A semidiscrete scheme for evolution equations with memory

We introduce a new mathematical framework for the time discretization of evolution equations with memory. As a model, we focus on an abstract version of the equation \begin{document}$ \partial_t u(t) - \int_0^\infty g(s) \Delta u(t-s)\, {{\rm{d}}} s = 0 $\end{document} with Dirichlet boundary conditions, modeling hereditary heat conduction with Gurtin-Pipkin thermal law. Well-posedness and exponential stability of the discrete scheme are shown, as well as the convergence to the solutions of the continuous problem when the time-step parameter vanishes.

Remark 1.1. Actually, in more generality and with no changes in the proofs, one could take µ only piecewise absolutely continuous with a finite number of jumps (or even an infinite number of jumps, provided that they do not accumulate anywhere on the line). In this case, µ can be supposed to be right-continuous (or left continuous). with Dirichlet boundary conditions, which serves as a model for heat propagation in a rigid isotropic homogeneous conductor with hereditary memory of Gurtin-Pipkin type [10]. Actually, it can be viewed as a fully hyperbolic (nonlocal) relaxation of the classical heat equation, formally recovered from (1.3) when memory kernel collapses at the Dirac mass at 0 + . It is worth noting that in (1.3) the temperature evolution is influenced by the past history of the temperature itself. This feature may be regarded as a more realistic description of physical reality [7,15,16,20].
The study of numerical schemes for the discretization of PDEs is an extremely fruitful field of modern Mathematics, particularly relevant for concrete applications. Nowadays, the literature on the subject is huge, covering a large variety of topics both in the deterministic and in the stochastic framework. Equations with memory have been widely investigated also from the numerical viewpoint: several problems concerning well-posedness, error estimates and asymptotic stability of numerical solutions have been addressed (see e.g. [5,11,12,13,14,19,21,22] and references therein). The importance of the discretization of equations with memory comes from their wide applicability in the study of a number of phenomena in several areas, ranging from Physics, Biology, Population Dynamics, and many more. Since these equations are much harder than their counterparts without memory (for instance, they are structurally nonlocal), having at disposal robust discrete schemes is possibly the only effective way to obtain significant results in view of concrete applications. Nevertheless, all the studies made so far deal with Volterra-type equations of the formu The latter is just a particular instance of (1.1), corresponding to a null initial past history of the state variable u (namely, u ≡ 0 for negative times). It should be noted that equation (1.1) is much more difficult to handle than (1.4). Indeed, a first thought in order to tackle (1.1) could be to fix just a particular initial past history of u, and then regard ∞ t g(s)Au(t − s) ds as a time-dependent source term. In which case, the system becomes nonautonomous. On the other hand, one would like to study the problem for different initial past histories of u, playing in this fashion the role of initial data. This however implies to take into account, besides the evolution of the variable u itself, the evolution of its past history as well. The strategy, first proposed by C.M. Dafermos in his pioneering paper [4], consists in introducing an auxiliary variable η accounting for the (integrated) past history of u, whose evolution takes place in a suitable Hilbert space M 0 called memory space. Within this approach, one can construct a solution semigroup acting on an extended phase space and exploit the powerful theory of dynamical systems. This allows to successfully address several issues regarding existence and uniqueness, continuous dependence, regularity and asymptotic behavior of solutions.
In this paper, we introduce a mathematical framework for the discretization of the evolution equation (1.1). To the best of our knowledge, this is the first attempt to discretize an equation with memory with infinite delay. And indeed, the difficulties in the continuous framework reflect in the discrete setting, making the problem intriguing and highly nontrivial. The first challenge, which is not encountered in the discretization of (1.4), consists in finding the discrete counterpart of the continuous memory space M 0 and the additional variable η. Secondly, one needs to set up a well-posed numerical scheme which, in addition, has to be robust as the discretization parameters converge to zero. Here, we propose an implicit method where only the time variable is discretized, thus allowing the exploitation of one's favorite spatial discretization when performing numerics. It is worth mentioning that our scheme does not rely on the linear structure of (1.1). Hence, it is conceivable that the framework developed in the present paper can be adapted to discretize nonlinear variants of the model. This task will be possibly the object of future projects. After the functional setting and a brief overview of the continuous case, we deal with the discretization of the memory space M 0 . Then, we introduce the semidiscrete scheme and we show its well-posedness, which translates into the existence of a discrete semigroup of solutions acting on a suitable extended discrete phase space. Next, we prove that the discrete semigroup decays exponentially to zero. Such an asymptotic behavior reflects the longterm properties of the continuous semigroup generated by (1.1), which decays exponentially to zero as well. In the last part of the work, we deal with the convergence of the discrete solutions to the continuous ones as the time-step parameter vanishes. This passage is essential in order to ensure that the scheme provides a good approximation of the continuous equation.
