Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions

We investigate a class of semilinear parabolic and elliptic problems with 
fractional dynamic boundary conditions. We introduce two new operators, the 
so-called fractional Wentzell Laplacian and the fractional Steklov operator, 
which become essential in our study of these nonlinear problems. Besides 
giving a complete characterization of well-posedness and regularity of 
bounded solutions, we also establish the existence of finite-dimensional 
global attractors and also derive basic conditions for blow-up.


1.
Introduction. Physical phenomena which requires the use of fractional operators is dubbed as anomalous diffusion in the sense that it exhibits deviations from normal diffusion, in that diffusion happens at either slower or faster scales [25]. From the probabilistic point of view and in analogy with the classical view of Brownian motion, the best way to understand anomalous diffusion is the random walk formalism: in the former particles make only small steps with finite probability such that the interaction between close neighbors is always short-ranged while in the latter particles are also allowed to take "arbitrarily" large steps (up to the system size) with a small finite probability for each such step, and so the interaction between particles is long-ranged. While normal diffusion is mostly ubiquitous to systems close to equilibrium, anomalous diffusion seems to be inherent in dynamical systems far from equilibrium (see [17,25,27,29]).
In recent years the mathematical community has commenced an intense agenda on understanding fractional Laplacian operators in the context of various applications that show up in optimization problems, nonlinear stochastic dynamics, phasetransition and anomalous-diffusion phenomena, crystal dislocation dynamics and many others (see [8,25] for a more complete list of references for potential applications). Many versions of the fractional Laplacian (−∆) s u(x) = C N,s P.V.
have been introduced and studied in connection with classical problems associated with the Laplace operator ∆, either on = R N or on bounded (sufficiently smooth) domains = Ω ⊂ R N , N ≥ 1. The case where |x − y| −N −2s has been replaced by a general symmetric kernel K(x, y) has been recently studied in [6,7,24]. We refer the reader to [2,6,7,16,24,28] for further literature which now appears to grow at a rapid rate but it is still full with many questions that remain to be answered. In the case of a bounded domain, by far the most predominant version of fractional Laplace operator in the literature is −∆ |Ω s , s ∈ (0, 1) , named the restricted fractional Laplacian in [28]; for this operator functions in = Ω are extended by zero outside of Ω, i.e., u = 0 on R N \Ω (in this case, one also simply refers to −∆ |Ω s as the fractional Dirichlet Laplacian and plays a similar role as the classical Laplace operator ∆ with the Dirichlet boundary condition u |∂Ω = 0; note that −∆ |Ω s should not be confused with a fractional power of the classical Dirichlet Laplacian, which has a spectral definition, and which is quite different). Besides, (in)homogeneous (nonlocal) Neumann boundary conditions can be also defined for (−∆) s in (1.1) but as in the previous case they must be posed on a "fat" collar surrounding the bounded set Ω (see [10,21]) due to the nonlocal character of the fractional Laplacian. An example of such a collar is R N \Ω (cf., for instance, [9]). For such an example, the definition of Neumann boundary condition requires that the function u is defined on all R N (or the "fat" collar surrounding Ω) and not only on Ω. From the view of applications, it seems quite impractical to deal with boundary conditions on the whole R N \Ω, especially when one has to deal with nonlinear (and even inhomogeneous) boundary conditions. Moreover, an initial datum must be also specified in the whole of R N , even though the evolution equation that governs the dynamics of the underlying physics is only posed in Ω. Thus, it is essential to find a definition of the fractional Laplacian that applies only to functions defined only on Ω and for which one has Dirichlet and fractional Neumann-Robin boundary conditions on ∂Ω, similar to the classical case. The socalled regional fractional Laplacian (−∆) s Ω , s ∈ (0, 1) has been first introduced in [2, 18,19] and also studied in detail in [30,31] (see Section 2). In this connection, a semilinear reaction-diffusion equation modelling long-range diffusion processes was also fully investigated in [16] when Ω is a bounded Lipschitz domain. It is worth emphasizing that all these Neumann problems for the various definitions of the fractional Laplace operator recover the classical Neumann problems in the limit case as s → 1.
