Existence of solutions for space-fractional parabolic hemivariational inequalities

This paper is devoted to the existence of solutions for space-fractional parabolic hemivariational inequalities by means of the well-known surjectivity result for multivalued ($S_+$) type mappings.


1.
Introduction. Fractional-order-in-space mathematical models, in which an integer-order differential operator is replaced by a corresponding fractional one, are becoming increasingly popular, since they provide an adequate description of many processes that exhibit anomalous diffusion. This is due to the fact that the nonlocal nature of the fractional operators enables one to capture the spatial heterogeneity that characterizes these processes. The study of problems in the framework of integro-differential equations is quite recent and has risen a great interest particularly in connection with problems involving nonlocal effects.

YONGJIAN LIU, ZHENHAI LIU AND CHING-FENG WEN
The aim of this paper is to prove the existence of at least one solution for nonlocal parabolic hemivariational inequalities as follows: where Ω c := R N \ Ω, Ω ⊂ R N is an open bounded set with Lipschitz boundary. The nonlocal operator L K defined as follows: where K : R N \ {0} → (0, +∞) is a function satisfying the assumption (A): (i) γK ∈ L 1 (R N ), where γ(x) = min{|x| 2 , 1}; (ii) there exists λ > 0 such that K(x) ≥ λ|x| −(N +2s) , ∀x ∈ R N \ {0}; (iii) K(x) = K(−x), ∀x ∈ R N \ {0}. A typical example for K is given by K(x) = |x| −(N +2s) for s ∈ (0, 1)(N > 2s). In this case L K is the fractional Laplace operator (−∆) s , which (up to normalization factor) is defined as ∂J(·) denotes the generalized subdifferential in the sense of Clarke (cf. [7,22]), f : We remark that the Dirichlet datum is given in Ω c = R N \ Ω and not simply on ∂Ω, consistently with the non-local character of the operator L K .
Recently, Liu and Tan [19] obtained an existence result for nonlocal elliptic hemivariational inequalities with Dirichlet boundary condition by use of pseudomonotone theory. While Teng [33] and Xi et al. [35] established multiplicity of weak solutions to nonlocal elliptic hemivariational inequalities by using the nonsmooth critical point theory and nonsmooth version of the three-critical-points theorem under the framework of the nonsmooth functional. However, to the best of our knowledge, the mathematical literature dedicated to the existence results for nonlocal parabolic hemivariatinal inequalities are still untreated and this fact is the motivation of the present work.
In this paper, we show the existence of at least one solution for the nonlocal parabolic hemivariational inequalities. The basic tools used in our paper are the surjectivity result for (S + ) and coercive operators, properties of the generalized subdifferential in the sense of Clarke. We believe that our result gives a natural approach to the theory of the nonlocal evolutional hemivariational inequalities. Furthermore, the hypotheses we assume on the nonlinear term are general and verifiable. We emphasize that our methods in this paper are also applicable to periodic and anti-periodic nonlocal evolutional hemivariational inequalities.
2. Mathematical framework. Let E be a real reflexive Banach space densely and continuously imbedded in a real Hilbert space H. Identifying H with its dual, we have E ⊆ H ⊆ E * ,where E * stands for the dual of E . The norm of any Banach space B is denoted by · B . The duality pairing between B and its dual B * is denoted by ·, · B .
We recall some preliminary material on function spaces and norms. Denote where u| Ω represents the restriction to Ω of function u(x).
It is easy to check that X and X 0 are norm linear spaces endowed with the norm: In X 0 , we may also use the norm with the inner product for u, v ∈ X 0 Then the two norms defined by (4) and (5) in X 0 are equivalent (see, [27] for details).
In the sequel, we denote by H s (Ω) the usual fractional Sobolev space endowed with the norm ( the Gagliardo norm) We remark that even in the model case in which K(x) = |x| −(N +2s) , the norm in (4) and (7) are not the same, because Ω × Ω is strictly contained in Q (this makes the classical fractional Sobolev space approach not sufficient for studying the problem). For further details on the fractional Sobolev spaces we refer to [25] and the references therein.
We stress that C 2 0 (Ω) ⊆ X 0 (see, e.g., [28]). So X 0 is non-empty and dense in L 2 (Ω). We may collect the useful facts on the space X 0 ( see [27], for more details) as follows.
where 2 * = 2N N −2s . Furthermore, the embedding is compact if p ∈ [1, 2 * ). Next, let us recall some useful facts from the theory of nonlinear operators of monotone type. Let L : D(L) ⊆ E → E * be a linear maximal monotone operator.
Definition 2.2. We say that a multivalued mapping A from D(A) ⊆ E → 2 E * has the (S + ) property with respect to D(L) if the following conditions hold: (i) The set Au is nonempty, bounded, closed and convex for each u ∈ D(A).
(ii) A is finitely weakly upper-semicontinuous, i.e., for each finite dimensional subspace S of E, A is an upper-semicontinuous mapping of D(A) ∩ S into 2 E * with E * given its weak topology. (iii) For any sequence {u i } in D(A) ∩ D(L) converging weakly to an element u of D(L) in E, Lu j converging weakly to Lu in E * and for any sequence The main tool we use in this paper is a surjectivity result for multivalued (S + ) type mappings (cf. [12,20,22]). For the convenience of the reader we include it here.
The next proposition provides basic properties of the generalized directional derivative and the generalized gradient.

