Well-posedness of history-dependent evolution inclusions with applications

In this paper, we study a class of evolution subdifferential inclusions involving history-dependent operators. We improve our previous theorems on existence and uniqueness and produce a continuous dependence result with respect to weak topologies under a weaker smallness condition. Two applications are provided to a frictional viscoelastic contact problem with long memory, and to a nonsmooth semipermeability problem.


Introduction
In this paper, we study a class of the Cauchy problems for evolution inclusions of subdifferential type in the framework of evolution triple of spaces. Recently, the problem has been studied in [21] without the convex subdifferential term and under the sign condition for the nonconvex potential, and in [8], where reformulated as a variational-hemivariational inequality, has been analyzed under a restrictive smallness hypothesis on the constants involved in operators A and ∂J [see problem (1.1), (1.2) below]. Here, we replace this restriction by a weaker condition and further examine the continuous dependence for such inclusions which was not studied before. In this way, the present paper is a continuation of [8]. Moreover, we provide a new application of our results to a semipermeability problem with nonmonotone and possibly multivalued subdifferential boundary conditions.
The main feature of the evolution inclusion under consideration is that both multivalued terms are generated by subdifferential operators which take their values in the dual space and not in a pivot space, and moreover, such inclusions depend on operators involved in the subdifferential maps and assumed to be history-dependent. We should note that Gasinski et al. [6] investigated an abstract first-order evolution inclusion in a reflexive Banach space extending several earlier work on parabolic hemivariational inequalities by Migórski [13], and Migórski and Ochal [15], and others.
The initial value problem for evolution inclusion under consideration reads as follows. Find w ∈ L 2 (0, T ; V ) with w ∈ L 2 (0, T ; V * ) such that w (t) + A(t, w(t)) + (R 1 w)(t) + M * ∂J(t, (Sw)(t), Mw(t)) +N * ∂ϕ(t, (Rw)(t), Nw(t)) f (t) a.e. t ∈ (0, T ), (1.1) w(0) = w 0 . (1. 2) The locally Lipschitz function ψ is called regular (in the Clarke sense) at u ∈ X if for all v ∈ X the one-sided directional derivative ψ (u; v) exists and satisfies ψ 0 (u; v) = ψ (u; v) for all v ∈ X. In what follows the generalized gradient of Clarke for a locally Lipschitz function and the subdifferential of a convex function will be denoted in the same way.
Recall that an operator A : X → X * is said to be demicontinuous if for all v ∈ X, the functional u → Au, v X * ×X is continuous, i.e., A is continuous as a map from X to X * endowed with the weak topology. Let 0 < T < ∞ and A : (0, T ) × X → X * . The Nemytskii (superposition) operator associated with A is the operator A : L 2 (0, T ; X) → L 2 (0, T ; X * ) defined by (Av)(t) = A(t, v(t)) for v ∈ L 2 (0, T ; X), a.e. t ∈ (0, T ).
A multivalued operator A : X → 2 X * is called coercive if either its domain D(A) = {u ∈ X | Au = ∅} is bounded or D(A) is unbounded and Given a set D in a normed space E, we define D E = sup{ x E | x ∈ D}. The space of linear and bounded operators from a normed space E to a normed space F is denoted by L(E, F ). It is endowed with the standard operator norm · L(E,F ) . For an operator L ∈ L(E, F ), we denote its adjoint by L * ∈ L(F * , E * ).