The purpose of this paper is to introduce dynamic boundary conditions for parabolic and elliptic equations which model anomalous diffusion processes in a bounded domain Ω with Lipschitz boundary ∂Ω. Strictly speaking, in the case when the first definition (1.1) is employed one could define a dynamic equation in Ω × (0, ∞) by means of the nonlocal Neumann derivative N s given as in [9]. However, as we already mentioned earlier this approach may pose at least one dilemma in the sense that in practice it is not always easy to collect an initial datum u 0 (x) = u (t = 0, x) in the whole of R N \Ω. More precisely this has a lot to do with how we measure u 0 in R N \Ω if we chose to work with the first definition (1.1); indeed, the problem of assigning values to u 0 in the whole of R N \Ω is not only one of a theoretical nature but may be also of some practical relevance since the values of u inside Ω are well-affected by values outside Ω due to the nonlocal nature of (−∆) s . It turns out that this is not the case for the regional fractional Laplacian (−∆) s Ω (see Section 2, for a rigorous definition). It is the latter operator that leads to the correct framework on which one can define the so-called fractional dynamic boundary condition on the boundary ∂Ω, and thus solve the previous conundrum contrary to what was stated in [ s Ω , see Section 2.3), one requires knowledge of the datum u |t=0 only on Ω ∪ ∂Ω. But in fact in this paper we shall do even more: we will investigate a class of semilinear reaction-diffusion problems with long-range interaction, of the form    ∂ t u + (−∆) s Ω u + f (u) = 0, in Ω × (0, ∞) , ∂ t u + C s N 2−2s u + δ(−∆) l Γ u + g (u) = 0, on ∂Ω × (0, ∞), u (0) = u 0 in Ω, u (0) = v 0 on ∂Ω. (1.4) Here, f and g are nonlinear functions, N 2−2s is called the fractional normal derivative (see Section 2.3), δ ∈ {0, 1} and (−∆) l Γ , l ∈ (0, 1) corresponds to a kind of fractional surface diffusion operator on ∂Ω. One particular interest in dealing with (1.4) is also probabilistic in that the corresponding linear operator for (1.4), the so-called fractional Wentzell Laplacian A s W,δ , bears a close relationship to a kind of stable (restricted) Lévy process in Ω = Ω ∪ ∂Ω such that any "particle" jumps between Ω and its complement R N \Ω are in fact suppressed (see [2], for a rigorous analysis involving censored stable processes and the regional fractional Laplacian (−∆) s Ω as "generators" of these processes in Ω). In plain physical terms, we may think of (1.4) as describing a nonlinear flow process in a closed bounded domain Ω with forced action from the boundary ∂Ω. In that case, the operator A s W,δ describes a particle jumping with intensity proportional to |x − y| −N −2s from a point x ∈ Ω to another y ∈ Ω such that when the particle hits the boundary at a point z ∈ ∂Ω it may "choose" (only if δ = 1) to jump to another point z ∈ ∂Ω with a strength proportional to |z − z| −(N −1+2l) , before it is "spewed" back into the region Ω. When such a fractional dynamic boundary condition is satisfied on ∂Ω (see (1.4)), we shall always refer to ∂Ω as a nonlocally reacting surface.
We now give a brief outline of the paper. Section 2 is divided into five subsections: in Section 2.1 we introduce the relevant definitions of fractional order Sobolev spaces while in Section 2.2 we recall the definition for the regional fractional Laplacian (−∆) s Ω and some results associated with a Dirichlet problem. Then in Section 2.3 we recall the notion of fractional-order normal derivative N 2−2s and a fundamental integration by parts formula (i.e., the corresponding Green identity in the fractional case). In Section 2.4, we define the fractional Wentzell operator associated with the corresponding linear problem for (1.4) and establish some crucial properties for it that will become important in the study of the nonlinear problem. The final Section 2.5 defines a new operator that we call the fractional Steklov operator. The latter is important in our investigation of an "elliptic-parabolic" problem which is similar to (1.4) in that its first equation is merely replaced by A rigorous analysis for the full nonlinear problems is performed in Sections 3 and 4 for the parabolic system (1.4) and the elliptic-parabolic system (see (4.1), for instance), respectively. Here our first goal is to derive strong and sharp results in terms of existence, regularity and stability of bounded solutions using a combination of nonlinear semigroup and energy methods. Then our second goal is to also show the existence of finite-dimensional global attractors for such problems in a nonreflexive setting by working instead with L ∞ -like solutions. We recall that the space L ∞ is the natural choice for parabolic problems since it actually carries a real physical meaning for most real-world applications; therefore, our preferred notion of solutions will typically be given in such spaces. In fact, it is this context in which our conditions on the nonlinearities become in fact optimal and sharp in the sense that global existence and regularity holds for such problems (see Sections 3,4). Besides, when such conditions are not satisfied then blow-up of some solutions may indeed occur (see Section 5). Finally, besides the semigroup approach of Section 4.1 we shall exploit in Section 4.2 a perturbation method to derive global existence results for the elliptic-parabolic problem, by viewing the latter as a singular perturbation of a sequence of parabolic problems of the form (1.4). We note that this method may be of some independent interest in the treatment of other nonlinear systems involving such fractional dynamic boundary conditions. 2. Intermediate results. In this section we introduce the function spaces needed to investigate our problem and we prove some intermediate results that will be used to obtain our main results. Ω Ω |u(x) − u(y)| 2 |x − y| N +2s dxdy < ∞} the fractional order Sobolev space endowed with the norm where Γ denotes the usual Gamma function. We let .
