Strong nonlinear functional-differential variational inequalities: problems without initial conditions

Problems without initial conditions for evolution equations and variational inequalities 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. In the case of linear and weakly nonlinear evolution equations and variational inequalities, standard initial conditions should be replaced with the behavior of the solution as the time variable goes to minus infinity. However, for some strongly nonlinear evolution equations and variational inequalities, this problem has a unique solution in the class of functions without behavior restriction as the time variable goes to minus infinity. In this study, the correctness of the problem without initial conditions for such types of variational inequalities from a new class, or more precisely, for sub-differential inclusions with functionals, is investigated. Moreover, estimates of solutions are obtained. The results are new and mostly theoretical.

1 Introduction

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.

2 Preliminaries

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 ${\left\{v_k\right\}}_{k=1}^{\infty }$ bounded in V, there exists an element v ∈ V and a subsequence ${\left\{v_{k_j}\right\}}_{j=1}^{\infty }$ such that $v_{k_j}\underset{j\to \infty }{\to }v$ 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

$\begin{array}{ll}V\subset H\subset V^{\prime }.& (1)\end{array}$

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 $w_m\underset{m\to \infty }{\to }w$ in C(S; X) if for each t₁, t₂ ∈ S, t₁ < t₂, sequence ${\left\{w_m{|}_{\left[t_1,t_2\right]}\right\}}_{m=1}^{\infty }$ converges to w|_{[t1, t2]} in C([t₁, t₂];X) (hereafter $\tilde{w}{|}_{\left[t_1,t_2\right]}$ is restriction of a function $\tilde{w}:S\to X$ to segment [t₁, t₂] ⊂ S).

Let r ∈ [1, ∞], r′ is dual to r, i.e., 1/r + 1/r′ = 1. Denote by $L_{\mathrm{\text{loc}}}^r(S;X)$ the linear space of classes of equivalent measurable functions w: S → X such that $w{|}_{\left[t_1,t_2\right]}\in L^r(t_1,t_2;X)$ 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 $L_{\mathrm{\text{loc}}}^r(S;X)$ 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 $L^r(t_1,t_2;X).$

By $D^{\prime }(\mathrm{\text{int}}S;V_w^{\prime })$, we mean the space of continuous linear functionals on D(intS) with values in $V_w^{\prime }$ (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 $V_w^{\prime }$ is the linear space V′ equipped with weak topology). It is easy to see (using (1)) that spaces $L_{\mathrm{\text{loc}}}^r(S;V)$, $L_{\text{loc}}^2(S;H)$, and $L_{\mathrm{\text{loc}}}^{r^{\prime }}(S;V^{\prime })$ can be identified with the corresponding subspaces of $D^{\prime }(\mathrm{\text{int}}S;V_w^{\prime })$ by rule ${\langle f,\phi \rangle }_D={\int }_{S}f(t)\phi (t)dt$, where 〈·, ·〉_D is the means the action of an element of $D^{\prime }(\mathrm{\text{int}}S;V_w^{\prime })$ on an element of D(intS), f is an element of one of spaces $L_{\mathrm{\text{loc}}}^r(S;V)$, $L_{\text{loc}}^2(S;H)$, $L_{\mathrm{\text{loc}}}^{r^{\prime }}(S;V^{\prime })$. In particular, this allows us to talk about derivatives w′ of functions w from $L_{\mathrm{\text{loc}}}^r(S;V)$ or $L_{\text{loc}}^2(S;H)$ in the perception of distributions $D^{\prime }(\mathrm{\text{int}}S;V_w^{\prime })$ and the belonging of such derivatives to $L_{\text{loc}}^{r^{\prime }}(S;V^{\prime })$ or $L_{\mathrm{\text{loc}}}^2(S;H)$.

Let us define the spaces

$H_{\text{loc}}^1(S;H):=\left\{w\in L_{\text{loc}}^2(S;H)\{w^{\prime }\in L_{\text{loc}}^2(S;H)\right\},$

$W_{\text{loc}}^{1,r}(S;V):=\left\{w\in L_{\text{loc}}^r(S;V)\{w^{\prime }\in L_{\text{loc}}^{r^{\prime }}(S;V^{\prime })\right\},r>1.$

From known results [see, e.g., Gajewski et al. [20]] it follows that $H_{\text{loc}}^1(S;H)\subset C(S;H)$ and $W_{\text{loc}}^{1,r}(S;V)\subset C(S;H)$, and for every w in $H_{\text{loc}}^1(S;H)$ or $W_{\text{loc}}^{1,r}(S;V)$ the function t → |w(t)|² is continuous on any segment of the interval S, and the following equality holds:

$\begin{array}{ll}\frac{d}{dt}|w(t){|}^2=2(w^{\prime }(t),w(t))\text{ for almost every (a.e.) }t\in S.& (2)\end{array}$

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:

$\begin{array}{ll}ab\le \epsilon |a{|}^r+{\epsilon }^{-1/(r-1)}|b{|}^{r^{\prime }}.& (3)\end{array}$

In particular,

$\begin{array}{ll}ab\le \epsilon |a{|}^2+{\epsilon }^{-1}|b{|}^2.& (4)\end{array}$

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 $v,w\in L^2(t_1,t_2;H)$, we have $(v(·),w(·))\in L^1\left(t_1,t_2\right)$ and

$\begin{array}{l}{\displaystyle {\int }_{t_1}^{t_2}}(w(t),v(t))dt\le ({\displaystyle {\int }_{t_1}^{t_2}}|v(t){|}^{2}dt{)}^{1/2}({\displaystyle {\int }_{t_1}^{t_2}}|w(t){|}^{2}dt{)}^{1/2}.\end{array}$

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 $v\in L^r(t_1,t_2;X)$ and $w\in L^{r^{\prime }}(t_1,t_2;X^{\prime })$, we have ${\langle w(·),v(·)\rangle }_X\in L^1\left(t_1,t_2\right)$ and

$\begin{array}{l}{\displaystyle {\int }_{t_1}^{t_2}}{\langle w(t),v(t)\rangle }_{X}dt\le \Vert w|{|}_{L^{r^{\prime }}(t_1,t_2;X^{\prime })}\Vert v|{|}_{L^r(t_1,t_2;X)}.\end{array}$

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

$z^{\prime }(t)+\beta (t)\chi \left(z(t)\right)\le 0for\text{ a.e. }t\in S,$

where $\beta \in L_{\text{loc}}^1(S;ℝ)$, β(t) ≥ 0 for a.e. t ∈ S, ${\int }_S\beta (t)dt=+\infty$; χ ∈ C([0, +∞)), χ(0) = 0, χ(s) > 0 if s > 0 and ${\int }_1^{+\infty }\frac{ds}{\chi (s)}<\infty$. Then z ≡ 0 on S.

PROPOSITION 2.5 [25]. Let Y be a Banach space with the norm ||·||_Y, and ${\left\{v_k\right\}}_{k=1}^{\infty }$ be a sequence of elements of Y that is weakly or *-weakly convergent to v in Y. Then $\underset{k\to \infty }{\underset{\_}{\lim }}$ ||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 $B_0{\subset }^c{B}_1\subset B_2$ (symbol ⊂ means continuous embedding and symbol ⊂^c means compact embedding). Then

$\begin{array}{ll}\left\{w\in L^r(0,T;B_0)|w^{\prime }\in L^q(0,T;B_2)\right\}{\subset }^c(L^r(0,T;B_1)\cap C(\left[0,T\right];B_2)).& (5)\end{array}$

Note that we understand embedding (5) as follows: if a sequence ${\left\{w_m\right\}}_{m=1}^{\infty }$ is bounded in the space $L^r(0,T;B_0)$, and the sequence ${\left\{w_m^{\prime }\right\}}_{m=1}^{\infty }$ is bounded in the space $L^q(0,T;B_2)$, then there exists a function $w\in L^r(0,T;B_1)\cap C(\left[0,T\right];B_2)$ and the subsequence ${\left\{w_{m_j}\right\}}_{j=1}^{\infty }$ of the sequence ${\left\{w_m\right\}}_{m=1}^{\infty }$ such that $w_{m_j}\underset{j\to \infty }{\to }w$ in C([0, T];B₂) and strongly in $L^r(0,T;B_1)$.

PROPOSITION 2.7. Let a sequence ${\left\{w_m\right\}}_{m=1}^{\infty }$ be bounded in the space $L_{\text{loc}}^r(S;V)$, where r > 1, and the sequence $\left\{w_m^{\prime }\right\}$ be bounded in the space $L_{\text{loc}}^2(S;H)$. Then there exists a function $w\in L_{\text{loc}}^r(S;V)$, $w^{\prime }\in L_{\text{loc}}^2(S;H)$, and a subsequence ${\left\{w_{m_j}\right\}}_{j=1}^{\infty }$ of the sequence ${\left\{w_m\right\}}_{m=1}^{\infty }$ such that $w_{m_j}\underset{j\to \infty }{\to }w$ in C(S; H) and weakly in $L_{\text{loc}}^r(S;V)$, and ${w^{\prime }}_{m_j}\underset{j\to \infty }{\to }{w}^{\prime }$ weakly in $L_{\text{loc}}^2(S;H)$.

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 ${\left\{w_m\right\}}_{m=1}^{\infty }$ to the segment [t₁, t₂], one can choose a subsequence that is convergent in C([t₁, t₂];H) and weakly in $L^r(t_1,t_2;V)$, and the sequence of derivatives of the elements of this subsequence is weakly convergent in $L^2(t_1,t_2;H)$. For each k ∈ ℕ, we choose a subsequence ${\left\{w_{m_{k,j}}\right\}}_{j=1}^{\infty }$ 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 ${\hat{w}}_k\in C(\left[T-k,T\right];H)\cap L^r(T-k,T;V),$ and the sequence ${\left\{w_{m_{k,j}}^{\prime }\right\}}_{j=1}^{\infty }$ is weakly convergent to the derivative ${\hat{w}}_k^{\prime }$ in L²(T − k, T; H). Making this choice, we ensure that the sequence ${\left\{w_{m_{k+1,j}}\right\}}_{j=1}^{\infty }$ was a subsequence of the sequence ${\left\{w_{m_{k,j}}\right\}}_{j=1}^{\infty }$. Now, according to the diagonal process, we select the desired subsequence as ${\left\{w_{m_{j,j}}\right\}}_{j=1}^{\infty },$ and we define the function w as follows: for each k ∈ ℕ, we take $w(t):={\hat{w}}_k(t)$ for t ∈ (T − k, T − k + 1).