Recall that the spaces (V, H, V * ) form an evolution triple of spaces, if V is a reflexive and separable Banach space, H is a separable Hilbert space, and the embedding V ⊂ H is dense and continuous. We introduce the following Bochner spaces V = L 2 (0, T ; V ), V * = L 2 (0, T ; V * ), and W = {w ∈ V | w ∈ V * }. It follows from standard results, see, e.g., [3,Section 8.4], that the space W endowed with the graph norm w W = w V + w V * is a separable and reflexive Banach space, and each element in W, after a modification on a set of null measure, can be identified with a unique continuous function on Finally, we state a fixed point result (see [9,Lemma 7] or [30, Proposition 3.1]) being a consequence of the Banach contraction principle. Lemma 1. Let X be a Banach space and 0 < T < ∞. Let F : L 2 (0, T ; X) → L 2 (0, T ; X) be an operator such that

History-dependent evolution inclusions
We start with the study of existence and uniqueness for abstract first-order evolution subdifferential inclusion in a general form. Our study is a continuation of paper [8] where a class of general dynamic history-dependent variational-hemivariational inequalities has been investigated. The aim is to provide an improved version of result in [8,Theorem 6] which actually holds under a more general smallness hypothesis.
We study the operator inclusions in the standard functional setting used for evolution problems which exploits the notion of an evolution triple of spaces (V, H, V * ). We use the notation V, V * and W, recalled in the previous section.
Here, ψ(t, ·) is a locally Lipschitz function and ∂ψ denotes its Clarke generalized gradient. We recall that a function w ∈ W is a solution of Problem 2, if there exists w * ∈ V * such that In the study of Problem 2, we need the following hypotheses. ·) is strongly monotone for a.e. t ∈ (0, T ), i.e., for a constant m A > 0, We have the following existence and uniqueness result. Proof. The proof follows the lines of [8,Theorem 6]. For this reason, we provide only an argument for the coercivity of the operator F : V → 2 V * defined by Fv = Av + Bv for v ∈ V, where A : V → V * and B : V → 2 V * are the Nemytskii operators corresponding to the translations of A(t, ·) and ∂ψ(t, ·) by the initial condition w 0 : for v ∈ V and a.e. t ∈ (0, T ). By H(A)(iii), (iv) and H(ψ)(iii), (iv), we have Using H(A)(iii) and H(ψ)(iii), we get Integrating this inequality on (0, T ), by the Hölder inequality, we deduce Combining (3.1) and (3.2), we obtain From the smallness hypothesis (H 1 ), we infer that F is a coercive operator. The rest of the proof of this theorem is analogous to [8,Theorem 6], and therefore, it is omitted here.
We pass now to the evolution inclusion with history-dependent operators which is the main object of our study in this paper.
In this problem, ∂J denotes the generalized gradient of a locally Lipschitz function J(t, z, ·) and ∂ϕ is the convex subdifferential of a convex and lower semicontinuous function ϕ(t, y, ·). Despite two subdifferential terms generated by convex and (in general) nonconvex functions, the inclusion involves three nonlinear operators R, R 1 and S called history-dependent ones.
We introduce the following hypotheses on the data of Problem 4. Let X, Y , Z and U be Banach spaces.
(ii) J(t, ·, v) is continuous on Z for all v ∈ X, a.e. t ∈ (0, T ). (iii) J(t, z, ·) is locally Lipschitz on X for all z ∈ Z, a.e. t ∈ (0, T ).  Proof. The proof is based on Theorem 3, uses some ideas from [8, Theorem 9] and consists of three steps.
Since m ψ = m J M 2 , hypothesis (H 3 ) implies condition (H 1 ). Therefore, by applying Theorem 3, we obtain that there exists a unique element w ξηζ ∈ W which solves the inclusion in Problem 2 with ψ replaced by ψ ξηζ . Moreover, by (3.3), it is clear that w ξηζ ∈ W is also a solution to the following problem: Step 2 We claim that a solution to the problem (3.4) is unique. Let w 1 , w 2 ∈ W be solutions to the problem (3.4). For simplicity, we skip the subscripts ξ, η and ζ in this part of the proof. Take w 2 (t) as the test function in the inclusion in (3.4) satisfied by w 1 , take w 1 (t) as the test function in the inclusions (3.4) for w 2 , and add the two resulting expressions. We have for a.e. t ∈ (0, T ). Next, we integrate the above inequality on (0, t), for all t ∈ [0, T ] and then use the integration by parts, H(A)(iv), H(J)(v), the monotonicity of the convex subdifferential and condition By the smallness condition (H 3 ), we conclude w 1 = w 2 . The solution to problem (3.4) is unique.