By definition, W s,2 0 (Ω) is the smallest closed subspace of W s,2 (Ω) containing D(Ω). We have the following result taken from [ and Ω has a Lipschitz continuous boundary, then every u ∈ W s,2 (Ω) is zero σ-a.e. on ∂Ω, where σ denotes the usual Lebesgue surface measure on ∂Ω. Therefore, to talk about traces of functions in W s,2 (Ω) that are not necessarily null on ∂Ω, it is not a restriction to assume that 1 2 < s < 1. We have the following result taken from [4,8].
Proposition 2.2. Let 1 2 < s < 1 and Ω ⊂ R N a bounded open set with a Lipschitz continuous boundary ∂Ω. Then the following assertions hold.
2.2. The Dirichlet problem for the regional fractional Laplacian. Let Ω ⊂ R N be a bounded open set with Lipschitz continuous boundary ∂Ω. Given a function g ∈ C(R N ), it is known that the Dirichlet problem for the fractional Laplacian is not well-posed due to the nonlocal character of (−∆) s (see e.g. [19] and the references therein). This follows from the fact that We recall that provided that the limit exists. Hence, the function u must coincide with g on all of R N \Ω in order to have a well-posed Dirichlet problem, that is, the following problem is well-posed instead. The situation turns out to be quite different for the regional fractional Laplace operator (−∆) s Ω , which is defined as follows: provided that the limit exists for all functions u ∈ L 1 s (Ω), where For the regional fractional Laplacian, it has been shown in [19,Theorem 7.6 and Corollary 7.7] that for every g ∈ C(∂Ω), there exists a unique function u ∈ C(Ω) solution of the Dirichlet problem (−∆) s Ω u = 0 in Ω, u = g on ∂Ω. Let W −s,2 (Ω) denote the dual of the reflexive Banach space W s,2 0 (Ω) and ·, · the duality map between the two spaces. We have the following result.
2.3. The fractional normal derivative. Let Ω ⊂ R N be a bounded domain of class C 1,1 with boundary ∂Ω. Let 1 2 < s < 1 and the constant where C 1,s is given by (2.1) with N = 1. Let the constant B N,s be such that (2.14) A simple calculation (see, e.g. [31]) shows that in fact B N,s = C s and depends on s only. The following integration by parts formula has been recently obtained in [18,Theorem 3.3].
Next, we introduce a weak formulation for the fractional normal derivative on non-smooth domains. (a) Let u ∈ W s,2 (Ω). We say that for all v ∈ D(Ω) and hence, for all v ∈ W s,2 0 (Ω) by density. In that case we write (−∆) s Ω u = w. (b) Let u ∈ W s,2 (Ω) such that (−∆) s Ω u ∈ L 2 (Ω). We say that u has a fractional normal derivative in L 2 (∂Ω) if there exists g ∈ L 2 (∂Ω) such that for all v ∈ W s,2 (Ω) ∩ C(Ω), hence for all v ∈ W s,2 (Ω) by density and in view of (2.2). In that case, the function g is uniquely determined by (2.17), we write C s N 2−2s u = g and then call g the fractional normal derivative of u.
Proposition 2.10. The bilinear symmetric form a δ Ω with domain W s,δl,2 (Ω) is a Dirichlet form in the space X 2 (Ω), that is, it is closed and submarkovian.
Proof. Let a δ Ω with domain W s,δl,2 (Ω) be the bilinear symmetric form in X 2 (Ω) defined in (2.24). First we show that the form a δ Ω is closed in X 2 (Ω). Indeed, let U n = (u n , u n | ∂Ω ) ∈ W s,δl,2 (Ω) be a sequence such that lim n,m→∞ (2.25) It follows from (2.25) that lim n,m→∞ u n − u m W s,2 (Ω) = 0. This implies that u n converges strongly to some function u ∈ W s,2 (Ω). It also follows from (2.25) that u n | ∂Ω is a Cauchy sequence in the Banach space to some function v. By uniqueness of the limit and the trace function, we (Ω) = 0 and this implies that the form a δ Ω is closed in X 2 (Ω). Next, we show that the form a δ Ω is submarkovian. Indeed, let ε > 0 and φ ε ∈ C ∞ (R) be such that (2.26)

CIPRIAN G. GAL AND MAHAMADI WARMA
We have shown that D ⊂ D ((−∆) s W,δ ) and the proof is finished.
We call (−∆) s W,δ , the realization of the regional fractional Laplace operator with the fractional Wentzell-Robin type boundary conditions.
s W,δ be the operator defined in (2.31). Then the following assertions hold.
(a) The operator −A s W,δ generates a submarkovian semigroup (e −tA s W,δ ) t≥0 on X 2 (Ω). The semigroup can be extended to contraction semigroups on X p (Ω) for every p ∈ [1, ∞], and each semigroup is strongly continuous if p ∈ [1, ∞) and bounded analytic if p ∈ (1, ∞). (b) The operator A s W,δ has a compact resolvent, and hence has a discrete spectrum.