3 Statement of the problem and formulation of main results

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

$({\mathcal{A}}_1)$ Φ(αv + (1 − α)w) ≤ αΦ(v) + (1 − α)Φ(w) ∀v, w ∈ V, ∀α ∈ [0, 1],

i.e., the functional Φ is convex;

$({\mathcal{A}}_2)$ $v_k\underset{k\to \infty }{\to }v\text{ in }V\Rightarrow \underset{k\to \infty }{\underset{\_}{\lim }}\Phi (v_k)\ge \Phi (v),$

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

$({\mathcal{A}}_3)$ there exist the constants p > 2 and K₁ > 0 such that

$\begin{array}{l}\Phi (v)\ge K_1\Vert v|{|}^p\forall v\in \text{dom}(\Phi );\end{array}$

moreover, Φ(0) = 0.

Recall [see, e.g., Showalter [4]] that for a functional Φ satisfying the conditions $({\mathcal{A}}_1)$ and $({\mathcal{A}}_2)$ its sub-differential is a mapping ∂Φ:V → 2^V′, defined as follows:

$\begin{array}{l}\partial \Phi (v):=\left\{v^{*}\in V^{\prime }|\Phi (w)\ge \Phi (v)+(v^{*},w-v)\forall w\in V\right\},v\in V,\end{array}$

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.,

$\begin{array}{l}(v_1^{*}-v_2^{*},v_1-v_2)\ge 0\forall \left[v_1,v_1^{*}\right],\left[v_2,v_2^{*}\right]\in \partial \Phi \end{array}$

and for every element $\left[v_1,v_1^{*}\right]\in V\times V^{\prime }$ we have the implication

$\begin{array}{l}(v_1^{*}-v_2^{*},v_1-v_2)\ge 0\forall \left[v_2,v_2^{*}\right]\in \partial \Phi \Rightarrow \left[v_1,v_1^{*}\right]\in \partial \Phi .\end{array}$

Suppose that the following condition holds:

$({\mathcal{A}}_4)$ there exist the constants q > 2 and K₂ > 0, K₃ > 0 such that

$\begin{array}{l}(v_1^{*}-v_2^{*},v_1-v_2)\ge K_2|v_1-v_2{|}^2+K_3|v_1-v_2{|}^q\forall \left[v_1,v_1^{*}\right],\left[v_2,v_2^{*}\right]\in \partial \Phi .\end{array}$

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

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

$|B(t,v_1)-B(t,v_2)|\le L|v_1-v_2|$

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 ($\mathcal{B}$) it follows that

$\begin{array}{ll}|B(t,v)|\le L|v|& (6)\end{array}$

for a.e. t ∈ S and for all v ∈ H.

Next, we will assume that the conditions $({\mathcal{A}}_1)$—$({\mathcal{A}}_4)$ and ($\mathcal{B}$) 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

$\begin{array}{ll}{u}^{\prime }(t)+\partial \Phi (u(t))+B(t,u(t))\ni f(t),t\in S,& (7)\end{array}$

where $f\in L_{\text{loc}}^{p^{\prime }}(S;V^{\prime })+L_{\text{loc}}^{q^{\prime }}(S;H)$ is given function.

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

1) $u\in W_{\text{loc}}^{1,p}(S;V)\cap L_{\text{loc}}^q(S;H)$;

2) u(t) ∈ D(∂Φ) for a.e. t ∈ S;

3) there exists a function $g\in L_{\text{loc}}^{p^{\prime }}(S;V^{\prime })+L_{\text{loc}}^{q^{\prime }}(S;H)$ such that, for a.e. t ∈ S, g(t) ∈ ∂Φ(u(t)) and

$\begin{array}{l}{u}^{\prime }(t)+g(t)+B(t,u(t))=f(t)\text{ in }{V}^{\prime }.\end{array}$

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

$\begin{array}{ll}L<K_2.& (8)\end{array}$

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

THEOREM 3.2. Let inequality (8) hold, and let $f\in L_{\text{loc}}^2(S;H)$. Then the problem P(Φ, B, f) has a unique solution. In addition, this solution belongs to the space $L_{\mathrm{\text{loc}}}^{\infty }(S;V)\cap H_{\text{loc}}^1(S;H)$, and for arbitrary t₁, t₂ ∈ S, t₁ < t₂, δ > 0 satisfies the estimates:

$\begin{array}{l}{\displaystyle \underset{t\in \left[t_1,t_2\right]}{max}}|u(t){|}^2+{\displaystyle {\int }_{t_1}^{t_2}}[|u(t){|}^2+|u(t){|}^q+\Vert u(t)|{|}^p]dt\le C_1[{\delta }^{-\frac{2}{q-2}}\\ +{\displaystyle {\int }_{t_1-\delta }^{t_2}}|f(t){|}^{2}dt],& (9)\end{array}$

$\begin{array}{l}\underset{t\in [t_1,t_2]}{\text{ess }\sup }\Vert u(t)){\Vert }^p+{\displaystyle {\int }_{t_1}^{t_2}|}{u}^{\prime }(t){|}^{2}dt\le C_2[\max \{{\delta }^{-\frac{2}{q-2}},{\delta }^{-\frac{q}{q-2}}\}\\ +{\displaystyle {\int }_{t_1-2\delta }^{t_2}|}f(t){|}^{2}dt+{\delta }^{-1}{\displaystyle {\int }_{t_1-2\delta }^{t_1}|}f(t){|}^{2}dt],& (10)\end{array}$

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

$\begin{array}{ll}{u}^{\prime }(t)+A(u(t))+B(t,u(t))=f(t),t\in S.& (11)\end{array}$

Note that condition $({\mathcal{A}}_3)$ implies the coercivity of operator A, i.e.,

$(A(v),v)\ge K_1\Vert v|{|}^p,v\in V.$

In addition, from condition $({\mathcal{A}}_4)$ follows the strong monotonicity of the operator A, i.e.,

$\begin{array}{l}(A(v_1)-A(v_2),v_1-v_2)\ge K_2|v_1-v_2{|}^2+K_3|v_1-v_2{|}^q\forall v_1,v_2\in V.\end{array}$

4 Proof of the main results

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 $g_i\in L_{\text{loc}}^{p^{\prime }}(S;V^{\prime })+L_{\text{loc}}^{q^{\prime }}(S;H)$ such that, for a.e. t ∈ S, g_i(t) ∈ ∂Φ(u_i(t)) and

$\begin{array}{ll}{u}_i^{\prime }(t)+g_i(t)+B(t,u_i(t))=f(t)\text{ in }{V}^{\prime },i=1,2.& (12)\end{array}$

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

$\begin{array}{ll}{w}^{\prime }(t)+g_1(t)-g_2(t)+B(t,u_1(t))-B(t,u_2(t))=0\text{ in }{V}^{\prime }.& (13)\end{array}$

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

$\begin{array}{l}(w^{\prime }(t),w(t))+(g_1(t)-g_2(t),u_1(t)-u_2(t))+(B(t,u_1(t))\\ -B(t,u_2(t)),u_1(t)-u_2(t))=0.& (14)\end{array}$

By condition $({\mathcal{A}}_4)$ and the fact that g_i(t) ∈ ∂Φ(u_i(t)), i = 1, 2, we have the inequality

$\begin{array}{ll}(g_1(t)-g_2(t),u_1(t)-u_2(t))\ge K_2|w(t){|}^2+K_3|w(t){|}^q\text{ for a.e. }t\in S.& (15)\end{array}$

By condition $(\mathcal{B}),$ for a.e. t ∈ S, we obtain

$\begin{array}{ll}(B(t,u_1(t))-B(t,u_2(t)),u_1(t)-u_2(t))\ge -L|w(t){|}^2.& (16)\end{array}$

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

$\begin{array}{ll}{(|w(t){|}^2)}^{\prime }+2K_3{\left(|w(t){|}^2\right)}^{q/2}\le 0\text{ for a.e. }t\in S.& (17)\end{array}$

From Equation (17), taking into account the condition q/2 > 1 and using Proposition 2.4 with $z(t):=|w(t){|}^2,\beta (t):=2K_3$ 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 $({\mathcal{A}}_1)$, $({\mathcal{A}}_2)$, 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 $\partial {\Phi }_H:H\to 2^H$ is the sub-differential of the functional Φ_H. In addition, the condition $({\mathcal{A}}_3)$ 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:

$\begin{array}{l}\frac{d}{dt}{\Phi }_H\left(w(t)\right)=(h(t),w^{\prime }(t)).\end{array}$

LEMMA 4.2 ([23, Proposition 3.12], [4, Proposition IV.5.2]). Let $\tilde{f}\in L^2(0,T;H)$ 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

$\begin{array}{ll}{w}^{\prime }(t)+\partial {\Phi }_H\left(w(t)\right)\ni \tilde{f}(t)\text{ in }H.& (18)\end{array}$

LEMMA 4.3. Let $\tilde{f}\in L^2(0,T;H)$ 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

$\begin{array}{ll}{w}^{\prime }(t)+\partial {\Phi }_H\left(w(t)\right)+B(t,w(t))\ni \tilde{f}(t)\text{ in }H,& (19)\end{array}$

i.e., there exists $\tilde{g}\in L^2(0,T;H)$ such that, for a.e. t ∈ (0, T], we have $\tilde{g}(t)\in \partial {\Phi }_H(w(t))$ and

