The aim of this study is to investigate problems without initial conditions for the evolution of functional-differential variational inequalities of a special form, so-called sub-differential inclusions with functionals. The partial case of this problem is a problem without initial conditions, or, in other words, the Fourier problem for integro-differential equations of the parabolic type.

Problem without initial conditions for evolution equations and variational inequalities (sub-differential inclusions) appear in the modeling of different non-stationary processes within many fields of science, such as ecology, economics, physics, cybernetics, etc., if these processes started a long time ago and initial conditions do not affect them in the actual time moment. Thus, we can assume that the initial time is minus infinity.

The research on the problem without initial conditions for the evolution equations and variational inequalities was conducted in the monographs [1–4], the papers [5–19], and others.

Note that the uniqueness of the solutions to the problem without initial conditions for linear and weak nonlinear evolution equations and variational inequalities is possible only under some restrictions on the behavior of solutions as the time variable changes to −∞. Moreover, in this case, to prove the existence of a solution, it is necessary to impose certain restrictions on the growth of the input data when the time variable goes to −∞. For the first time, it was strictly justified by Tychonoff [5] in the case of the heat equation. Later, similar results for various evolution equations and variational inequalities were obtained in monographs [1–4], papers [6–8, 12, 14, 16–19], and others.

However, as was shown by Bokalo [9], a problem without initial conditions for some strongly nonlinear parabolic equations has a unique solution in the class of functions without behavior restriction as the time variable changes to −∞. Furthermore, similar results were obtained in studies [10, 13, 15] (see also references therein) for strongly nonlinear evolution equations and in Bokalo [11] for evolution variational inequalities.

Note that the problem without initial conditions for weakly nonlinear functional-differential variational inequalities was investigated only in the study [17]. There, the existence and uniqueness of the solution to this problem were proved under certain restrictions on its behavior and the growth of the input data when the time variable is directed to −∞. As we know, the problem without initial conditions for strongly nonlinear functional-differential variational inequalities without restrictions on the behavior of the solution and the growth of the input data when the time variable is directed to −∞ has not been considered in the literature, and this serves as one of the motivations for the study of such problems.

The outline of this study is as follows: Section 2 comprises notations, definitions of needed function spaces, and auxiliary results. In Section 3, we set the problem statement and provide our key findings. The proof of the main results is kept in Section 4. Comments on the main results are given in Section 5. Section 6 provides conclusions.

Let V be a separable reflexive real Banach space with norm ||·||, and H be a real Hilbert space with the scalar products (·, ·) and norms |·|, respectively. Suppose that V ⊂ H with dense, continuous, and compact injection, i.e., the closure of V in H coincides with H, and there exists a constant λ > 0 such that λ|v|² ≤ ||v||² for all v ∈ V, and for every sequence {vk}k=1∞ bounded in V, there exists an element v ∈ V and a subsequence {vkj}j=1∞ such that vkj→j→∞v strongly in H.

Let V′ and H′ be the dual spaces of V and H, respectively. Suppose the space H′ (after appropriate identification of functionals) is a subspace of V′. Identifying the spaces H and H′ by the Riesz-Fréchet representation theorem, we obtain dense and continuous embeddings

Note that in this case 〈g, v〉 = (g, v) for every v ∈ V, g ∈ H ⊂ V′, where 〈g, v〉 is the means the action of an element g ∈ V′ on an element of v ∈ V, i.e., 〈·, ·〉 is canonical product for the duality pair [V′, V]. Therefore, we can use the notation (·, ·) instead of 〈·, ·〉, and we will do it in the future.

Let T > 0 be an arbitrary fixed real number, and let S: = (−∞, T], and intS: = (−∞, T).

We introduce some spaces for functions and distributions. Let X be an arbitrary Banach space with the norm ||·||_X. By C(S; X) we mean the linear space of continuous functions defined on S with values in X. We say that wm→m→∞w in C(S; X) if for each t₁, t₂ ∈ S, t₁ < t₂, sequence {wm|[t1,t2]}m=1∞ converges to w|_{[t1, t2]} in C([t₁, t₂];X) (hereafter w~|[t1,t2] is restriction of a function w~:S→X to segment [t₁, t₂] ⊂ S).