(c) Let U n ∈ D(A s W,δ ) ⊂ W s,δl,2 (Ω) be an eigenfunction associated with λ W s,n . Then, for every V ∈ W s,δl,2 (Ω), a δ Ω (U n , V ) = λ s,n (U n , V ) X 2 (Ω) which translates to A s W,δ U n = λ W s,n U n . Let λ > 0 be a real number. Since λ ∈ ρ(−A s W,δ ), we have that . Since for every F ∈ X 2 Ω and λ > 0, it follows from (2.34) that there exists a constant M > 0 such that This completes the proof of part (c).
, defines an equivalent norm on D(A s W,δ ). Using (2.34) with p = 2, q = ∞, for t ∈ (0, 1) and the contractivity of e −tA s W,δ for t > 1, for U ∈ D(A s W,δ ), we deduce The first integral is finite if and only if γ < 4. Using the expression of γ given in (2.35) and(2.36), we get the condition given on dimension N in the hypothesis of the assertion (d).

CIPRIAN G. GAL AND MAHAMADI WARMA
(e) As in part (d), if α > 0, then (αI + A s W,δ ) −1 F D(A s W,δ ) , for F ∈ X 2 (Ω), defines an equivalent norm on D(A s W,δ ). Using (2.34) with p = 2 and q ∈ (2, ∞) and the contractivity of e −tA s W,δ for t > 1, for U ∈ D(A s W,δ ), we deduce once again that provided that the first integral is finite, i.e., γ < 4q q−2 . Using the expression of γ in (2.35) and(2.36), we get the condition given on dimension N in the hypothesis of the assertion (e). The proof of the theorem is finished.
if δ = 0, where we recall that a δ Ω is given in (2.24). We will simply say that U is a weak solution of (2.40).
We have the following result as a consequence of Theorem 2.12.
Proposition 2.15. The form F λ is continuous and elliptic. Let D s,λ be the linear self-adjoint operator on L 2 (∂Ω) associated with F λ in the sense that, for any φ ∈ W s− 1 2 ,2 (∂Ω), 48) where N 2−2s u is to be understood in the sense of Definition 2.7 and C s is the constant given in (2.13).
The operator D s,λ is called the fractional Dirichlet-to-Neumann map.

(2.54)
It is easy to see that there is a constant C 2 > 0 such that . Moreover, Combining (2.53) and (2.55) we find that .
3. Parabolic equations with fractional dynamic boundary conditions. In this section, we shall consider a semilinear parabolic equation associated with the fractional Wentzell operator A s W,δ , of the form where we recall that s ∈ (1/2, 1) , l ∈ (0, 1) and δ ∈ {0, 1} . As usual, no additional connection between v 0 and u 0 is required at this point.
In what follows strong solutions to problem (3.1) are defined via nonlinear semigroup theory for bounded initial data and satisfy the differential equations almost everywhere in t > 0. We first introduce the rigorous notion of (global) weak solutions to the problem (3.1) as in the classical case for the semilinear parabolic equation with classical diffusion ∆ and the corresponding dynamic boundary conditions. For the sake of simplicity of notation, the symbol ·, · stands for the duality pairing between Banach space X and its dual X * . The classical notion of a weak energy solution in the space X 2 (Ω) is given by the following. Definition 3.1. Let p, q > 1. The function U = (u, u| ∂Ω ) is said to be a weak solution of (3.1) if, for a.e. t ∈ (0, T ) , for any T > 0, the following properties are valid: • Regularity: where p = p/ (p − 1) and q = q/ (q − 1) . • Variational identity: for the weak solutions the following equality ∂ t U (t) , ξ + a δ Ω (U (t) , ξ) + f (u (t)) , ξ + g (u (t)) , ξ = 0 (3.3) holds for all ξ ∈ W s,δl,2 (Ω) ∩ X p,q Ω , a.e. t ∈ (0, T ). Finally, we have, in the space X 2 (Ω), U (0) = (u 0 , v 0 ) almost everywhere.
• Energy identity: weak solutions satisfy the following identity Remark 3.2. Note that by (3.2), U ∈ C w [0, T ] ; X 2 (Ω) , that is, the space of all X 2 (Ω)-valued weakly continuous functions on the interval [0, T ]. Therefore the initial value U (0) = (u 0 , v 0 ) is meaningful when (u 0 , v 0 ) ∈ X 2 (Ω). As usual, v 0 is some function in L 2 (∂Ω) and there need not be any connection between v 0 and the boundary trace of u 0 , which may not exist for u 0 ∈ L 2 (Ω).
Finally, our notion of (global) strong solution is as follows.
Definition 3.3. Let (u 0 , v 0 ) ∈ X ∞ (Ω) be given. A weak solution U is "strong" if, in addition, it fulfills the regularity properties: such that U ∈ L ∞ ([η, T ]; D(A s W,δ )), for any T > η > 0. This section consists of two main parts. At first we will establish the existence and uniqueness of a (local) strong solution on a finite time interval using the theory of monotone operators exploited in [15] and [16]. Then exploiting a Moser iteration argument we show that the local solution is actually a global solution. In the second part, we establish the existence of a compact (finite dimensional) global attractor for problem (3.1).