$\begin{array}{ll}{w}^{\prime }(t)+\tilde{g}(t)+B(t,w(t))=\tilde{f}(t)\text{ in }H.& (20)\end{array}$

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

$M:=\left\{w\in C(\left[0,T\right];H)|w(0)=w_0\right\}.$

Consider M with the metric

$\rho (w_1,w_2)={\displaystyle \underset{t\in \left[0,T\right]}{max}}[e^{-\alpha t}|w_1(t)-w_2(t)|],w_1,w_2\in M.$

The metric space (M, ρ) is complete. Now let us consider an operator A: M → M defined as follows: for any given function $\tilde{w}\in M$, it defines a function $\hat{w}\in M\cap H^1(0,T;H)$ such that, for a.e. t ∈ (0, T], $\hat{w}(t)\in D(\partial {\Phi }_H)$ and

$\begin{array}{ll}{\hat{w}}^{\prime }(t)+\partial {\Phi }_H(\hat{w}(t))\ni \tilde{f}(t)-B(t,\tilde{w}(t))\text{ in }H.& (21)\end{array}$

Clearly, variational inequality (21) coincides with variational inequality (18) after replacing $\tilde{f}(t)$ by $\tilde{f}(t)-B(t,\tilde{w}(t))$, and w(0) = w₀ by $\hat{w}(0)=w_0$. 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 ${\tilde{w}}_1,{\tilde{w}}_2$ be arbitrary functions from M, and ${\hat{w}}_1:=A{\tilde{w}}_1$, ${\hat{w}}_2:=A{\tilde{w}}_2$. According to Equation (21) there exist functions ${\hat{g}}_1$ and ${\hat{g}}_2$ from L²(0, T; H) such that for every j ∈ {1, 2} and for a.e. t ∈ (0, T] we have ${\hat{g}}_j(t)\in \partial {\Phi }_H({\hat{w}}_j(t))$ and

$\begin{array}{ll}{\hat{w}}_j^{\prime }(t)+{\hat{g}}_j(t)=\tilde{f}(t)-B(t,{\tilde{w}}_j(t)),& (22)\end{array}$

while ${\hat{w}}_j(0)=w_0$.

Subtracting identity (22) for j = 2 from identity (22) for j = 1, and, for a.e. t ∈ (0, T], multiplying the obtained identity by ${\hat{w}}_1(t)-{\hat{w}}_2(t)$, we get

$\begin{array}{c}({({\hat{w}}_1(t)-{\hat{w}}_2(t))}^{\prime },{\hat{w}}_1(t)-{\hat{w}}_2(t))+({\hat{g}}_1(t)-{\hat{g}}_2(t),{\hat{w}}_1(t)-{\hat{w}}_2(t))\\ =-(B(t,{\tilde{w}}_1(t))-B(t,{\tilde{w}}_2(t)),{\hat{w}}_1(t)-{\hat{w}}_2(t))\\ \text{ for a.e.}\end{array}$

$\begin{array}{ll}t\in (0,T],& (23)\end{array}$

$\begin{array}{ll}{\hat{w}}_1(0)-{\hat{w}}_2(0)=0.& (24)\end{array}$

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:

$\begin{array}{l}({({\hat{w}}_1(t)-{\hat{w}}_2(t))}^{\prime },{\hat{w}}_1(t)-{\hat{w}}_2(t))=\frac{1}{2}(|{\hat{w}}_1(t)-{\hat{w}}_2(t){|}^2{)}^{\prime }.\end{array}$

As a result, we get the equality

$\begin{array}{l}\frac{1}{2}|{\hat{w}}_1(\sigma )-{\hat{w}}_2(\sigma ){|}^2+{\displaystyle \underset{0}{\overset{\sigma }{\int }}}({\hat{g}}_1(t)-{\hat{g}}_2(t),{\hat{w}}_1(t)-{\hat{w}}_2(t))dt\\ =-{\displaystyle \underset{0}{\overset{\sigma }{\int }}}(B(t,{\tilde{w}}_1(t))-B(t,{\tilde{w}}_2(t)),{\hat{w}}_1(t)-{\hat{w}}_2(t))dt.& (25)\end{array}$

By condition $({\mathcal{A}}_4)$, for a.e. t ∈ (0, T], we have the inequality

$\begin{array}{ll}({\hat{g}}_1(t)-{\hat{g}}_2(t),{\hat{w}}_1(t)-{\hat{w}}_2(t))\ge K_2|{\hat{w}}_1(t)-{\hat{w}}_2(t){|}^2.& (26)\end{array}$

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

$\begin{array}{l}|(B(t,{\tilde{w}}_1(t))-B(t,{\tilde{w}}_2(t)),{\hat{w}}_1(t)-{\hat{w}}_2(t))|\\ \le |B(t,{\tilde{w}}_1(t))-B(t,{\tilde{w}}_2(t))||{\hat{w}}_1(t)-{\hat{w}}_2(t)|\\ \le L|{\tilde{w}}_1(t)-{\tilde{w}}_2(t)\Vert {\hat{w}}_1(t)-{\hat{w}}_2(t)|\le \epsilon |{\hat{w}}_1(t)-{\hat{w}}_2(t){|}^2\\ +{\epsilon }^{-1}{L}^2|{\tilde{w}}_1(t)-{\tilde{w}}_2(t){|}^2,& (27)\end{array}$

where ε > 0 is an arbitrary.

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

$\begin{array}{l}|{\hat{w}}_1(\sigma )-{\hat{w}}_2(\sigma ){|}^2+2(K_2-\epsilon ){\displaystyle \underset{0}{\overset{\sigma }{\int }}}|{\hat{w}}_1(t)-{\hat{w}}_2(t){|}^{2}dt\\ \le 2{\epsilon }^{-1}{L}^2{\displaystyle \underset{0}{\overset{\sigma }{\int }}}|{\tilde{w}}_1(t)-{\tilde{w}}_2(t){|}^{2}dt.& (28)\end{array}$

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

$\begin{array}{ll}|{\hat{w}}_1(\sigma )-{\hat{w}}_2(\sigma ){|}^2\le C_3{\displaystyle {\int }_0^{\sigma }}|{\tilde{w}}_1(t)-{\tilde{w}}_2(t){|}^{2}dt,\sigma \in (0,T],& (29)\end{array}$

where $C_3:=2K_2^{-1}{L}^2$.

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

$\begin{array}{c}{e}^{-2\alpha \sigma }|{\hat{w}}_1(\sigma )-{\hat{w}}_2(\sigma ){|}^2\le C_3{e}^{-2\alpha \sigma }{\displaystyle {\int }_0^{\sigma }}{e}^{2\alpha t}{e}^{-2\alpha t}|{\tilde{w}}_1(t)-{\tilde{w}}_2(t){|}^{2}dt\\ \le C_3{e}^{-2\alpha \sigma }{\displaystyle \underset{t\in \left[0,T\right]}{max}}[e^{-\alpha t}|{\tilde{w}}_1(t)-{\tilde{w}}_2(t)|{]}^2{\displaystyle \underset{0}{\overset{\sigma }{\int }}}{e}^{2\alpha t}dt\\ =\frac{C_3}{2\alpha }(1-e^{-2\alpha \sigma })[\rho ({\tilde{w}}_1,{\tilde{w}}_2){]}^2\le \frac{C_3}{2\alpha }[\rho ({\tilde{w}}_1,{\tilde{w}}_2){]}^2,\\ \sigma \in (0,T].& (30)\end{array}$

From Equation (30), it easily follows that

$\rho ({\hat{w}}_1,{\hat{w}}_2)\le \sqrt{C_3/(2\alpha )}\rho ({\tilde{w}}_1,{\tilde{w}}_2).$

From this, choosing α > 0 such that inequality C₃/(2α) < 1 holds, we obtain that operator A: M → M is a contraction. Hence, we may apply the Banach fixed-point theorem [24, Theorem 5.7] and deduce that there exists a unique function w ∈ M ∩ H¹(0, T; H) such that Aw = w, i.e., we have proved over the statement, i.e., Lemma 4.3.

Step 2 (solution approximations). Let us consider the next problem: to find a function $u\in H_{\text{loc}}^1(S;H)$ such that, for a.e., t ∈ S, u(t) ∈ D(∂Φ_H) and

$\begin{array}{ll}{u}^{\prime }(t)+\partial {\Phi }_H\left(u(t)\right)+B(t,u(t))\ni f(t)\text{ in }H.& (31)\end{array}$

We call this problem the problem P(Φ_H, B, f). The solution of the problem P(Φ_H, B, f) is the solution of the problem P(Φ, B, f). We prove the existence of a solution to the problem P(Φ_H, B, f).