Let r ∈ [1, ∞], r′ is dual to r, i.e., 1/r + 1/r′ = 1. Denote by Llocr(S;X) the linear space of classes of equivalent measurable functions w: S → X such that w|[t1,t2]∈Lr(t1,t2;X) for each t₁, t₂ ∈ S, t₁ < t₂. We say that a sequence {w_m} is bounded (strongly, weakly, or *-weakly convergent, respectively, to w) in Llocr(S;X) if, for each t₁, t₂ ∈ S, t₁ < t₂, the sequence {w_m|_{[t1, t2]}} is bounded (strongly, weakly, or *-weakly convergent, respectively, to w|_{[t1, t2]}) in Lr(t1,t2;X).

By D′(intS;Vw′), we mean the space of continuous linear functionals on D(intS) with values in Vw′ (hereafter, D(intS) is the space of test functions, i.e., the space of infinitely differentiable on intS functions with compact supports, equipped with the corresponding topology, and Vw′ is the linear space V′ equipped with weak topology). It is easy to see (using (1)) that spaces Llocr(S;V), Lloc2(S;H), and Llocr′(S;V′) can be identified with the corresponding subspaces of D′(intS;Vw′) by rule 〈f,φ〉D=∫Sf(t)φ(t)dt, where 〈·, ·〉_D is the means the action of an element of D′(intS;Vw′) on an element of D(intS), f is an element of one of spaces Llocr(S;V), Lloc2(S;H), Llocr′(S;V′). In particular, this allows us to talk about derivatives w′ of functions w from Llocr(S;V) or Lloc2(S;H) in the perception of distributions D′(intS;Vw′) and the belonging of such derivatives to Llocr′(S;V′) or Lloc2(S;H).

Let us define the spaces

From known results [see, e.g., Gajewski et al. [20]] it follows that Hloc1(S;H)⊂C(S;H) and Wloc1,r(S;V)⊂C(S;H), and for every w in Hloc1(S;H) or Wloc1,r(S;V) the function t → |w(t)|² is continuous on any segment of the interval S, and the following equality holds:

In this study, we use the following well-known facts:

Proposition 2.1 [Corollaries from Young's inequality, Gajewski et al. [20]]. Let r > 1, ε > 0 be arbitrary, and r′ such that 1/r + 1/r′ = 1. Then, for all a, b ∈ ℝ, following inequality holds:

In particular,

Proof. Inequality (3) is a corollary from standard Young's inequality: a b ≤ |a|^r/r + |b|^r′/r′, if we note that r > 1 and r′ > 1. Inequality (4) we get from inequality (3) with r = 2.     □

Proposition 2.2 [Cauchy-Bunyakovsky-Schwarz inequality, Gajewski et al. [20]]. Let t₁, t₂ ∈ ℝ, and t₁ < t₂. Then, for v,w∈L2(t1,t2;H), we have (v(·),w(·))∈L1(t1,t2) and

Proposition 2.3 [Hölder's inequality, Gajewski et al. [20]]. Let r ∈ [1, ∞], r′ be a conjugated to r (i.e., 1/r + 1/r′ = 1), t₁, t₂ ∈ ℝ, t₁ < t₂. Suppose that X is a Banach space and X′ is a dual of X, 〈·, ·〉_X is the action of an element of X′ on an element of X. Then, for v∈Lr(t1,t2;X) and w∈Lr′(t1,t2;X′), we have 〈w(·),v(·)〉X∈L1(t1,t2) and

Proposition 2.4 [Lemma 1.1 [9]]. Let z: S → ℝ be a nonnegative and absolutely continuous on each interval of S function that satisfies differential inequality

where β∈Lloc1(S;ℝ), β(t) ≥ 0 for a.e. t ∈ S, ∫Sβ(t)dt=+∞; χ ∈ C([0, +∞)), χ(0) = 0, χ(s) > 0 if s > 0 and ∫1+∞dsχ(s)<∞. Then z ≡ 0 on S.