Proof. The proof is similar to that of [16, Lemma 3.3] and hence omitted.
Now we state the first main theorem of this section.
Theorem 3.5. Assume that the functions f, g ∈ C 1 loc (R) satisfy for all τ ∈ R, for some constants c f , c g ∈ R, C f , C g > 0. Furthermore, we assume that the first eigenvalue λ W,δ,1 for the eigenvalue problem is positive, that is, λ W,δ,1 > 0. Then the following assertions hold.
(b) Finally, for any two strong solutions U 1 , U 2 the following estimate holds: , for t ≥ 0, for some constants C, L > 0.
Proof. To prove the first part (a), one can proceed exactly as in the proof of [16,Theorem 3.7] with some minor modifications. In this proof, c > 0 will denote a constant that is independent of t, T max , m, k and initial data, which only depends on the other structural parameters of the problem. Such a constant may vary even from line to line. Moreover, we shall denote by Q τ (m) a monotone nondecreasing function in m of order τ, for some nonnegative constant τ, independent of m. More precisely, Q τ (m) ∼ cm τ as m → +∞. We now give some brief details. Indeed, . By Theorem 2.12 we know that −A s W,δ = −∂Φ δ Ω , i.e., A s W,δ equals the subdifferential of the proper convex and lower-semicontinuous functional Φ δ Ω defined on X 2 (Ω) by and −A s W,δ generates a strongly continuous (linear) semigroup (e −tA s W,δ ) t≥0 of contraction operators on X 2 (Ω). Finally, e −tA s W,δ is non-expansive on X ∞ (Ω) and A s W,δ is strongly accretive on X 2 (Ω). As usual, we can rewrite (3.1) in functional form as where we have set F (U ) = (f (u) , g (u| ∂Ω )). The existence of locally bounded solution can now be sought as the fixed point of the following map in the Banach space for some 0 < T * ≤ T and R * > 0 (see, e.g., [16,Theorem 3.7]). This local solution can certainly be (uniquely) extended on a right maximal time interval [0, The regularity of the "fixed-point" solution in Definition 3.3 can be ascertained by exploiting some regularity results for nonhomogeneous equations (see again [16,Theorem 3.7, Step 1]). We briefly explain why we can take T max = ∞ because of condition (3.6). To this end, let m ≥ 1 and consider the function X m+1 (Ω) . First, notice that E m is well-defined on (0, T max ) because U is bounded in Ω × (0, T max ). Since U is a strong solution on (0, T max ), the function E m (t) is also differentiable for a.e. t ∈ (0, T max ). For strong solutions and t ∈ (0, T max ), integration by parts yields the following standard energy identity Assumptions (3.6)-(3.7) together with the application of Gronwall's inequality gives the following estimate for t ∈ (0, T max ) , for some C = C (f, g) > 0, depending on the constants from (3.6) and |Ω| , σ (∂Ω). Next, one can check that E m (t) satisfies a local recursive relation which can be used to perform an iterative argument exactly as in [16,Theorem 3.7, Step 3]. We have, owing to [16,Lemma 3.4] the following inequality: .
As usual, set m k + 1 = 2 k , k ∈ N, and define As in the proof of [16,Theorem 3.7], we obtain after some computations and standard estimates, for every ε > 0, (3.14) and for some positive constant γ > 0 independent of k, for a sufficiently small 0 < ≤ 1 4 . Next, owing to Lemma 3.4, we have for some ζ > 0 independent of U, k. We can now combine (3.17) with (3.16) for t ∈ (0, T max ) . Integrating (3.18) over (0, t), we infer from the Gronwall inequality (see, e.g., [ 20) which together with the energy estimate (3.11) gives T max = +∞ so that a strong solution is in fact global. Having proved that the solution U is globally (and uniformly in T ) bounded on [0, T ] for any T > 0, estimate (3.8) can be proved in a standard way. This completes the proof of the theorem.
Remark 3.6. If both c f , c g < 0 in (3.6), then automatically λ W,δ,1 > 0. Therefore, our assumption λ W,δ,1 > 0 is more general since it may also allow other cases in which either c f > 0, c g < 0, c f < 0, c g > 0 or c f , c g > 0. Furthermore, it is worth emphasizing that even without the assumption (3.7), a global bound in C [0, T ] ; X ∞ (Ω) (for any T > 0) can still be derived with any constants c f , c g > 0 in (3.6), with the exception that in this case the bound is no longer uniform with respect to time and grows like e CT , for some C > 0. The condition on the first eigenvalue of (3.7) turns out to be crucial for the attractor theory (see below).
We note that under the assumptions of Theorem 3.5, the parabolic system (3.1) defines a (nonlinear) continuous semigroup where U is the (unique) strong solution in the sense of Definition 3.3. As usual, the existence of a weak energy solution in the sense of Definition 3.1 follows from classical arguments. We shall only state this result without proof but refer the reader to [13,16] for further details.