At first, we construct a sequence of functions, that, in some perception, approximates the solution of the problem P(Φ_H, B, f). For each k ∈ ℕ we put ${\hat{f}}_k(t):=f(t)$ for t ∈ S_k: = (T − k, T] and let us consider the problem of finding a function ${\hat{u}}_k\in H^1(S_k;H)$ such that ${\hat{u}}_k(T-k)=0$ and, for a.e. t ∈ S_k, we have ${\hat{u}}_k(t)\in D(\partial {\Phi }_H)$ and

$\begin{array}{ll}{\hat{u}}_k^{\prime }(t)+\partial {\Phi }_H\left({\hat{u}}_k(t)\right)+B(t,{\hat{u}}_k(t))\ni {\hat{f}}_k(t)\text{ in }H.& (32)\end{array}$

The existence of a unique solution to problem (32) implies Lemma 4.3. Note that sub-differential inclusion in (32) means that there exists a function ${\hat{g}}_k\in L^2(S_k;H)$ such that, for a.e., t ∈ S_k, we have ${\hat{g}}_k(t)\in \partial {\Phi }_H({\hat{u}}_k(t))$ and

$\begin{array}{ll}{{\hat{u}}_k}^{\prime }(t)+{\hat{g}}_k(t)+B(t,{\hat{u}}_k(t))={\hat{f}}_k(t)\text{ in }H.& (33)\end{array}$

Note that D(∂Φ_H) ⊂ dom(Φ_H) = dom(Φ) ⊂ V, and thus ${\hat{u}}_k(t)\in V$ for a.e. t ∈ S_k. According to the definition of the subdifferential of a functional and the fact that ${\hat{g}}_k(t)\in \partial \Phi ({\hat{u}}_k(t))$, we have

$\Phi (0)\ge \Phi ({\hat{u}}_k(t))+({\hat{g}}_k(t),0-{\hat{u}}_k(t))\text{ for a.e. }t\in S_k.$

From this and condition $({\mathcal{A}}_3)$ we obtain

$\begin{array}{ll}({\hat{g}}_k(t),{\hat{u}}_k(t))\ge \Phi ({\hat{u}}_k(t))\ge K_1\Vert {\hat{u}}_k(t)|{|}^p\text{ for a.e. }t\in S_k.& (34)\end{array}$

Since the left side of this chain of inequalities belongs to $L^1(S_k)$, then ${\hat{u}}_k$ belongs to $L^p(S_k;V)$.

For each k ∈ ℕ, we extend functions ${\hat{f}}_k,{\hat{u}}_k$, and ${\hat{g}}_k$ by zero for the entire interval S and denote these extensions by f_k, u_k, and g_k, respectively. From the above, it follows that, for each k ∈ ℕ, the function u_k belongs to L^p(S; V), its derivative $u_k^{\prime }$ belongs to L²(S; H), and, for a.e. t ∈ S, g_k(t) ∈ ∂Φ_H(u_k(t)) and [see Equation (33)],

$\begin{array}{ll}{u}_k^{\prime }(t)+g_k(t)+B(t,u_k(t))=f_k(t)\text{ in }H.& (35)\end{array}$

Step 3 (estimates of solution approximations). To demonstrate the convergence ${\left\{u_k\right\}}_{k=1}^{\infty }$ to the solution of the problem P(Φ_H, B, f), we need some estimates of the functions u_k, k ∈ ℕ.

Let the function ${\theta }_{*}\in C^1(ℝ)$ such that θ_*(t) = 0 if t ∈ (−∞, −1], ${\theta }_{*}(t)=e^{\frac{t^2}{t^2-1}}$ if t ∈ (−1, 0), θ_*(t) = 1 if t ∈ [0, +∞) [see Bokalo [9]]. Obviously, ${\theta }_{*}^{\prime }(t)\ge 0$ for arbitrary t ∈ ℝ, and for any 0 < ν < 1, we have

$\begin{array}{ll}{\displaystyle \underset{t\in (-1,0)}{sup}}\frac{{\theta }_{*}^{\prime }(t)}{{\theta }_{*}^{\nu }(t)}=C_4,& (36)\end{array}$

where C₄ > 0 is a constant depending on ν only.

Let t₁, t₂, and δ be arbitrary real fixed numbers such that t₁, t₂ ∈ S, t₁ < t₂, δ > 0. We put

$\begin{array}{ll}\theta (t):={\theta }_{*}(\frac{t-t_1}{\delta }),t\in S.& (37)\end{array}$

It is clear that θ(t) = 0 if t ∈ (−∞, t₁ − δ], 0 < θ(t) < 1 if t ∈ (t₁ − δ, t₁), θ(t) = 1 if t ∈ [t₁, +∞), and ${\theta }^{\prime }(t)={\delta }^{-1}{\theta }_{*}^{\prime }((t-t_1)/\delta )\ge 0$ for every t ∈ ℝ.

Let k ∈ ℕ. Obviously, $\theta u_k\in H^1(S;H)$. For each t ∈ S, multiply the identity (35) scalar by θ(t)u_k(t) and integrate from t₁ − δ to τ ∈ [t₁, t₂]. As a result, we obtain

$\begin{array}{l}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(u_k^{\prime }(t),u_k(t))dt+{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(g_k(t),u_k(t))dt\\ +{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(B(t,u_k(t)),u_k(t))dt={\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(f_k(t),u_k(t))dt.& (38)\end{array}$

From this, taking into account (2) and using the integration-by-parts formula, we transform the first term on the left side of the equality (38) as follows:

$\begin{array}{l}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(u_k^{\prime }(t),u_k(t))dt=\frac{1}{2}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(|u_k(t){|}^2{)}^{\prime }dt=\frac{1}{2}|u_k(\tau ){|}^2\\ -\frac{1}{2}{\displaystyle {\int }_{t_1-\delta }^{t_1}}{\theta }^{\prime }(t)|u_k(t){|}^{2}dt.& (39)\end{array}$

Then from Equation (38), using Equation (39), we receive

$\begin{array}{l}|u_k(\tau ){|}^2+2{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(g_k(t),u_k(t))dt={\displaystyle {\int }_{t_1-\delta }^{t_1}}{\theta }^{\prime }(t)|u_k(t){|}^{2}dt\\ -2{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(B(t,u_k(t)),u_k(t))dt+2{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(f_k(t),u_k(t))dt.& (40)\end{array}$

Since (0, 0) ∈ ∂Φ_H and (g_k(t), u_k(t)) ∈ ∂Φ_H for a.e. t ∈ S, from condition $({\mathcal{A}}_4)$ we get

$\begin{array}{ll}(g_k(t),u_k(t))\ge K_2|u_k(t){|}^2+K_3|u_k(t){|}^q\text{ for a.e. }t\in S.& (41)\end{array}$

According to the definition of u_k and g_k and using the inequality (34), we obtain

$\begin{array}{ll}(g_k(t),u_k(t))\ge \Phi \left(u_k(t)\right)\ge K_1\Vert u_k(t)|{|}^p\text{ for a.e. }t\in S.& (42)\end{array}$

Let us estimate the second term on the left-hand side of equality (40), using inequalities (41) and (42), in this way:

$\begin{array}{l}2{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(g_k(t),u_k(t))dt\ge 2(\sigma +(1-\sigma )){\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(g_k(t),u_k(t))dt\\ \ge 2\sigma K_2{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k(t){|}^{2}dt+2\sigma K_3{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k(t){|}^{q}dt\\ +2(1-\sigma )K_1{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)\Vert u_k(t)|{|}^{p}dt+2(1-\sigma ){\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)\Phi \left(u_k(t)\right)dt,& (43)\end{array}$

where σ ∈ (0, 1) is arbitrary.

Using the inequality (34) (with r = q/2, r′ = q/(q − 2)), we estimate the first term on the right-hand side of Equation (40) as follows:

$\begin{array}{l}{\displaystyle {\int }_{t_1-\delta }^{\tau }}{\theta }^{\prime }(t)|u_k(t){|}^{2}dt={\displaystyle {\int }_{t_1-\delta }^{t_1}}{\theta }^{\prime }(t){\theta }^{-\frac{2}{q}}(t)·{\theta }^{\frac{2}{q}}(t)|u_k(t){|}^{2}dt\\ \le {\epsilon }_1{\displaystyle {\int }_{t_1-\delta }^{t_1}}\theta (t)|u_k(t){|}^{q}dt+{\epsilon }_1^{-\frac{2}{q-2}}{\displaystyle {\int }_{t_1-\delta }^{t_1}}{\left({\theta }^{\prime }(t){\theta }^{-\frac{2}{q}}(t)\right)}^{\frac{q}{q-2}}dt,& (44)\end{array}$

where ε₁ > 0 is an arbitrary number.

Based on Equation (36), it is easy to demonstrate that

${\displaystyle {\int }_{t_1-\delta }^{t_1}}{\left({\theta }^{\prime }(t)·{\theta }^{-\frac{2}{q}}(t)\right)}^{\frac{q}{q-2}}dt=$

${\displaystyle {\int }_{t_1-\delta }^{t_1}}({\delta }^{-1}·{\theta }_{*}^{\prime }((t-t_1)/\delta )·{\theta }_{*}^{-\frac{2}{q}}((t-t_1)/\delta ){)}^{\frac{q}{q-2}}dt$

$=[(t-t_1)/\delta =s,t=\delta s+t_1,dt=\delta ds]={\delta }^{-\frac{2}{q-2}}$

${\displaystyle {\int }_{-1}^0}({\theta }_{*}^{\prime }(s)·{\theta }_{*}^{-\frac{2}{q}}(s){)}^{\frac{q}{q-2}}ds$

$\begin{array}{ll}\le C_4^{\frac{q}{q-2}}·{\delta }^{-\frac{2}{q-2}},& (45)\end{array}$

where C₄ is constant from Equation (36) with ν = 2/q (note that C₄ depends on q only).

So from Equation (44) using Equation (45), we obtained

$\begin{array}{ll}{\displaystyle {\int }_{t_1-\delta }^{\tau }}{\theta }^{\prime }(t)|u_k(t){|}^{2}dt\le {\epsilon }_1{\displaystyle {\int }_{t_1-\delta }^{t_1}}\theta (t)|u_k(t){|}^{q}dt+C_5{({\epsilon }_1\delta )}^{-\frac{2}{q-2}},& (46)\end{array}$

where $C_5:=C_4^{\frac{q}{q-2}}$ depends on q only.

Let us estimate the second term on the right-hand side of equality (40). Using (6), we receive

$\begin{array}{l}|{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|B(t,u_k(t)),u_k(t))dt|\le {\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|B(t,u_k(t))||u_k(t)|dt\\ \le L{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k(t){|}^{2}dt.& (47)\end{array}$

Let us estimate the third term on the right-hand side of equality (40), using inequality (4):

$\begin{array}{l}{\displaystyle {\int }_{t_1-\delta }^{\tau }\theta }(t)(f_k(t),u_k(t))dt\le {\displaystyle {\int }_{t_1-\delta }^{\tau }\theta }(t)|f_k(t)||u_k(t))|dt\\ \le {\epsilon }_2{\displaystyle {\int }_{t_1-\delta }^{\tau }\theta }(t)|u_k(t){|}^{2}dt+{\epsilon }_2^{-1}{\displaystyle {\int }_{t_1-\delta }^{\tau }\theta }(t)|f_k(t){|}^{2}dt,& (48)\end{array}$

where ε₂ > 0 is an arbitrary constant.