Proposition 2.5 [25]. Let Y be a Banach space with the norm ||·||_Y, and {vk}k=1∞ be a sequence of elements of Y that is weakly or *-weakly convergent to v in Y. Then lim_k→∞ ||v_k||_Y ≥ ||v||_Y.

Proposition 2.6 [Aubin theorem, Aubin [21]]. Let r > 1 and q > 1 be given numbers. Suppose that B₀, B₁, and B₂ are Banach spaces such that B0⊂cB1⊂B2 (symbol ⊂ means continuous embedding and symbol ⊂^c means compact embedding). Then

Note that we understand embedding (5) as follows: if a sequence {wm}m=1∞ is bounded in the space Lr(0,T;B0), and the sequence {wm′}m=1∞ is bounded in the space Lq(0,T;B2), then there exists a function w∈Lr(0,T;B1)∩C([0,T];B2) and the subsequence {wmj}j=1∞ of the sequence {wm}m=1∞ such that wmj→j→∞w in C([0, T];B₂) and strongly in Lr(0,T;B1).

Proposition 2.7. Let a sequence {wm}m=1∞ be bounded in the space Llocr(S;V), where r > 1, and the sequence {wm′} be bounded in the space Lloc2(S;H). Then there exists a function w∈Llocr(S;V), w′∈Lloc2(S;H), and a subsequence {wmj}j=1∞ of the sequence {wm}m=1∞ such that wmj→j→∞w in C(S; H) and weakly in Llocr(S;V), and w′mj→j→∞w′ weakly in Lloc2(S;H).

Proof. From Proposition 2.6 for q = 2, B₀ = V, B₁ = B₂ = H, we have that, for every t₁, t₂ ∈ S, t₁ < t₂, from the sequence of restrictions of the elements {wm}m=1∞ to the segment [t₁, t₂], one can choose a subsequence that is convergent in C([t₁, t₂];H) and weakly in Lr(t1,t2;V), and the sequence of derivatives of the elements of this subsequence is weakly convergent in L2(t1,t2;H). For each k ∈ ℕ, we choose a subsequence {wmk,j}j=1∞ of the given sequence that is convergent in C([T − k, T];H) and weakly in L^r(T − k, T; V) to some function w^k∈C([T-k,T];H)∩Lr(T-k,T;V), and the sequence {wmk,j′}j=1∞ is weakly convergent to the derivative w^k′ in L²(T − k, T; H). Making this choice, we ensure that the sequence {wmk+1,j}j=1∞ was a subsequence of the sequence {wmk,j}j=1∞. Now, according to the diagonal process, we select the desired subsequence as {wmj,j}j=1∞, and we define the function w as follows: for each k ∈ ℕ, we take w(t):=w^k(t) for t ∈ (T − k, T − k + 1).

Let Φ:V → ℝ_∞: = (−∞, +∞) be a proper functional, i.e., dom(Φ): = {v ∈ V:Φ(v) < +∞} ≠ ∅, which satisfies the conditions:

(A1) Φ(αv + (1 − α)w) ≤ αΦ(v) + (1 − α)Φ(w) ∀v, w ∈ V, ∀α ∈ [0, 1],

i.e., the functional Φ is convex;

(A2) vk→k→∞v   in    V ⇒   lim_k→∞  Φ(vk)≥Φ(v),

i.e., the functional Φ is lower semicontinuous;

(A3) there exist the constants p > 2 and K₁ > 0 such that

moreover, Φ(0) = 0.