In the final part of this section, we focus on the long-term analysis for our problem (3.1). We proceed to investigate its asymptotic properties using the notion of an exponential attractor. We begin with the following.
• G δ attracts the images of all bounded subsets of X ∞ (Ω) at an exponential rate, namely, there exists two constants ρ > 0, C > 0 such that for every bounded subset B of X ∞ (Ω). Here, dist H denotes the standard Hausdorff semidistance between sets in a Banach space H; • G δ has finite fractal dimension in X ∞ (Ω).
The next result gives the existence of such an attractor. Theorem 3.9. Let the assumptions of Theorem 3.5 be satisfied. Then (S δ (t) , X ∞ (Ω) has an exponential attractor G δ in the sense of Definition 3.8.
Since the exponential attractor always contains the global attractor, as a consequence of Theorem 3.9 we immediately have the following.
Theorem 3.10. The semigroup S δ (t) associated with the parabolic problem (3.1) possesses a global attractor A δ , bounded in X ∞ (Ω)∩D(A s W,δ ), compact in X 2 (Ω) and of finite fractal dimension in the X ∞ (Ω)-topology. This attractor is generated by all complete bounded trajectories of (3.1), that is, A δ = K δ|t=0 , where K δ is the set of all strong solutions U = (u, u| ∂Ω ) which are defined for all t ∈ R + and bounded in the X ∞ (Ω) ∩ D(A s W,δ )-norm. Our construction of an exponential attractor is based on the following abstract result [11,Proposition 4.1].
Proposition 3.11. Let H,V,V 1 be Banach spaces such that the embedding V 1 → V is compact. Let B be a closed bounded subset of H and let S : B → B be a map. Assume also that there exists a uniformly Lipschitz continuous map T :

23)
for some constant 0 ≤ γ < 1 2 and K ≥ 0. Then, there exists a (discrete) exponential attractor M d ⊂ B of the semigroup {S(n) := S n , n ∈ Z + } with discrete time in the phase space H, which satisfies the following properties: • semi-invariance: S (M d ) ⊂ M d ; • compactness: M d is compact in H; • exponential attraction: dist H (S n B, M d ) ≤ Ce −αn , for all n ∈ N and for some α > 0 and C ≥ 0, where dist H denotes the standard Hausdorff semidistance between sets in H; • finite-dimensionality: M d has finite fractal dimension in H.
Remark 3.12. The constants C and α, and the fractal dimension of M d can be explicitly expressed in terms of L, K, γ, B H (and hence, in terms of the Sobolev-Poincaré constants involved in the previous Poincaré inequalities) and Kolmogorov's κ-entropy of the compact embedding V 1 → V, for some κ = κ (L, K, γ). We recall that the Kolmogorov κ-entropy of the compact embedding V 1 → V is the logarithm of the minimum number of balls of radius κ in V necessary to cover the unit ball of V 1 .
We will prove the main theorem by carrying first a sequence of dissipative estimates for the strong solution and then applying Proposition 3.11 to our situation at the end. Lemma 3.13. Under the assumptions of Theorem 3.5, there exists a sufficiently large radius R > 0 independent of time and the initial data, such that the ball is an absorbing set for S δ (t) in X ∞ Ω . More precisely, for any bounded set B ⊂ X ∞ Ω , there exists a time t * = t * (B) > 0 such that S δ (t) B ⊂ B δ , for all t ≥ t * .

CIPRIAN G. GAL AND MAHAMADI WARMA
Proof. We claim that the existence of an absorbing set in the topogy of X ∞ (Ω) is a consequence of the following recursive inequality for E m k (t): , for all k ≥ 1, (3.24) where the sequence {t k } k∈N is defined recursively t k = t k−1 − µ/2 k , k ≥ 1, t 0 = τ .
Here we recall that C = C (µ) > 0 and κ > 0 are independent of k and that C (µ) is uniformly bounded in µ if µ ≥ 1. We recall that (3.24) is in fact a generic property that follows directly from (3.18) (see, e.g., [16,13] and the references therein). Iterating in (3.24) with respect to k ≥ 1, we obtain Taking the 2 k -th root on both sides of (3.25) and letting k → +∞ allows us to obtain the claim in view of the X 2 (Ω)-estimate (3.11) and the fact that the series in (3.25) are convergent. Indeed, the existence of an absorbing ball in the topology of X 2 Ω for the semigroup S δ (t) together with (3.25) gives an absorbing ball for S δ (t) in the space X ∞ Ω . In order to get the existence of a bounded absorbing set in D a δ Ω = W s,δl,2 (Ω) we argue as follows. Testing (3.3) with ξ = (∂ t u (t) , ∂ t u (t) | ∂Ω ) (note that such a test function is allowed by the regularity of the strong solution) we find d dt a δ Ω (U (t) , U (t)) + 2 f (u (t)) , 1 L 2 (Ω) + 2 (g (u (t)) , 1) L 2 (∂Ω) (3.26) for all t ≥ t 1 (B (0, R δ ) denotes a X ∞ (Ω)-ball of radius R δ > 0, centered at 0).