From Equation (40), using Equations (43), and (46)–(48), we receive

$\begin{array}{l}|u_k(\tau ){|}^2+2(\sigma K_2-L-{\epsilon }_2){\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k(t){|}^{2}dt+(2\sigma K_3-{\epsilon }_1)\\ {\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k(t){|}^{q}dt+2(1-\sigma )K_1{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)\Vert u_k(t)|{|}^{p}dt\\ +2(1-\sigma ){\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)\Phi \left(u_k(t)\right)dt\\ \le C_5{({\epsilon }_1\delta )}^{-\frac{2}{q-2}}+2{\epsilon }_2^{-1}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|f_k(t){|}^{2}dt.& (49)\end{array}$

In Equation (49), using condition (8), we choose σ ∈ (0, 1) such that the inequality σK₂ − L > 0 holds, and then we take ε₁ = σK₃, ε₂ = (σK₂ − L)/2. As a result, we get

$\begin{array}{l}|u_k(\tau ){|}^2+{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)\left[|u_k(t){|}^2+|u_k(t){|}^q+\Vert u_k(t)|{|}^p+\Phi \left(u_k(t)\right)\right]dt\\ \le C_6{\delta }^{-\frac{2}{q-2}}+C_7{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|f_k(t){|}^{2}dt,& (50)\end{array}$

where C₆, C₇ are positive constants dependent on K₁, K₂, K₃, L, and q only.

Since τ ∈ [t₁, t₂] is arbitrary, from Equation (50) and the definition of θ, we obtain

$\begin{array}{l}{\displaystyle \underset{t\in \left[t_1,t_2\right]}{max}}|u_k(t){|}^2+{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}|u_k(t){|}^{2}dt+\\ {\displaystyle {\int }_{t_1}^{t_2}}|u_k(t){|}^{q}dt+{\displaystyle {\int }_{t_1}^{t_2}}\Vert u_k(t)|{|}^{p}dt+{\displaystyle {\int }_{t_1}^{t_2}}\Phi \left(u_k(t)\right)dt\\ \le 2C_6{\delta }^{-\frac{2}{q-2}}+2C_7{\displaystyle {\int }_{t_1-\delta }^{t_2}}|f_k(t){|}^{2}dt.& (51)\end{array}$

From Equation (50) and the definition of f_k, since t₁, t₂ ∈ S and δ > 0 are all arbitrary, it follows that

$\begin{array}{l}\text{the sequence }\left\{u_k\right\}\text{ is bounded in }{L}_{\text{loc}}^{\infty }(S;H),L_{\text{loc}}^2(S;H),L_{\text{loc}}^q(S;H),\\ \text{and }{L}_{\text{loc}}^p(S;V),and& (52)\end{array}$

$\begin{array}{ll}\text{the sequence}\left\{\Phi \left(u_k\right)\right\}\text{ is bounded in }{L}_{\text{loc}}^1(S).& (53)\end{array}$

Step 4 (estimates of derivatives of solution approximations). Now let us find estimates of $u_k^{\prime }$, k ∈ ℕ. Let t₁, t₂, and δ be arbitrary real numbers such that t₁, t₂ ∈ S, t₁ < t₂, and δ > 0. θ is a function defined above. We multiply equality (35) for almost every t ∈ S scalar by $\theta (t)u_k^{\prime }(t)$ and integrate the resulting equality from t₁ − δ to τ ∈ [t₁, t₂]:

$\begin{array}{l}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k^{\prime }(t){|}^{2}dt+{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(g_k(t),u_k^{\prime }(t))dt\\ +{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(B(t,u_k(t)),u_k^{\prime }(t))dt\\ ={\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(f_k(t),u_k^{\prime }(t))dt.& (54)\end{array}$

Since $g_k\in L^2(t_1-\delta ,t_2;H)$ and g_k(t) ∈ ∂Φ_H(u_k(t)) for a. e. t ∈ (t₁ − δ, t₂), Lemma 4.1 implies that the function Φ_H(u_k(·)) is continuous on [t₁ − δ, t₂] and

$\begin{array}{ll}({\Phi }_H(u_k(t)){)}^{\prime }=(g_k(t),u_k^{\prime }(t))\text{ for a.e. }t\in (t_1-\delta ,t_2).& (55)\end{array}$

Taking into account Equation (55), we can estimate the second term on the left side of Equation (54) as follows:

$\begin{array}{l}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(g_k(t),u_k^{\prime }(t))dt={\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)({\Phi }_H(u_k(t)){)}^{\prime }dt\\ ={\Phi }_H\left(u_k(\tau )\right)-{\displaystyle {\int }_{t_1-\delta }^{\tau }}{\theta }^{\prime }(t){\Phi }_H\left(u_k(t)\right)dt\\ \ge {\Phi }_H\left(u_k(\tau )\right)-{\displaystyle \underset{t\in \left[t_1-\delta ,t_1\right]}{max}}{\theta }^{\prime }(t){\displaystyle \underset{t_1-\delta }{\overset{t_1}{\int }}}{\Phi }_H\left(u_k(t)\right)dt.& (56)\end{array}$

By inequality (4) with ε = 4, taking into Equation (6), we receive

$\begin{array}{l}|{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(B(t,u_k(t)),u_k^{\prime }(t))dt|\le {\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|B(t,u_k(t))||u_k^{\prime }(t)|dt\\ \le L{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k(t)\Vert u_k^{\prime }(t)|dt\le 4L^2{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k(t){|}^{2}dt\\ +\frac{1}{4}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k^{\prime }(t){|}^{2}dt,& (57)\end{array}$

$\begin{array}{l}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)(f_k(t),u_k^{\prime }(t))dt\le 4{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|f_k(t){|}^{2}dt\\ +\frac{1}{4}{\displaystyle {\int }_{t_1-\delta }^{\tau }}\theta (t)|u_k^{\prime }(t){|}^{2}dt.& (58)\end{array}$

From Equation (54), using Equations (56)–(58) and

$\begin{array}{l}{\displaystyle \underset{t\in \left[t_1-\delta ,t_1\right]}{max}}{\theta }^{\prime }(t)={\delta }^{-1}{\displaystyle \underset{t\in \left[t_1-\delta ,t_1\right]}{max}}{\theta }_{*}^{\prime }((t-t_1)/\delta )\le C_8{\delta }^{-1},\\ C_8:={\displaystyle \underset{s\in \left[-1,0\right]}{max}}{\theta }_{*}^{\prime }(s),\end{array}$

we have

$\begin{array}{l}\frac{1}{2}{\displaystyle {\int }_{t_1}^{\tau }}|u_k^{\prime }(t){|}^{2}dt+{\Phi }_H\left(u_k(\tau )\right)\le 4{\displaystyle {\int }_{t_1-\delta }^{\tau }}|f_k(t){|}^{2}dt\\ +4L^2{\displaystyle {\int }_{t_1-\delta }^{\tau }}|u_k(t){|}^{2}dt+C_8{\delta }^{-1}{\displaystyle {\int }_{t_1-\delta }^{t_1}}{\Phi }_H\left(u_k(t)\right)dt.& (59)\end{array}$

Since τ ∈ [t₁, t₂] is arbitrary, from Equation (59) by the definition of Φ_H and condition $({\mathcal{A}}_3)$ (remind that u_k(t) ∈ V for a.e. t ∈ S), we have

$\begin{array}{l}{\text{less sup }}_{t\in \left[t_1,t_2\right]}\Vert u_k(t)|{|}^p+{\displaystyle {\int }_{t_1}^{t_2}}|u_k^{\prime }(t){|}^{2}dt\\ \le C_9[{\displaystyle {\int }_{t_1-\delta }^{t_2}}|f_k(t){|}^{2}dt+{\displaystyle {\int }_{t_1-\delta }^{t_2}}|u_k(t){|}^{2}dt+{\delta }^{-1}{\displaystyle {\int }_{t_1-\delta }^{t_1}}\Phi \left(u_k(t)\right)dt],& (60)\end{array}$

where C₉ > 0 is a positive constant dependent on K₁ and L only.

From Equation (60), taking into account (51), we obtain

$\begin{array}{l}\underset{t\in [t_1,t_2]}{\text{ess }\sup }\Vert u_k(t)){\Vert }^p+{\displaystyle {\int }_{t_1}^{t_2}|}{u}_{k^{\prime }}(t){|}^{2}dt\le C_{10}[{\delta }^{-\frac{2}{q-2}}+{\delta }^{-\frac{q}{q-2}}\\ +{\displaystyle {\int }_{t_1-2\delta }^{t_2}|}{f}_k(t){|}^{2}dt+{\delta }^{-1}{\displaystyle {\int }_{t_1-2\delta }^{t_1}|}{f}_k(t){|}^{2}dt],& (61)\end{array}$

where C₁₀ > 0 is a positive constant dependent on K₁, K₂, K₃, L, and q only.

From the estimate (4) and the definition of f_k, since t₁, t₂ ∈ S and δ > 0 are arbitrary, it implies that

$\begin{array}{ll}\text{the sequence }{\left\{u_k\right\}}_{k=1}^{+\infty }\text{ is bounded in }{L}_{\text{loc}}^{\infty }(S;V),& (62)\end{array}$

$\begin{array}{ll}\text{the sequence }{\left\{u_k^{\prime }\right\}}_{k=1}^{+\infty }\text{ is bounded in }{L}_{\text{loc}}^2(S;H).& (63)\end{array}$

From Equations (6) and (51) we have

$\begin{array}{l}{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}|B(t,u_k(t)){|}^{2}dt\le L^2{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}|u_k(t){|}^{2}dt\le C_{11}\\ (1+{\displaystyle \underset{t_1-1}{\overset{t_2}{\int }}}|f_k(t){|}^{2}dt)\le C_{12},& (64)\end{array}$

where C₁₁, C₁₂ are positive constants independent on k ∈ ℕ.