Define the integral functional
Proposition 2.5. [13] Under the assumption (H), the functional J in (9) is locally Lipschitz and the following inequalities hold: and Then W with the norm u W = u X + ∂u ∂t X * is a Banach space. Furthermore, the imbedding W ⊆ L 2 (Q) is compact and the embedding W ⊆ C([0, T ], L 2 (Ω)) is continuous (for more details, see [36]).

Main results.
Definition 3.1. u ∈ W is said to be a weak solution to (1) with f ∈ X * and u 0 ∈ L 2 (Ω) if there exists w ∈ ∂J(u) such that We define a linear functional K : X → X * by Ku ∈ X * such that (Ku)(v) =: T 0 R n vL K u dxdt, ∀v ∈ X , which is well-defined by Proposition 3.2 below. In this way, we may say that u ∈ W is a weak solution to problem (1) with f ∈ X * and u 0 ∈ L 2 (Ω) if there exists w ∈ ∂J(u) such that ∂u ∂t + Ku + w = f u(x, 0) = u 0 (x) in Ω.
Proposition 3.2. K : X → X * is a linear bounded strongly monotone operator.
Proof. By means of Assumption A, we have for all u, v ∈ X , which implies K : X → X * is a linear bounded strongly monotone operator and The proof is complete. (13), we also may say that u ∈ W is said to be a weak solution to (1) with f ∈ X * and u 0 ∈ L 2 (Ω) if there exists w ∈ ∂J(u) such that