Here and below, f and g denote the primitives of f and g, respectively, such that f (0) = g (0) = 0. The application of the uniform Gronwall's lemma together with (3.11) and the existence of an absorbing set for S δ (t) in the space X ∞ Ω yields the existence of a time t * = t * (B) ≥ 1 (B is any bounded set of initial data contained in X ∞ Ω ) such that for some constant C > 0 independent of time and the initial data. This final estimate implies the existence of a bounded absorbing set in W s,δl,2 (Ω) and the final claim follows.
Next we carry some estimates for the difference of any two strong solutions, estimates which will become crucial in the final proof.

29)
for some ω, L > 0, M, K, C ≥ 0, all independent of t and U i .
Proof. Recall that the injection D a δ Ω = W s,δl,2 (Ω) → X 2 Ω is compact and continuous. Owing to Lemma 3.13, we also have for some constant C f,g > 0 which depends only on f, g and on the constant from (3.30). Integrating the foregoing inequality in time entails the desired estimate (3.28) owing to the Gronwall inequality and the fact that · X 2 (Ω) ≤ C · D(a δ Ω ) , for some C > 0. Finally, we observe that for any test function ξ ∈ D(a δ Ω ) there holds since f, g ∈ C 1 loc (R), owing to (3.30). This estimate together with (3.8) gives the desired control on the time derivative in (3.29).
The last ingredient we need is the uniform Hölder continuity of the time map t → S δ (t) U 0 in the X ∞ Ω -norm, namely, Lemma 3.16. Let the assumptions of Theorem 3.5 be satisfied. Consider U (t) = S δ (t) U 0 with U 0 ∈ B δ . Then the following estimate holds:
Proof. Exploiting the bound (3.30), by comparison in (3.3), we have as in the proof of Lemma 3.15 that (W s,δl,2 (Ω)) * dt ≤ C T , for any T > 0. This estimate entails the inequality By a duality argument, (3.32) and the uniform bound (3.30) further yield Inequality (3.31) is a consequence of (3.33) and the X 2 → X ∞ smoothing property (3.25). More precisely, the derivation of the X 2 → X ∞ continuous dependance estimate for the difference U (t) − U (τ ) of any two strong solutions U (t) , U (τ ) is actually reduced to the same iteration procedure leading to (3.25) (cf. the proof of Lemma 3.13). The proof is completed.
We can now finish the proof of Theorem 3.9, using the abstract scheme of Proposition 3.11.
Proof of Theorem 3.9. First, we construct the exponential attractor E d of the discrete map S δ (T * ) on B δ (the above constructed absorbing ball in X δ ), for a sufficiently large T * . Indeed, let B 1,δ = [∪ t≥T * S δ (t) B δ ] X 2 , where [·] X 2 denotes the closure in the space X 2 Ω and then set B δ := S δ (1) B 1,δ . Thus, B δ is a semiinvariant closed but also compact (for the X 2 -metric) subset of the phase space X ∞ Ω and S δ (T * ) : B δ → B δ , provided that T * is large enough. Then, we apply Proposition 3.11 on the set B δ with H = X 2 Ω and S = S δ (T * ) , with T * > 0 large enough so that M e −ωT * < 1 2 (see (3.28)). Besides, letting we have that V 1 → V is compact owing to the fact that D(a δ Ω ) → X 2 Ω is compact. Secondly, define T : B δ → V 1 to be the solving operator for (3.1) on the time interval [0, T * ] such that TU 0 := U ∈ V 1 , with U (0) = U 0 ∈ B δ . Due to Lemma 3.15 and (3.29), we have the global Lipschitz continuity (3.22) of T from B δ to V 1 , and (3.28) gives us the basic estimate (3.23) for the map S = S δ (T * ). Therefore, the assumptions of Proposition 3.11 are verified and, consequently, the map S = S δ (T * ) possesses an exponential attractor E d on B δ . In order to construct the (continuous) exponential attractor G δ for the semigroup S δ (t) with continuous time, we note that this semigroup is Lipschitz continuous with respect to the initial data in the topology of X 2 (Ω) (in fact it is also Lipschitz continuous with respect to the metric topology of X ∞ (Ω), owing to the X 2 → X ∞ smoothing property). Moreover, by Lemma 3.16 the map (t, U 0 ) → S δ (t) U 0 is also uniformly Hölder continuous on [0, T * ] × B δ , where B δ is endowed with the metric topology of X ∞ (Ω). Hence, the desired exponential attractor G δ for the continuous semigroup S δ (t) can be obtained by the standard formula Finally, the finite-dimensionality of G δ in X ∞ (Ω) follows from the finite dimensionality of E d in X 2 (Ω) and the X 2 → X ∞ smoothing property. The remaining properties of G δ are also immediate. Theorem 3.9 is now proved.