From Equations (35), (63), and (64) and the definition of f_k, we get that

$\begin{array}{ll}\text{the sequence }{\left\{g_k\right\}}_{k=1}^{+\infty }\text{ is bounded in }{L}_{\text{loc}}^2(S;H).& (65)\end{array}$

Step 5 (passing the limit). Since V is reflexive Banach space, H is Hilbert space, and V embeds in H by compact injection, from Equations (52), (62), (63), (65), and Proposition 2.7, we have the existence of functions $u\in L_{\text{loc}}^{\infty }(S;V)\cap L_{\text{loc}}^q(S;H)\cap H_{\text{loc}}^1(S;H)$, $g\in L_{\text{loc}}^2(S;H)$, and a subsequence of the sequence ${\left\{u_k,g_k\right\}}_{k=1}^{+\infty }$ (until denoted by ${\left\{u_k,g_k\right\}}_{k=1}^{+\infty }$) such that

$\begin{array}{lll}{u}_k{\to }_{k\to \infty }u& *\text{-weakly in }{L}_{\text{loc}}^{\infty }(S;V),\text{ and weakly in }{L}_{\text{loc}}^p(S;V),& (66)\end{array}$

$\begin{array}{lll}{u}_k{\to }_{k\to \infty }u& \text{weakly in }{L}_{\text{loc}}^q(S;H),\text{ and weakly in }{H}_{\text{loc}}^1(S;H),& (67)\end{array}$

$\begin{array}{lll}{u}_k{\to }_{k\to \infty }u& \text{in }C(S;H),& (68)\end{array}$

$\begin{array}{lll}{g}_k{\to }_{k\to \infty }g& \text{weakly in }{L}_{\text{loc}}^2(S;H).& (69)\end{array}$

From Equation (68) and condition $(\mathcal{B})$, for each t₀ < T, we have

$\begin{array}{l}{\displaystyle \underset{t_0}{\overset{T}{\int }}}|B(t,u_k(t))-B(t,u(t)){|}^{2}dt\le L^2{\displaystyle \underset{t_0}{\overset{T}{\int }}}|u_k(t)-u(t){|}^{2}dt{\displaystyle \underset{k\to \infty }{\to }}0.\end{array}$

Thus, we obtain

$\begin{array}{ll}B(·,u_k(·)){\to }_{k\to \infty }B(·,u(·))\text{ strongly in }{L}_{\text{loc}}^2(S;H).& (70)\end{array}$

Let v ∈ H, φ ∈ C(S) be arbitrary while suppφ is compact. For a.e. t ∈ S, we multiply equality (35) by v and φ(t), and then integrate in t on S. As a result, we obtain equality

$\begin{array}{l}{\displaystyle \underset{S}{\int }}(u_k^{\prime }(t),v)\phi (t)dt+{\displaystyle \underset{S}{\int }}(g_k(t),v)\phi (t)+{\displaystyle \underset{S}{\int }}(B(t,u_k(t)),v)\phi (t)dt\\ ={\displaystyle \underset{S}{\int }}(f_k(t),v)\phi (t)dt,k\in \mathbb{N}.& (71)\end{array}$

We pass to the limit in Equation (71) as k → ∞, taking into account (67), (69), (70), and the convergence of ${\left\{f_k\right\}}_{k=1}^{\infty }$ to f in $L_{\text{loc}}^2(S;H)$. As a result, since v, φ are arbitrary, for a.e. t ∈ S, we obtain the equality

$\begin{array}{l}{u}^{\prime }(t)+g(t)+B(t,u(t))=f(t)\text{ in }H.\end{array}$

Step 6 (proof that u(t) ∈ D(∂Φ_H) and g(t) ∈ ∂Φ_H(u(t)) for a. e. t ∈ S). Let k ∈ ℕ be an arbitrary number. Since u_k(t) ∈ D(∂Φ_H) and g_k(t) ∈ ∂Φ_H(u_k(t)) for a.e. t ∈ S, applying the monotonicity of the sub-differential ∂Φ_H, we obtain that for a.e. t ∈ S the following inequality holds:

$\begin{array}{ll}(g_k(t)-v^{*},u_k(t)-v)\ge 0\forall \left[v,v^{*}\right]\in \partial {\Phi }_H.& (72)\end{array}$

Let τ ∈ S and h > 0 be arbitrary numbers. We integrate (72) in t from τ−h to τ:

$\begin{array}{ll}{\displaystyle {\int }_{\tau -h}^{\tau }}(g_k(t)-v^{*},u_k(t)-v)dt\ge 0\forall \left[v,v^{*}\right]\in \partial {\Phi }_H.& (73)\end{array}$

Now we pass to the limit in Equation (73) as k → ∞, according to Equations (68) and (69). As a result, we obtain

$\begin{array}{ll}{\displaystyle {\int }_{\tau -h}^{\tau }}(g(t)-v^{*},u(t)-v)dt\ge 0\forall \left[v,v^{*}\right]\in \partial {\Phi }_H.& (74)\end{array}$

The monograph [25, Theorem 2] and Equation (74) imply that for every $\left[v,v^{*}\right]\in \partial {\Phi }_H$ there exists a set of measure zero R_[v,v*] ⊂ S such that for all $\tau \in S\backslash R_{\left[v,v^{*}\right]}$ we have u(τ) ∈ V, g(τ) ∈ H

$\begin{array}{ll}0\le {\displaystyle \underset{h\to +0}{lim}}\frac{1}{h}{\displaystyle {\int }_{\tau -h}^{\tau }}(g(t)-v^{*},u(t)-v)dt=(g(\tau )-v^{*},u(\tau )-v)\ge 0.& (75)\end{array}$

Let us demonstrate that there exists a set of measure zero R ⊂ S such that

$\begin{array}{ll}\forall \tau \in S\backslash R:(g(\tau )-v^{*},u(\tau )-v)\ge 0\forall \left[v,v^{*}\right]\in \partial {\Phi }_H.& (76)\end{array}$

Since V and H are separable spaces, there exists a countable set F⊂∂Φ_H, which is dense in ∂Φ_H. Denote $R:={\cup }_{\left[v,v^{*}\right]\in F}{R}_{\left[v,v_{*}\right]}$. Since the set F is countable and any countable union of sets of measure zero is a set of measure zero, then R is a set of measure zero.

Therefore, for any τ ∈ S\R inequality (76) holds for every [v, v^*] ∈ F. Let $\left[\hat{v},{\hat{v}}^{*}\right]$ be an arbitrary element from ∂Φ_H. Then from the density F in ∂Φ_H we have the existence of a sequence ${\left\{\left[v_l,v_l^{*}\right]\right\}}_{l=1}^{\infty }\subset F$ such that v_l → v in V, $v_l^{*}\to v^{*}$ in H, and for every τ ∈ S\R

$\begin{array}{ll}(g(\tau )-v_l^{*},u(\tau )-v_l)\ge 0\forall l\in \mathbb{N}.& (77)\end{array}$

Thus, passing to the limit in inequality (77) as l → ∞, we obtain (g(τ)−v^*, u(τ)−v) ≥ 0 for every τ ∈ S\R. Hence, we have Equation (76), i.e., for a.e. t ∈ S, the following holds:

$\begin{array}{l}(g(t)-v^{*},u(t)-v)\ge 0\forall \left[v,v^{*}\right]\in \partial {\Phi }_H.\end{array}$

From this, according to the maximal monotonicity of ∂Φ_H, we obtain that [u(t), g(t)] ∈ ∂Φ_H for a.e. t ∈ S, i.e., u(t) ∈ D(∂Φ_H) and g(t) ∈ ∂Φ_H(u(t)) for a.e. t ∈ S. Thus, function u is the solution of the problem P(Φ, B, f), and therefore P(Φ_H, B, f).

Step 7 (completion of proof). Estimates (9) and (10) of the solution of the problem P(Φ, B, f) follow directly from estimates (51) (given that ${\int }_{t_1}^{t_2}\Phi \left(u_k(t)\right)dt\ge 0$) and (4), convergence (66)–(68) and Proposition 2.5.     □

5 Comments on the main results

Let us introduce an example of the problem that is studied here. Let n ∈ ℕ, Ω be a bounded domain in ℝⁿ, ∂Ω be the boundary of Ω, and ∂Ω be the piecewise surface. We put Q: = Ω × S, Σ: = ∂Ω × S, and Ω_t: = Ω × {t} ∀t ∈ S. For an arbitrary measurable set F⊂ℝ^k, where k = n or k = n + 1, and r ∈ [1, ∞], let L^r(F) be the standard Lebesgue space with norm $\Vert ·|{|}_{L^r(F)}$. Let $L_{\text{loc}}^r(\overline{Q})$ be the linear space of classes of equivalent functions defined on Q such that their restrictions on any bounded measurable set Q′⊂Q belong to L^r(Q′). For r ∈ (1, ∞), we denote by $W^{1,r}(\Omega )=\left\{v\in L^r(\Omega )|v_{x_i}\in L^r(\Omega ),i=\overline{1,n}\right\}$ the standard Sobolev space with norm $\Vert v|{|}_{W^{1,r}(\Omega )}:=(\Vert v{\Vert }_{L^r(\Omega )}^r+\Vert \nabla v{\Vert }_{L^r(\Omega )}^r{)}^{1/r}$, where ∇u: = (u_x1, …, u_xn) [see, e.g., Brezis [24]].