Remark 2. By
Firstly, we are going to discuss the problem (1) with zero-initial condition, i.e., ∂u ∂t + Ku + w = f u(x, 0) = 0 in Ω. (14) For this purpose, we define Here ∂u ∂t stands for the generalized derivative of u, i.e., Then, L : D(L) ⊆ X → X * defined by (15) is a closed densely linear maximal monotone operator (cf. [36]). Therefore, to get a solution of the problem (1) with zero-initial condition means to solve the following abstract equation:  (8) and (10), respectively, then K + ∂J : X → 2 X * is coercive.
Proof. We shall firstly show that Ku + ∂J(u) has the (S + ) property with respect to D(L). By Proposition 2.4, we observe that ∂J is nonempty, convex, weak-compact subset of X * . Then for each u ∈ X , Ku + ∂J(u) is nonempty, bounded, closed and convex subset of X * . Moreover, Ku + ∂J(u) is upper semicontinuous from X to w − X * .
For any sequence {u k } in D(L) converging weakly to an element u of D(L) in X , Lu k converging weakly to Lu in X * and for any sequence w k in X * with w k ∈ ∂J(u k ) for each k ≥ 1, if the following condition holds we have to show that the strong convergence of {u k } to u and there exists a subsequence {w n k } of {w k } such that {w n k } converges weakly to w ∈ ∂J(u), which means that K + ∂J has the (S + ) property with respect to D(L).
For this purpose, we obtain from (16) lim sup By Proposition 2.1, we have X 0 ⊆ L 2 (Ω) ⊆ (X 0 ) * and the embedding X 0 → L 2 (Ω) is compact. Therefore, by Remark 1, we have that the imbedding W → L 2 (Q) is compact. Hence, u k → u strongly in L 2 (Q). Applying Theorem 2.2 in [6], we have On the other hand, we see that {w k } is bounded in L 2 (Q) from (11). Furthermore, {w k } is bounded in X * . So we may assume that and lim Then, from (17) we have lim sup k→∞ Ku k , u k − u X ≤ 0.
Therefore, we obtain from u k → u weakly in X lim sup By the proof of Proposition 3.2, Ku k → Ku strongly in X * .
In the following, we show that K + ∂J : X → X * is coercive. By Proposition 2.1 and 2.5, we have from (11) and (18) inf{ Ku + w, u X |w ∈ ∂J(u)} If 1 ≤ p < 2, or p = 2 with c 1 c(2) < 1, then which implies that K + ∂J : X → 2 X * is coercive. The proof is complete. Therefore, from Theorem 2.3, we get then by [36] the periodic operator L : D(L) ⊆ X → X * defined by (22) is closed densely linear maximal monotone and by [18] the anti-periodic operator L : D(L) ⊆ X → X * defined by (23) is also closed densely linear maximal monotone. Therefore, the periodic problem (1) and anti-periodic problem (1) are also solvable.
Therefore, from Theorem 3.4, we get Corollary 1. Under the assumption (H) with 1 ≤ p < 2 or p = 2 such that c 1 c(2) < 1, the nonlocal nonlinear hemivariational inequality (1) with periodic and anti-periodic condition has at least a weak solution.
Now we turn to the initial-valued problem (1). Assume that u 0 ∈ X 0 . Let J(u) = J(u − u 0 ), and f = f + K(u 0 ). Then it is easy to see that J and f satisfy the same conditions as J and f , respectively. Therefore, by Theorem 3.4, we have Corollary 2. Under the assumption (H) with 1 ≤ p < 2 or p = 2 such that c 1 c(2) < 1, the nonlocal nonlinear hemivariational inequality (1) with u 0 ∈ X 0 has at least a weak solution.
Proof. Let u 0 ∈ L 2 (Ω). Since X 0 is dense in L 2 (Ω), we can find a sequence {u 0n } ⊂ X 0 such that u 0n converges to u 0 in L 2 (Ω). For each n ≥ 1, by Corollary 2, there exists u n ∈ W and w n ∈ ∂J(u n ) such that Therefore, we have (Ω) + Ku n , u n X + w n , u n X = f, u n X . By virtue of the proof of Proposition 2.5 and Proposition 3.2, we obtain Since c 1 c(2) < 1, by use of the Young inequality, we get where C 1 and C 2 are positive constants. By u 0n → u 0 in L 2 (Ω), we have that sup n { u 0n 2 L 2 (Ω) } < +∞. Therefore {u n } is a bounded sequence in X by (25). From (H), we infer that the boundedness of {u n } in X implies the boundedness of { w n X * }. So in virtue of (24), we get ∂u n ∂t X * ≤ Ku n X * + w n X * + f X * ≤ Const., which implies that there exists a subsequence, again denoted by {u n } such that u n → u weakly in X .
∂u n ∂t → ∂u ∂t weakly in X * .
So u is a weak solution of the problem (1) under the assumption u 0 ∈ L 2 (Ω). The proof is complete.