Step 3 Consider operator Λ : where w ξηζ ∈ W denotes the unique solution to the problem (3.4) corresponding to (ξ, η, ζ). By an argument similar to the one used in [8, Theorem 9 and Lemma 3], we deduce that there exists a unique fixed point (ξ * , η * , ζ * ) of Λ, i.e., Let w ξ * η * ζ * ∈ W be the unique solution to the problem (3.4) corresponding to (ξ * , η * , ζ * ). By definition of operator Λ, we have Finally, we use these equalities in (3.4), and conclude w ξ * η * ζ * is the unique solution of Problem 4. This completes the proof of the theorem.

A continuous dependence result
In this section, we provide a new continuous dependence result for Problem 4. We study the continuity in the weak topologies of the map which to the right-hand side and initial condition in Problem 4 assigns its unique solution.
We will prove first the following a priori estimate on a solution.

Proposition 6. Under hypotheses of Theorem
Proof. Let us denote by w ∈ W a solution to Problem 4. This means that there are ξ ∈ L 2 (0, T ; X * ) and η ∈ L 2 (0, T ; U * ) such that We take the duality with w(t) in (4.1) to get In the estimates below, we use several times the Hölder inequality, Young's inequality ab ≤ ε 2 2 a 2 + 1 2ε 2 b 2 with ε > 0, and elementary inequality (a+b Well-posedness of history-dependent evolution Page 9 of 22 114 By assumption H(ϕ)(iv) and (4.6), we obtain The latter combined with (4.3) and the monotonicity of the convex subdifferential implies Similarly, by hypotheses H(J)(iv) and (v), (4.2), and (4.8), we get Next, exploiting (4.7), we have On the other hand, a simple calculation gives Subsequently, we integrate (4.5) on (0, t) for all t ∈ [0, T ], use integration by parts formula in [4, Proposition 8.4.14], H(A)(iii) and (iv), and inequalities (4.9)-(4.13), to deduce for all t ∈ [0, T ]. Applying the Gronwall lemma, see, e.g., [19, Lemma 2.7], we deduce the desired estimate on the term w V . By (4.1), we obtain the bound on w V * , and finally also on the norm of the solution w in C(0, T ; H). This proves the estimate in the statement of the proposition and completes the proof.
To provide a result on the continuous dependence, we need stronger versions of hypotheses introduced in the previous section. In particular, operator A will be assumed to be time independent and the weaklyweakly continuous which obviously implies the demicontinuity in H(A)(ii). All hypotheses introduced below are clearly satisfied in applications in Sects. 5 and 6. Then, w n (t) + Aw n (t) + (R 1 w n )(t) + M * ξ n (t) + N * η n (t) = f n (t) a.e. t ∈ (0, T ), (4.14) ξ n (t) ∈ ∂J(t, (Sw n )(t), Mw n (t)) a.e. t ∈ (0, T ), (4.15) η n (t) ∈ ∂ϕ(t, (Rw n )(t), Nw n (t)) a.e. t ∈ (0, T ), (4.16) w n (0) = w n 0 . (4.17) By Proposition 6 combined with (H 4 )(iii) and weak convergences of {f n } and {w n 0 }, the sequence {w n } is uniformly bounded in W. By the reflexivity of W, we can find a subsequence, denoted in the same way, such that w n → w weakly in W with w ∈ W, as n → ∞. We will prove that w is the unique solution in W to Problem 4 corresponding to (f, w 0 ). Using an argument similar to [34,Lemma 13], from hypothesis H(A) 1 , we known that Aw n → Aw weakly in V * , as n → ∞. Next, we prove the following claims.
Hypothesis H(J)(iv) implies that the sequence {ζ n } is bounded in X * . Hence, from the reflexivity of X * , without any loss of generality, we may assume that ζ n → ζ weakly in X * . Since D is weakly closed, we have ζ ∈ D, and by condition ζ n ∈ ∂J(t, z n , x n ), we get ζ n , w X * ×X ≤ J 0 (t, z n , x n ; w) for all w ∈ X.