Recall [see, e.g., Showalter [4]] that for a functional Φ satisfying the conditions (A1) and (A2) its sub-differential is a mapping ∂Φ:V → 2^V′, defined as follows:

and the domain of the sub-differential ∂Φ is the set D(∂Φ): = {v ∈ V|∂Φ(v) ≠ ∅}. We identify the subdifferential ∂Φ with its graph, assuming that [v, v^*] ∈ ∂Φ if and only if v^* ∈ ∂Φ(v), i.e., ∂Φ = {[v, v^*] | v ∈ D(∂Φ), v^* ∈ ∂Φ(v)}. R. Rockafellar in study [22, Theorem A] proves that the sub-differential ∂Φ is a maximal monotone operator, i.e.,

and for every element [v1,v1*]∈V×V′ we have the implication

Suppose that the following condition holds:

(A4) there exist the constants q > 2 and K₂ > 0, K₃ > 0 such that

Assume that B(t, ·):H → H, t ∈ S, is a given family of operators that satisfy the condition:

(B) for any v ∈ H the mapping B(·, v):S → H is measurable, and there exists a constant L ≥ 0 such that following inequality holds:

for a.e. t ∈ S, and all v₁, v₂ ∈ H; in addition, B(t, 0) = 0 for a.e. t ∈ S.

Remark 3.1. From the condition (B) it follows that

for a.e. t ∈ S and for all v ∈ H.

Next, we will assume that the conditions (A1)—(A4) and (B) are fulfilled, and p′ and q′ are such that 1/p + 1/p′ = 1, 1/q + 1/q′ = 1.

Let us consider the evolution variational inequality, or, in other words, subdifferential inclusion

where f∈Llocp′(S;V′)+Llocq′(S;H) is given function.

Definition 3.1. The solution of variational inequality (7) is called a function u: S → V that satisfies the following conditions:

The problem of finding a solution to variational inequality (7) for given Φ, B, and f is called the problem P(Φ, B, f), and the function u is called its solution.

We consider the existence and uniqueness of the solution to the problem P(Φ, B, f). The main results of this study are the following two theorems:

Theorem 3.1. Suppose that

Then the problem P(Φ, B, f) has at most one solution.

Theorem 3.2. Let inequality (8) hold, and let f∈Lloc2(S;H). Then the problem P(Φ, B, f) has a unique solution. In addition, this solution belongs to the space Lloc∞(S;V)∩Hloc1(S;H), and for arbitrary t₁, t₂ ∈ S, t₁ < t₂, δ > 0 satisfies the estimates:

where C₁, C₂ are positive constants depending on K₁, K₂, K₃, and q only.

Remark 3.2. If Φ is such that dom(Φ): = V and ∂Φ(v) = {A(v)}, v ∈ V, where A: V → V′ is some operator, then variational inequality (7) will be functional-differential equation

Note that condition (A3) implies the coercivity of operator A, i.e.,

In addition, from condition (A4) follows the strong monotonicity of the operator A, i.e.,

Proof. [Proof of the Theorem 3.1] Assume the contrary. Let u₁ and u₂ be two solutions to the problem P(Φ, B, f). Then for every i ∈ {1, 2} there exists function gi∈Llocp′(S;V′)+Llocq′(S;H) such that, for a.e. t ∈ S, g_i(t) ∈ ∂Φ(u_i(t)) and

We put w: = u₁ − u₂. From equalities (12), for a.e. t ∈ S, we obtain

Multiplying equality (13) scalar by w(t), for a.e. t ∈ S, we obtain

By condition (A4) and the fact that g_i(t) ∈ ∂Φ(u_i(t)), i = 1, 2, we have the inequality

By condition (B), for a.e. t ∈ S, we obtain

By Equations (2), (8), (15), and (16), from Equation (14) we get such differential inequality

From Equation (17), taking into account the condition q/2 > 1 and using Proposition 2.4 with z(t):=|w(t)|2,β(t):=2K3 for all t ∈ S, and χ(s): = s^q/2 for all s ∈ [0, +∞), we receive |w(t)|² = 0 for all t ∈ S, i.e., u₁ = u₂ a.e. on S. The resulting contradiction completes the proof of the uniqueness of the solution to the problem P(Φ, B, f).

Proof. [Proof of the Theorem 3.2] We divide the proof into seven steps.