2. Functional setting. For r ∈ R, we introduce the compactly nested family of Hilbert spaces (the index r will be omitted whenever zero) In particular, we have the compact embedding H 1 H, along with the Poincaré inequality where λ 1 > 0 is the first eigenvalue of A. If r > 0, it is understood that H −r denotes the completion of the domain, so that H −r is the dual space of H r . Accordingly, the symbol ·, · also stands for duality product between H r and H −r . Next, we consider the memory spaces of square summable H r+1 -valued functions on R + with respect to the measure µ(s)ds (again, r will be omitted whenever zero) M r 0 = L 2 µ (R + ; H r+1 ) endowed with the weighted inner product and norm Due to the Poincaré inequality (2.1), we have the continuous (but not compact) inclusion M 0 ⊂ M −1 0 . Finally, we define the phase space of our problem In the last part of the paper, we will also make use of the "asymmetric" energy spaces equipped with the natural product norm.
3. The continuous case. Since u = u(t) is assigned for t ≤ 0, we can interpret as an initial datum. Accordingly, when s > t, we have the equality This leads us to the following definition of (weak) solution. (ii) The function η is given by  where the prime denotes the distributional derivative with respect to the internal variable s of the function η (see [9]). Indeed, such a linear equation usually appears in the definition of solutions in problems with memory. However, the alternative Definition 3.1 adopted in this paper, and firstly introduced in [2], presents several advantages. In particular, it is more flexible when performing regularization schemes (since the variable η, which belongs to a space of functions with values in a Hilbert space, is automatically regularized via u). Besides, and this is important in view of our purposes, it can be discretized in a very natural way.
Well-posedness and asymptotic stability of the problem above have been analyzed in several works (see e.g. [1,3,6,8,15] and references therein), In particular, the following results have been established.
• For every T > 0 and every z ∈ H 0 there exists a unique weak solution (u(t), η t ) on [0, T ] with initial datum z, in the sense of Definition 3.1. Hence, the problem generates a linear C 0 -semigroup • Assuming the Dafermos condition (1.2), the semigroup S 0 (t) is exponentially stable, namely, there exist ν > 0 and C ≥ 1 such that In fact, as shown in [3], the Dafermos condition (1.2) is only sufficient in order for exponential stability to occur.

4.
Discretization of the memory space. Throughout the paper we agree to denote For an arbitrarily given τ > 0, we introduce the nonnegative nonincreasing (and vanishing at infinity) sequence Then, we consider the discrete memory space endowed with the weighted inner product and norm Finally, we define the discrete phase space normed by (u, η) 2 Hτ = u 2 + η 2 Mτ . A word of warning. Along the paper, the Young and Poincaré inequalities, as well as the continuous and the discrete Hölder inequalities, will be used several times, often without explicit mention. In particular, recalling (4.2), for all η ∈ M τ we have the estimate We conclude the section by showing that the sequence µ k fulfills a discrete version of the Dafermos condition (1.2).
holds for every k ∈ N + .
Proof. From the very definition of the sequence µ k , we infer that It is also apparent that (1.2) is equivalent to for every t ≥ 0 and almost every s > 0. Hence, and the conclusion follows.

5.
The semidiscrete scheme. The main goal of this paper is the analysis of a (semi) discrete version of the evolution problem (1.1). To this end, we first give the exact discrete formulation of the Definition 3.1 of solution.
5.1. The scheme. For an arbitrarily given time-step parameter τ > 0, we partition the plane (t, s) into a square grid with cells of the form where n and k play the roles of the time variable t ≥ 0 and the "internal" variable s > 0, respectively. Next, we sample the state variable (u, η), making use of an implicit approximation scheme. More precisely, given any initial datum we plan to solve recursively the equation Here and in what follows, we will always assume n ∈ N and k ∈ N + , unless otherwise specified. In fact, equality (5.1) is to be correctly interpreted in the weak form for every test function ϕ ∈ H 1 . Extending the definition of η n k for k = 0 as is verified for all n ∈ N and all k ∈ N + .
