability estimate
holds for all .
Let v and be solutions for and , respectively. The difference of both equations (1.1a) then reads
Lemma 4.3 implies for all ,
We test in equation (5.1) with . To prove (i), the monotonicity of A and Young’s inequality imply
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
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 and 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.
We consider , and with corresponding solutions v and . As , the first assumption of Theorem 5.1 is fulfilled, which shows
for all . 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 , 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 . For , let denote the operator related to the kernel and let denote the corresponding solution to problem (1.1). Moreover, assume that . Then for all and corresponding solutions there holds
for almost all t.
Taking the difference of the equations (1.1a) for and , respectively, and testing with leads to
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 , we first have to extend the functions considered to the interval for a fixed . We then prove the formula on an arbitrary interval with
At the end, instead of a fixed α, we consider a sequence with and as . If we considered a sequence of positive , we would not be able to identify the limit properly.
We fix . The centered Steklov average for a function , with X being an arbitrary Banach space, is defined by
where and . It is well known (see [16, Theorem 9, p. 49]) that in X for almost all . In addition, it is easy to show that as well as, by using Fubini’s theorem,
and thus in .
First, note that both and are in , and take the same value at . Hence, the difference fulfills the initial condition . We assume both and to be extended by for , and thus the difference to be extended by 0 for . We conclude that and similarly . Relation (3.1) still holds for both and for almost all .
it is easy to show that is bounded in . Therefore, there exists a subsequence, again denoted by h, such that in
In view of (3.1), we have
such that provide . Thus, in contrast to the difference itself, the Steklov average is an element of
which provides the regularity needed.
For all with we find
The first term on the right-hand side tends to zero as since
in as , the second one since is bounded in , and
in as . The latter follows from the linearity of .
Due to the regularity properties of , we are allowed to split the last integrand in (5.3). We use
as well as for any integrable function z on . We rewrite the third term as
Taking the limit and employing (5.2) results in
for almost all , namely for those that are Lebesgue points of and . Finally, we consider a sequence with and as . We remark that and . Due to the choice of the extension for , equation (5.4) shows for that
for almost all , and thus
It is easy to show that , which proves the assertion. ∎
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We could have also used this method to prove Lemma 4.3, but it only provides the integration-by-parts formula to hold on for almost all .
About the article
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”.