Let p > 2 and K be a nonempty convex closed set in W^{1, p}(Ω), which contains 0. We consider the problem: find a function $u\in L_{\text{loc}}^p(\overline{Q})$ such that $u_{x_i}\in L_{\text{loc}}^p(\overline{Q})$, $i=\overline{1,n}$, $u_t\in L_{\text{loc}}^2(\overline{Q})$, and, for a.e. t ∈ S, we have u(·, t) ∈ K and

${\displaystyle \underset{{\Omega }_t}{\int }}[u_t(v-u)+|\nabla u{|}^{p-2}\nabla u\nabla (v-u)+|u{|}^{p-2}u(v-u)+a(x)u(v-u)$

$\begin{array}{ll}+(v-u){\displaystyle \underset{\Omega }{\int }}b(x,y,t)u(y,t)dy]dx\ge {\displaystyle \underset{{\Omega }_t}{\int }}f(v-u)dx\forall v\in K,& (78)\end{array}$

where $f\in L_{\text{loc}}^2(\overline{Q}),$ a ∈ L^∞(Ω), and S ∋ t → b(·, ·, t) ∈ L²(Ω × Ω) are given.

This problem is called problem (78), and a function u is its solution.

Note that in cases K = W^{1, p}(Ω), this problem is equivalent to the problem of finding a weak solution to a problem without initial conditions for a nonlinear integro-differential parabolic equation:

$\begin{array}{l}{u}_t-\text{div}(|\nabla u{|}^{p-2}\nabla u)+|u{|}^{p-2}u+a(x)u+{\displaystyle \underset{\Omega }{\int }}b(x,y,t)u(y,t)dy\\ =f(x,t),(x,t)\in Q,\end{array}$

$\begin{array}{l}\frac{\partial u}{\partial \nu }=0.\end{array}$

We remark that problem (78) can be written more abstractly. Indeed, after appropriate identification of functions and functionals, we have continuous and dense embedding

$W^{1,p}(\Omega )\subset L^2(\Omega )\subset {(W^{1,p}(\Omega ))}^{\prime },$

where (W^{1, p}(Ω))′ is dual to W^{1, p}(Ω) space. Clearly, for any h ∈ L²(Ω) and v ∈ W^{1, p}(Ω), we have 〈h, v〉 = (h, v), where 〈·, ·〉 is the notation for action of element of (W^{1, p}(Ω))′ on element of W^{1, p}(Ω), and (·, ·) is a scalar product in L²(Ω). Thus, we can use the notation (·, ·) instead of 〈·, ·〉.

Now, we denote V: = W^{1, p}(Ω), H: = L²(Ω) and define operators A: V → V′ and B(t, ·):H → H, t ∈ S, as follows:

$\begin{array}{ll}(A(v),w)={\displaystyle \underset{\Omega }{\int }}[|\nabla v{|}^{p-2}\nabla v\nabla w+|v{|}^{p-2}vw+avw]dx,v,w\in V,& (79)\end{array}$

$\begin{array}{ll}B(t,v)(·):={\displaystyle \underset{\Omega }{\int }}b(·,y,t)v(y)dy,v\in H,t\in S.& (80)\end{array}$

Then problem (78) can be rewritten as follows: find a function $u\in L_{\text{loc}}^p(S;V)$ such that $u^{\prime }\in L_{\text{loc}}^2(S;H)$ and, for a.e. t ∈ S, we have u(t) ∈ K and

$\begin{array}{ll}(u^{\prime }(t)+A(u(t))+B(t,u(t)),v-u(t))\ge (f(t),v-u(t))\forall v\in K,& (81)\end{array}$

where $f\in L_{\text{loc}}^2(S;H)$ is given function.

We remark that, for a.e. t ∈ S, variational inequality (81) can be written as

$\begin{array}{l}(u^{\prime }(t)+A(u(t))+B(t,u(t))-f(t),v-u(t))+I_K(v)-I_K(u(t))\end{array}$

$\begin{array}{ll}\ge 0\forall v\in V,& (82)\end{array}$

where

$\begin{array}{ll}{I}_K(v):=\{\begin{array}{l}0,\text{ if }v\in K,\\ +\infty ,\text{ if }v\in V\backslash K.\end{array}& (83)\end{array}$

We can write inequality (82) as follows:

$\begin{array}{c}{I}_K(v)\ge I_K(u(t))+(-u^{\prime }(t)-A(u(t))-B(t,u(t))+f(t),v-u(t))\\ \forall v\in V.& (84)\end{array}$

The functional I_K from V to ℝ_∞ is proper, convex and lower semicontinuous. By the definition of the subdifferential $\partial I_K:V\to 2^{V^{\prime }}$ inequality (84) is equivalent to inclusion

$\partial I_K(u(t))\ni -u^{\prime }(t)-A(u(t))-B(t,u(t))+f(t),$

i.e.,

$\begin{array}{ll}{u}^{\prime }(t)+A(u(t))+\partial I_K(u(t))+B(t,u(t))\ni f(t).& (85)\end{array}$

We define

$\begin{array}{ll}\text{Ψ}(v):={\displaystyle \underset{\Omega }{\int }}[p^{-1}(|\nabla v{|}^p+|v{|}^p)+2^{-1}a|v{|}^2]dx,v\in V,& (86)\end{array}$

and

$\begin{array}{ll}\Phi (v):=\text{Ψ}(v)+I_K(v),v\in V.& (87)\end{array}$

The functionals Ψ and Φ from V to ℝ_∞ are proper, convex and lower semicontinuous. As easy to demonstrate, we have ∂Ψ(v) = {A(v)}⊂V′ for each v ∈ V, and

$\begin{array}{ll}\partial \Phi (v):=A(v)+\partial I_K(v),v\in V.& (88)\end{array}$

From the above [see, in particular, Equations (85) and (88)], it follows that the problem (79) can be written as such a sub-differential inclusion: find a function $u\in L_{\text{loc}}^p(S;V)$ such that $u^{\prime }\in L_{\text{loc}}^2(S;H)$ and, for a.e. t ∈ S, u(t) ∈ D(∂Φ) and

$\begin{array}{ll}{u}^{\prime }(t)+\partial \Phi (u(t))+B(t,u(t))\ni f(t)\text{ in }H.& (89)\end{array}$

So problem (78) is a partial case of the problem P(Φ, B, f). Based on this, let's illustrate the main results of this study (see Theorems 1, 2).

COROLLARY 5.1. Let the following condition hold:

$\begin{array}{ll}{\text{ess sup }}_{t\in S}\Vert b(·,·,t)|{|}_{L^2(\Omega \times \Omega )}<{\text{ess inf}}_{x\in \Omega }a(x).& (90)\end{array}$

Then problem (78) has a unique solution. In addition, it belongs to the space $L_{\mathrm{\text{loc}}}^{\infty }(S;W^{1,p}(\Omega ))\cap H_{\text{loc}}^1(S;L^2(\Omega ))$ and for arbitrary t₁, t₂ ∈ S, t₁ < t₂, δ > 0 satisfies the estimates:

$\begin{array}{ll}{\displaystyle \underset{t\in \left[t_1,t_2\right]}{max}}{\displaystyle \underset{\Omega }{\int }}|u(x,t){|}^{2}dx+{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}{\displaystyle \underset{\Omega }{\int }}[|u(x,t){|}^2+|u(x,t){|}^p+|\nabla u(x,t){|}^p]dxdt& (91)\end{array}$

$\begin{array}{ll}\le C_{15}[{\delta }^{-\frac{2}{q-2}}+{\displaystyle \underset{t_1-\delta }{\overset{t_2}{\int }}}{\displaystyle \underset{\Omega }{\int }}|f(x,t){|}^{2}dxdt],& (92)\end{array}$

$\begin{array}{l}{\text{ess }sup}_{t\in \left[t_1,t_2\right]}{\displaystyle \underset{\Omega }{\int }}[|u(x,t){|}^p+|\nabla u(x,t){|}^p]dx+{\displaystyle \underset{t_1}{\overset{t_2}{\int }}}{\displaystyle \underset{\Omega }{\int }}|u_t(x,t){|}^{2}dxdt\\ \le C_{16}[max\left\{{\delta }^{-\frac{2}{q-2}},{\delta }^{-\frac{q}{q-2}}\right\}+{\displaystyle \underset{t_1-2\delta }{\overset{t_2}{\int }}}{\displaystyle \underset{\Omega }{\int }}|f(x,t){|}^{2}dxdt\end{array}$

$\begin{array}{ll}+{\delta }^{-1}{\displaystyle \underset{t_1-2\delta }{\overset{t_1}{\int }}}{\displaystyle \underset{\Omega }{\int }}|f(x,t){|}^{2}dxdt],& (93)\end{array}$

where C₁₅, C₁₆ are positive constants depending on ${\mathrm{\text{ess sup }}}_{t\in S}\Vert b(·,·,t)|{|}_{L^2(\Omega \times \Omega )}$, ess inf_{x ∈ Ω}a(x), and p only.

Proof. [Proof of Corollary 5.1] We need to demonstrate that functional Φ, defined in Equations (83)—(87), and family of operators B(t, ·), t ∈ S, defined in Equation 80, satisfy the conditions of Theorems 1, 2.

Writing the functional Ψ defined in Equation (86) in the form

$\begin{array}{ll}\text{Ψ}(v)=p^{-1}\Vert v{\Vert }_{W^{1,p}(\Omega )}^p+2^{-1}{\displaystyle \underset{\Omega }{\int }}a|v{|}^{2}dx,v\in W^{1,p}(\Omega ),& (94)\end{array}$

we obtain that the functional Ψ is proper and dom(Ψ) = W^{1, p}(Ω).