The proof of the lemma is actually nothing but a direct calculation, and is left to the reader. It is worth observing that the (formal) position (5.4) actually reflects the boundary condition (3.1) of the continuous model.
which is nothing but the classical first-order upwind scheme [17] for the advection equation (3.4). Besides, due to the definition of µ k , the discrete equation (5.1) is the backward Euler approximation of the integrodifferential equation (3.2), the integral being computed by the rectangle method.

5.2.
Well-posedness. The first step is showing that the scheme is well posed.
Proof. Assume that (u n , η n ) has been defined up to m ∈ N. Then, substituting (5.5) into (5.1) and exploiting (4.2), we obtain It is immediate to verify that the right-hand side of the identity above belongs to the space H −1 . Indeed, due to (2.1) and (5.4), where the latter estimate follows from the (discrete) Hölder inequality. Therefore At this point, we merely define η m+1 through (5.5) and we apply Lemma 5.1. It remains to show that η m+1 belongs to M τ . Indeed, since the sequence µ k is nonincreasing, and using again (5.4), we infer that In particular, the sequence E n is nonincreasing.
Proof. Choosing ϕ = 2τ u n+1 in (5.3) (recall that u n+1 ∈ H 1 ), and using the identity we get In light of (5.4) and (5.5), the last term can be rewritten as Hence, Neglecting the term P n ≥ 0, the conclusion follows.
Theorem 5.4 produces an immediate corollary.
Corollary 5.5. The linear map S τ : H τ → H τ acting as is bounded (hence continuous).
6. Exponential stability. The aim of this section is to prove the exponential decay of the discrete semigroup S n τ , and to discuss the result in the limit τ → 0. 6.1. Exponential stability of the discrete semigroup. Our main theorem reads as follows.
The proof requires the introduction of the auxiliary energy-type functional Observe that, on account of (2.1), Lemma 6.2. For every n ∈ N, the functional Ψ n satisfies the inequality Proof. By means of direct calculations, we get the identity Invoking (5.1), the first term on the right-hand side can be written as Moreover, as µ k ↓ 0, we infer from (5.4)-(5.5) that Collecting the estimates above, the proof is finished.
At this point, we introduce the positive quantity ε = ε(τ ) = min 1 2 Next, we consider the functional Λ n = E n + εΨ n .
We have now all the ingredients to prove Theorem 6.1.
6.2. Passage to the limit. Here we show that the conclusion of Theorem 6.1 is consistent with the exponential decay of the continuous semigroup S 0 (t), formally (at this stage) obtained by letting τ → 0. Namely, we want to render precise the following statement: the fact that S n τ ≤ 2q n τ implies, in the limit τ → 0, that S 0 (t) ≤ 2e −ωt for some ω > 0. The symbols · denote the operator norms in the respective spaces.
Proposition 6.4. There exists a constant ω > 0 independent of τ > 0 such that, for every fixed t ≥ 0, we have where n τ ∈ N is the unique integer satisfying t ≤ τ n τ < t + τ.
Proof. For simplicity, we proceed within the further assumption µ(0) = lim s→0 + µ(s) < ∞, ensuring in particular that the function ε(τ ) defined in (6.3) satisfies Nevertheless, the result is valid even when µ is not bounded about zero (see Remark 6.5 below). Owing to Theorem 6.1, the inequality S nτ τ ≤ 2q nτ τ holds, where q τ is given by (6.6). Since τ n τ → t as τ → 0, invoking (6.7) we get The result is proved. In this way, in the definition of ε(τ ), the quantity µ 0 (τ ) is replaced by a certain function of τ that converges to µ(s * ) > 0 as τ → 0. Then one repeats exactly the previous argument. We leave the full details to the interested reader.

7.
Convergence. Finally, we show that the solutions of the discrete model converge to the solutions of the continuous one when the time-step parameter τ > 0 vanishes. In other words, for every fixed t ≥ 0, we want to prove the convergence (in some suitable sense) S nτ τ → S 0 (t) as τ → 0, where, as before, t ≤ τ n τ < t + τ . To this end, we need to introduce the discretization operator D τ acting on M 0 as In the next Lemma 7.2 we will see that D τ maps M 0 into M τ .
we consider the solution (u n , η n ) to the discrete scheme (5.1)-(5.2) corresponding to the initial datum (u 0 , η 0 ) ∈ H τ . Then, calling Λ(t) = max(1 − |t|, 0), we introduce the linear interpolating function Recalling the definition of the spaces V r 0 given in (2.2), the main result of the section reads as follows.
The remaining of the paper is devoted to the proof of Theorem 7.1. In what follows, T > 0 is (arbitrary but) fixed. 7.2. Uniform bounds. The first step is proving estimates on u τ which are independent of the time-step parameter τ . We first need a result on the operator D τ introduced earlier.