Step 1 (auxiliary statements). We define the functional Φ_H:H → ℝ_∞ by the rule: Φ_H(v): = Φ(v) if v ∈ V, and Φ_H(v): = +∞ otherwise. Note that conditions (A1), (A2), Lemma IV.5.2, and Proposition IV.5.2 of the monograph [4] imply that Φ_H is a proper, convex, and lower semicontinuous functional on H, dom(Φ_H) = dom(Φ) ⊂ V and ∂Φ_H = ∂Φ ∩ (V × H), where ∂ΦH:H→2H is the sub-differential of the functional Φ_H. In addition, the condition (A3) implies that 0 ∈ ∂Φ_H(0).

The following statements will be used in the sequel:

Lemma 4.1 [[4, Lemma IV.4.3]]. Let −∞ < a < b < +∞, and w ∈ H¹(a, b; H), g ∈ L²(a, b; H) such that g(t) ∈ ∂Φ_H(w(t)) for a.e. t ∈ (a, b). Then the function Φ_H(w(·)) is absolutely continuous on the interval [a, b] and for any function h:[a, b] → H such that, for a.e. t ∈ (a, b), h(t) ∈ ∂Φ_H(w(t)), and the following equality holds:

Lemma 4.2 ([23, Proposition 3.12], [4, Proposition IV.5.2]). Let f~∈L2(0,T;H) and w₀ ∈ dom(Φ). Then there exists a unique function w ∈ C([0, T];H) ∩ H¹(0, T; H) such that w(0) = w₀ and, for a.e., t ∈ (0, T], w(t) ∈ D(∂Φ_H) and

Lemma 4.3. Let f~∈L2(0,T;H) and w₀ ∈ dom(Φ). Then there exists a unique function w ∈ C([0, T];H) ∩ H¹(0, T; H) such that w(0) = w₀ and, for a.e. t ∈ (0, T], w(t) ∈ D(∂Φ_H) and

i.e., there exists g~∈L2(0,T;H) such that, for a.e. t ∈ (0, T], we have g~(t)∈∂ΦH(w(t)) and

Proof. [Proof of Lemma 4.3] Let α > 0 be an arbitrary fixed number, and set

Consider M with the metric

The metric space (M, ρ) is complete. Now let us consider an operator A: M → M defined as follows: for any given function w~∈M, it defines a function w^∈M∩H1(0,T;H) such that, for a.e. t ∈ (0, T], w^(t)∈D(∂ΦH) and

Clearly, variational inequality (21) coincides with variational inequality (18) after replacing f~(t) by f~(t)-B(t,w~(t)), and w(0) = w₀ by w^(0)=w0. Thus, using Lemma 4.2, we get that operator A is well-defined. Let us demonstrate that the operator A is a contraction for some α > 0. Indeed, let w~1,w~2 be arbitrary functions from M, and w^1:=Aw~1, w^2:=Aw~2. According to Equation (21) there exist functions g^1 and g^2 from L²(0, T; H) such that for every j ∈ {1, 2} and for a.e. t ∈ (0, T] we have g^j(t)∈∂ΦH(w^j(t)) and

while w^j(0)=w0.

Subtracting identity (22) for j = 2 from identity (22) for j = 1, and, for a.e. t ∈ (0, T], multiplying the obtained identity by w^1(t)-w^2(t), we get

We integrate equality (23) by t from 0 to σ ∈ (0, T], taking into account (24) and that [see Equation (2)] for a.e. t ∈ (0, T]. The following holds:

As a result, we get the equality

By condition (A4), for a.e. t ∈ (0, T], we have the inequality

Taking into account condition (B) and inequality (4) for a.e. t ∈ (0, T], we obtain

where ε > 0 is an arbitrary.

From Equation (25), according to Equations (26) and (27), we have

Choosing ε = K₂, from Equation (28) we obtain

where C3:=2K2-1L2.

After multiplying inequality (30) by e^−2ασ, we obtain

From Equation (30), it easily follows that