Note that for arbitrary r ≥ 2, function $F_r(\xi )=|\xi {|}^r$, ξ ∈ ℝⁿ, is convex. Indeed, for all α ∈ [0, 1], we have

$F_r(\alpha \xi +(1-\alpha )\eta )=|\alpha \xi +(1-\alpha )\eta {|}^r\le {(\alpha |\xi |+(1-\alpha )|\eta |)}^r$

$\begin{array}{ll}\le \alpha |\xi {|}^r+(1-\alpha )|\eta {|}^r=\alpha F_r(\xi )+(1-\alpha )F_r(\eta ),\xi ,\eta \in {ℝ}^n.& (95)\end{array}$

Here we used the convex function $g_r(s)=s^r$, s ∈ [0, +∞), since $g_r^{″}(s)=r(r-1)s^{r-2}>0$ for all s ∈ (0, +∞).

From Equation (95), with r = p and r = 2, it is easy to see that functional Ψ is convex, hence functional Φ satisfies the condition (${\mathcal{A}}_1$).

Let $v_k\underset{k\to \infty }{\to }v$ in W^{1, p}(Ω). Then $\Vert v_k|{|}_{W^{1,p}(\Omega )}\underset{k\to \infty }{\to }\Vert v|{|}_{W^{1,p}(\Omega )}$ and $v_k\underset{k\to \infty }{\to }v$ in L²(Ω). From this, it follows:

$\begin{array}{ll}\Vert v_k{\Vert }_{W^{1,p}(\Omega )}^p{\to }_{k\to \infty }\Vert v{\Vert }_{W^{1,p}(\Omega )}^p,& (96)\end{array}$

$|{\displaystyle \underset{\Omega }{\int }}a|v_k{|}^{2}dx-{\displaystyle \underset{\Omega }{\int }}a|v{|}^{2}dx|\le {\displaystyle \underset{\Omega }{\int }}a|v_k^2-v^2|dx={\displaystyle \underset{\Omega }{\int }}a|v_k+v\Vert v_k-v|dx$

$\begin{array}{ll}\le \text{ess sup }a·(\Vert v_k|{|}_{L^2(\Omega )}+\Vert v|{|}_{L^2(\Omega )})·\Vert v_k-v|{|}_{L^2(\Omega )}{\to }_{k\to \infty }0.& (97)\end{array}$

From Equations (94), (96), and (97), it follows that the functional Ψ is lower semicontinuous, hence functional Φ satisfies the condition (${\mathcal{A}}_2$).

Since a > 0 a.e. on Ω, then [see Equation (94)]

$\text{Ψ}(v)\ge p^{-1}\Vert v{\Vert }_{W^{1,p}(\Omega )}^p,v\in W^{1,p}(\Omega ).$

Hence, given that I_K(v) ≥ 0, v ∈ V, condition (${\mathcal{A}}_3$) holds with $K_2:=p^{-1}$.

It is easy to show that

$\partial \text{Ψ}(v)=\left\{A(v)\right\}\subset {(W^{1,p}(\Omega ))}^{\prime }\forall v\in W^{1,p}(\Omega ),$

where A(·) is defined in Equation (79).

Then for any $v_1,v_2\in W^{1,p}(\Omega )$ we have

$(A(v_1)-A(v_2),v_1-v_2)={\displaystyle \underset{\Omega }{\int }}[(|\nabla v_1{|}^{p-2}\nabla v_1-|\nabla v_2{|}^{p-2}\nabla v_2)$

$(\nabla v_1-\nabla v_2)$

$\begin{array}{ll}+(|v_1{|}^{p-2}{v}_1-|v_2{|}^{p-2}{v}_2)(v_1-v_2)+a|v_1-v_2{|}^2]dx.& (98)\end{array}$

Since the function $F_p(\xi )=|\xi {|}^r$, ξ ∈ ℝⁿ, is convex, from the convexity criterion we have

$\begin{array}{ll}(\nabla F_p(\xi )-\nabla F_p(\eta ))(\xi -\eta )\ge 0,\xi ,\eta \in {ℝ}^n.& (99)\end{array}$

Since $\nabla F_p(\xi )=p|\xi {|}^{p-2}\xi ,\xi \in {ℝ}^n,$ then from Equation (99) it follows:

$\begin{array}{ll}{\displaystyle \underset{\Omega }{\int }}[(|\nabla v_1{|}^{p-2}\nabla v_1-|\nabla v_2{|}^{p-2}\nabla v_2)(\nabla v_1-\nabla v_2)dx\ge 0.& (100)\end{array}$

By Bokalo [9], for arbitrary s₁, s₂ ∈ ℝ, the inequality

$\begin{array}{l}(|s_1{|}^{p-2}{s}_1-|s_2{|}^{p-2}{s}_2)(s_1-s_2)\ge 2^{2-p}|s_1-s_2{|}^p\end{array}$

holds. Hence, for all v₁, v₂ ∈ L_p(Ω), we have

$\begin{array}{ll}{\displaystyle \underset{\Omega }{\int }}(|v_1{|}^{p-2}{v}_1-|v_2{|}^{p-2}{v}_2)(v_1-v_2)dx\ge 2^{2-p}{\displaystyle \underset{\Omega }{\int }}|v_1-v_2{|}^{p}dx.& (101)\end{array}$

Using Hölder's inequality (see Proposition 2.3) with r = p/2, we have this chain of inequalities:

${\displaystyle \underset{\Omega }{\int }}|v_1-v_2{|}^{2}dx\le ({\displaystyle \underset{\Omega }{\int }}{1}^{r^{\prime }}dx{)}^{\frac{1}{r^{\prime }}}({\displaystyle \underset{\Omega }{\int }}|v_1-v_2{|}^{p}dx{)}^{\frac{1}{r}}=$

${({\text{mes}}_n\Omega )}^{\frac{p-2}{p}}({\displaystyle \underset{\Omega }{\int }}|v_1-v_2{|}^{p}dx{)}^{\frac{2}{p}}.$

From this, we obtain

$\begin{array}{l}{\displaystyle \underset{\Omega }{\int }}|v_1-v_2{|}^{p}dx\ge ({\text{mes}}_n\Omega {)}^{\frac{2-p}{2}}({\displaystyle \underset{\Omega }{\int }}|v_1-v_2{|}^{2}dx{)}^{\frac{p}{2}}\\ =({\text{mes}}_n\Omega {)}^{\frac{2-p}{2}}\Vert v_1-v_2{\Vert }_{L^2(\Omega )}^p.& (102)\end{array}$

From Equations (101), (102) it follows:

$\begin{array}{l}{\displaystyle \underset{\Omega }{\int }}(|v_1{|}^{p-2}{v}_1-|v_2{|}^{p-2}{v}_2)(v_1-v_2)dx\end{array}$

$\begin{array}{ll}\ge 2^{2-p}({\text{mes}}_n\Omega {)}^{\frac{2-p}{2}}\Vert v_1-v_2{\Vert }_{L^2(\Omega )}^p.& (103)\end{array}$

Also, we have

$\begin{array}{ll}{\displaystyle \underset{\Omega }{\int }}a|v_1-v_2{|}^{2}dx\ge ({\text{ess inf}}_{\Omega }a){\displaystyle \underset{\Omega }{\int }}|v_1-v_2{|}^{2}dx.& (104)\end{array}$

Hence, from Equation (98), using Equations (100), (103), and (104), we have

$\begin{array}{l}(A(v_1)-A(v_2),v_1-v_2)\ge K_2\Vert v_1-v_2{\Vert }_{L^2(\Omega )}^2+K_3\Vert v_1-v_2{\Vert }_{L^2(\Omega )}^p,\\ v_1,v_2\in W^{1,p}(\Omega ),& (105)\end{array}$

where K₂: = ess inf_Ωa, $K_3:=2^{2-p}({\mathrm{\text{mes}}}_n\Omega {)}^{\frac{2-p}{2}}.$

From Equation (94) and the monotonicity of I_K(·) it follows condition (${\mathcal{A}}_4$) with q = p.

Let us prove that condition ($\mathcal{B}$) holds. Since Equation (80), we have for almost all t ∈ S and for all $v_1,v_2\in L^2(\Omega )$:

$\Vert B(t,v_1)(·)-B(t,v_2)(·)|{|}_{L^2(\Omega )}=||{\displaystyle \underset{\Omega }{\int }}b(·,y,t)(v_1(y)-v_2(y))dy|{|}_{L^2(\Omega )}$

$\le {\displaystyle \underset{\Omega }{\int }}|v_1(y)-v_2(y)|·\Vert b(·,y,t)|{|}_{L^2(\Omega )}dy\le \Vert b(·,·,t)|{|}_{L^2(\Omega \times \Omega )}·$

$\Vert v_1-v_2|{|}_{L^2(\Omega )}\le L\Vert v_1-v_2|{|}_{L^2(\Omega )},$

where $L:={\mathrm{\text{ess sup }}}_{t\in S}\Vert b(·,·,t)|{|}_{L^2(\Omega \times \Omega )}$, i.e., condition ($\mathcal{B}$) holds.

From the above, it follows that in this case, condition (8) has form (90). Estimates (91) and (93) are derived directly from estimates (9) and (10).

6 Conclusion

We investigated the problem without initial conditions for some strictly nonlinear functional-differential variational inequalities in the form of sub-differential inclusions with functionals. The conditions for the existence of a unique solution to this problem in the absence of restrictions on the solution's behavior and the growth of input data when the time variable is directed to −∞ have been obtained. There are also estimates of the solution to the researched problem provided.

The results obtained here can be used to study mathematical models in many fields of science, such as ecology, economics, physics, cybernetics, etc.

In the future, it would be worthwhile to obtain similar results for functional-differential variational inequalities that do not have the form of subdifferential inclusions with functionals.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

Author contributions

MB: Conceptualization, Investigation, Methodology, Validation, Writing – original draft, Writing – review & editing. IS: Investigation, Methodology, Validation, Writing – original draft, Writing – review & editing. TB: Investigation, Validation, Writing – original draft, Writing – review & editing.

Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this study.

Acknowledgments

The authors thank all the reviewers for their constructive suggestions.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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