Let {(y n , u n )} ⊂ (∂ϕ) − (E) and (y n , u n ) → (y, u) in Y × U , as n → ∞. We can find {ρ n } ⊂ U * such that ρ n ∈ ∂ϕ(t, y n , u n ) ∩ E for each n ∈ N. It is clear from H(ϕ)(iv) that the sequence {ρ n } is bounded in U * , which by the reflexivity of U * entails, at least for a subsequence, ρ n → ρ weakly in U * . Obviously, ρ ∈ E and ρ n , w U * ×U ≤ ϕ(t, y n , w) − ϕ(t, y n , u n ) for all w ∈ U.
Passing to the weak limit in H in (4.17), we obtain w(0) = w 0 . Applying convergences w n → w , (4.18), (4.19), (4.25), and f n → f weakly in V * , we take the limit in w n + Aw n + R 1 w n + M * ξ n + N * η n = f n in V * . Hence, The latter combined with (4.23), (4.24) and w(0) = w 0 imply that that w ∈ W is a solution to Problem 4 corresponding to (f, w 0 ). Since the solution is unique, we conclude that the whole sequence {w n } converges weakly in W to w. This completes the proof of the theorem.

Application to a frictional contact problem
In this section, we provide new continuous dependence results to a dynamic viscoelastic contact problem with friction which in its weak form leads to a history-dependent evolution inclusion analyzed in Sect. 4. Note that existence and uniqueness result for this problem has been obtained in [8] under a more restrictive smallness condition while the continuous dependence in weak topologies for this problem has not been studied before. We shortly recall the necessary notation and refer to [8,19,30] for a detailed explanation and a discussion on mechanical interpretation. Let Ω ⊂ R d , d = 2, 3, be a regular domain occupied in its reference configuration by a viscoelastic body. Its boundary Γ consists of three disjoint measurable parts Γ D , Γ N and Γ C , such that m(Γ D ) > 0. The body is clamped on Γ D (the displacement field vanishes there), the surface tractions act on Γ N , and Γ C is a contact surface. All indices i, j, k, l run between 1 and d and, unless stated otherwise, the summation convention over repeated indices is applied. We use u = (u i ), σ = (σ ij ) and ε(u) = (ε ij (u)) to denote the displacement vector, the stress tensor and linearized strain tensor, respectively. The latter is defined by where u i,j = ∂u i /∂x j . For a vector field, we use the notation v ν and v τ for the normal and tangential components of v on ∂Ω given by v ν = v · ν and v τ = v − v ν ν, where ν stands for the outward unit normal on the boundary. The normal and tangential components of the stress field σ on the boundary are defined by σ ν = (σν) · ν and σ τ = σν − σ ν ν, respectively. The symbol S d stands for the space of symmetric matrices of order d, and the canonical inner products on R d and S d are given by We are interested in the evolution process of the mechanical state of the body, in the finite time interval. The classical formulation of the contact problem is stated as follows.  Now, we recall the weak formulation of Problem 8. To this end, we need the classical Hilbert spaces with their standard inner products and norms. The symbol γ represents the norm of the trace operator γ : V → L 2 (Γ; R d ).
For v ∈ H 1 (Ω; R d ), we use the same symbol v for the trace of v on Γ and we use the notation v ν and v τ for its normal and tangential traces.
For the weak formulation of the problem and further discussion, we need the following hypotheses.
For the potential function j ν , we assume For the damper coefficient k and the friction bound F b , we assume H(f 0 ) : the densities of body forces, surface tractions and the initial data satisfy Finally, we define f : (0, T ) → V * by for all v ∈ V and a.e. t ∈ (0, T ). Under the above notation, we obtain the following weak formulation of Problem 8 in terms of the displacement. and subsequently, u ν (t) = t 0 w ν (s) ds + u 0ν and u τ (t) = t 0 w τ (s) ds + u 0τ for t ∈ (0, T ). Using these relations, Problem 9 in terms of velocity can be equivalently formulated as follows. and w ν (s) ds + u 0ν for all w ∈ V, t ∈ (0, T ), (5.14) Further, consider the boundary potentials J : (0, T ) × Z × X → R and ϕ : (0, T ) × Y × U → R defined by With the notation above, we formulate the following history-dependent evolution inclusion associated with Problem 10.