Lemma 7.2. The discretization operator D τ maps M 0 into M τ (in fact, onto), and Proof. Making use of (4.1) and the definition of ξ k , Since in the last formula above, for every fixed k, the variable s belongs to (kτ, (k + 1)τ ] and µ is nonincreasing, it is clear that µ(s) ≤ µ(kτ ), for k ≥ 1. Thus, for all s ∈ (kτ, (k + 1)τ ], owing once more to the fact that µ is nonincreasing, we obtain and the conclusion follows. Proof. Let t ∈ [0, T ] be arbitrarily fixed, and let n = n(t, τ ) ∈ N be such that t ∈ [nτ, (n + 1)τ ).
It is immediate to check that, being u τ (t) a convex combination of u n and u n+1 , Since E n is nonincreasing, by Lemma 7.2 we get , implying that u τ is bounded in L ∞ (0, T ; H) uniformly with respect to τ . Moreover, for t = nτ , the time derivative of u τ readṡ u τ (t) = u n+1 − u n τ .
As a consequence, exploiting (5.1) together with (4.2), (5.7) and the discrete Hölder inequality, ]. Invoking (5.8) and Lemma 7.2, the right-hand side can be estimated as , meaning thatu τ is bounded in L ∞ (0, T ; H −1 ) uniformly with respect to τ . The next step is passing (7.1) to the limit. To this end, calling the following holds. Proof. Since u τn ∈ L ∞ (0, T ; H), and so u τn ∈ L ∞ (0, T ; M −1 0 ) (viewed as a constant function of the variable s), the representation formula (7.1) is completely equivalent to the assertion that the function η τn is the (unique) mild solution in the sense of Pazy [18] to the linear problem where now T is the infinitesimal generator of the right-translation semigroup on M −1 0 (see [9]). Accordingly, The same argument shows that as well. In order to prove (7.4), for every fixed t ∈ [0, T ], we estimate Due to the Dafermos condition (1.2), the function s 2 µ(s) is summable. In particular, there exists a structural constant c > 0 such that Finally, appealing to (7.2), yielding the conclusion. 7.4. Identifying the limit. We now show that (v, ζ) coincides with the solution (u, η) of the continuous problem (3.2)-(3.3) with initial datum z = (w, ξ).
An auxiliary result is needed, stated again as a lemma.
Lemma 7.6. As τ → 0, we have the convergencė The proof of Lemma 7.6 is rather long and technical, and it is postponed to the final section of this work.
Since for t = nτu an exploitation of equation (5.1) together with (4.1) yield the equalitẏ Accordingly, the conclusion of Lemma 7.6 is attained if we show that as τ → 0, uniformly with respect to t. To this end, we split the series as and we estimate separately the three terms. In what follows, C > 0 will denote a generic constant independent of t, τ , and depending only on the H 0 -norm of the initial datum z = (w, ξ) and on the structural quantities of the problem.
Lemma 8.1. We have the inequality Proof. Let n > 0 (otherwise the term f t (τ ) does not appear), and let 1 ≤ k ≤ n and s ∈ I k−1 be arbitrarily chosen. Being s < nτ ≤ t, from the representation formula (7.1) we learn that Combining the two relations above, Making use of Lemma 7.3 we readily see that We conclude that as claimed.
Lemma 8.2. We have the inequality Proof. The argument is similar. Invoking (7.1), we find the control Exploiting the first equation in (5.2) and arguing as in the proof of Lemma 8.1 (the details are left to the reader), the first two integrals are controlled by Next, since µ is nonincreasing, we infer from the continuous Hölder inequality and the Poincaré inequality (2.1) that The proof is complete.
In order to estimate the last term h t (τ ), we introduce the further quantity The following holds. Finally, recalling that η 0 = D τ (ξ), we readily see that the remaining term is less than or equal to r t (τ ).
Collecting Lemmas 8.1-8.3, we arrive at Since the first two terms above vanish as τ → 0, to complete the proof of Lemma 7.6 we just need to prove the final Lemma 8.4. The limit lim τ →0 r t (τ ) = 0 holds uniformly with respect to t.
Proof. Along the proof, it is convenient to highlight the dependence of r t (τ ) on the second component ξ of the initial datum z. Accordingly, we will write r t (τ, ξ), and we will denote by C 0 a generic constant independent of ξ (as well as of t, τ ). The proof requires a number of steps.
We conclude that as desired.