+N * ∂ϕ(t, (Rw)(t), Nw(t)) f (t) a.e. t ∈ (0, T ), Since B is linear and continuous, we deduce R 11 v n → R 11 v weakly in V * . Also since R 12 is linear and continuous, it is also weakly-weakly continuous, and therefore, R 12 v n → R 12 v weakly in V * . We conclude that R 1 is weakly-weakly continuous, history-dependent, and clearly, R 1 0 is bounded in L 2 (0, T ; V * ). In this way, condition (H 4 ) is verified. The conclusion of the theorem follows now from Theorem 7, which completes the proof.
Observe that Problems 10 and 11 are equivalent. This follows form the facts that every solution to Problem 11 is a solution to Problem 10, and that both problems have unique solutions. Thus, w ∈ W solves inequality in Problem 10 if and only if it is solves inclusion in Problem 11. We apply Theorem 12 to deduce the following well-posedness result for variational-hemivariational inequality in Problem 9. It shows the continuous dependence of the solution to the contact problem with respect to the densities of applied forces and the initial data. where {u n } and u are unique solutions to Problem 9 corresponding to (f n 0 , f n N , u n 0 , w n 0 ) and (f 0 , f N , u 0 , w 0 ), respectively.
Proof. Assume (5.20) and (5.21). Let f n , f ∈ V * be elements defined by (5.9) corresponding to (f n 0 , f n N ) and (f 0 , f N ), respectively. Since the map (f 0 , f n ) → f is linear and continuous, we have f n → f weakly in V * . Combining this with hypothesis w n 0 → w 0 weakly in V , by Theorem 12, we infer w n → w weakly in W, where u n (t) = t 0 w n (s) ds + u n 0 and u(t) = t 0 w(s) ds + u 0 for all t ∈ [0, T ], see (5.10). Clearly, we have u n → u weakly in W, and analogously as in the proof of (5.19), we obtain t 0 w n (s) ds → t 0 w(s) ds weakly in V , for all t ∈ [0, T ]. The latter together with hypothesis (5.21) implies (5.22). This completes the proof.

Application to a semipermeability problem
In this section, we illustrate the applicability of results in Sect. 4 in analysis of a semipermeability problem. Our aim is to provide the weak formulation of the problem which will be a variational-hemivariational inequality without history-dependent operators and to establish its well-posedness.
The semipermeability boundary conditions describe behavior of various types of membranes, natural and artificial ones and arise in models of heat conduction, electrostatics, hydraulics and in the description of flow of Bingham's fluids, where the solution represents temperature, electric potential and pressure. These boundary conditions were first examined by Duvaut and Lions [5] in the convex setting, where semipermeability relations were assumed to be monotone and they led to variational inequalities. More generally, nonmonotone semipermeability conditions can be modeled by the Clarke generalized gradient, see, e.g., [6,15,25,26] and the references therein.

Problem 14.
Find u = u(x, t) such that ∂u ∂t where A represents a linear operator A : V → V * , ∂u ∂νA denotes the conormal derivative with respect to operator A, and ν stands for the unit outward normal on the boundary. Problem 14 has been studied in [6] under more restrictive hypotheses and a different weak formulation.
To provide the weak formulation of Problem (14), we introduce assumptions on the data of the problem. Let V = {v ∈ H 1 (Ω) | v = 0 on Γ c }, H = L 2 (Ω), V = L 2 (0, T ; V ), W = { u ∈ V | u ∈ V * }. We denote by i : V → H the embedding operator and by γ : V → L 2 (Γ) the trace operator. For v ∈ H 1 (Ω), we always write v instead of iv and γv.
We need the following hypotheses on the data.