4. Elliptic equations with fractional dynamic boundary conditions. In this section, we wish to investigate a simple prototype of initial-boundary value problem, of the form on ∂Ω. In what follows we shall exploit two different basic approaches based on semigroup and perturbation methods to derive a well-posed theory and long-term results for problem (4.1).

A semigroup approach.
Our goal is to argue as in Section 3 for the parabolic problem (3.1) by employing instead the results stated in Proposition 2.17 for the fractional Steklov operator associated with (4.1). Indeed, letting λ = 0 we observe that (4.1) can be recast as a semilinear parabolic equation on ∂Ω for the fractional Steklov operator B δ 0 as follows: For the latter problem, we shall argue exactly as in the proof of Theorem 3.5 for the parabolic problem. For notion of (global) strong solution for (4.2) we shall use the following.
Let v 0 ∈ L ∞ (∂Ω) be given. A strong solution u of (4.2) fulfills the regularity properties ), for any T > η > 0. The equation (4.2) is satisfied in the following sense: The corresponding Poincaré-Young inequality for the quadratic form F δ 0 , as defined in (2.50) is as follows.
Proof. The proof follows by a contradiction argument (see [16,Lemma 3.3] for further details).
The corresponding result on global existence of strong solutions is the following.
(a) For every v 0 ∈ L ∞ (∂Ω), there exists a unique strong solution u of (4.1) in the sense of Definition 4.1. (b) Finally, for any two strong solutions u 1 , u 2 the following estimate holds: , for t ≥ 0, for some constants C, L > 0.
Proof. As in the proof of Theorem 3.5, owing to Proposition 2.17, the existence of a locally bounded (generalized) solution can be sought as a fixed point of the map in the Banach space for some 0 < T * ≤ T and R * > 0 (see, e.g., [16,Theorem 3.7]). Of course, the local solution can be (uniquely) extended by continuity on a right maximal time interval [0, T max ), with T max > 0 depending on U 0 X ∞ (Ω) . To conclude that the solution u belongs to the class in Definition 4.1, we argue as in [16] by further setting θ (t) := −g (u (t)) , for u ∈ C ([0, T max ) ; L ∞ (∂Ω)) and notice that u is also the "generalized" solution of . We can then employ [16, Theorem 3.5] to observe that the "generalized" solution u has the additional regularity ∂ t u ∈ L 2 (η, T max ); L 2 (∂Ω) , for every η > 0, such that together with g ∈ C 1 loc (R) and u ∈ C ([0, T max ) ; L ∞ (∂Ω)) it yields Thus, we can apply [16,Theorem 3.6] to deduce that such that the solution u is Lipschitz continuous on [η, T max ), for every η > 0. Thus, we have obtained a locally-defined strong solution in the sense of Definition 3.3. Next, we can exploit a Moser iteration argument to prove that one can take T max = ∞. As usual, this iteration will be performed for the L m+1 (∂Ω)-norm of the solution u but first we notice that on (0, T max ), integration by parts in (4.2) yields d dt The application of Gronwall's inequality together with the assumptions of the theorem give for some C = C (g) > 0, depending on the constants from (4.3) and σ (∂Ω). Furthermore, one can check that E m (t) satisfies a local recursive relation which can be used to perform an iterative argument exactly as in [16,Theorem 3.7, Step 3]. We have, owing to [16,Lemma 3.4] the following inequality: 2 ) (4.10) . We emphasize that (4.10) is the same inequality as that of (3.12) if we set E m (t) := u (t) m+1 L m+1 (∂Ω) in (4.10), owing to the fact that F δ 0 and a δ Ω coincide as quadratic forms (compare (2.24) and (2.50)). Thus, one can still establish the estimate (3.15) and the analogue of inequality (3.19) for the elliptic problem (4.1) by using instead Lemma 4.2. The iteration argument yields as usual the estimate sup t∈(0,Tmax) Once we recall (4.9), the foregoing estimate easily allows us to reach the claim, i.e., the function u ∈ C([0, T ] ; L ∞ (∂Ω)), for any T > 0.
To prove the second part, we set u = u 1 −u 2 and observe that the strong solution u satisfies for some constant L > 0 depending only on g and the C([0, T ] ; L ∞ (∂Ω))-norm of u i . Integrating this inequality over (η, t) , for every t ≥ η, we infer from the Gronwall inequality that . Letting η → 0 + and recalling that u is also continuous, gives for any t ≥ 0. Integrating (4.11) over (0, t) and exploiting once more the previous inequality, we easily arrive at the stability estimate (4.4). The proof is finished.
Under the full force of Theorem 4.3, the elliptic-parabolic system (4.1) defines a (nonlinear) continuous semigroup where u is the (unique) strong solution in the sense of Definition 4.1.
The last results concern the long-term behavior for the problem (4.1) in terms of finite dimensional global and exponential attractors.
Theorem 4.4. Let the assumptions of Theorem 4.3 be satisfied. Then the semigroup (T δ (t) , L ∞ (∂Ω)) has an exponential attractor E δ in the following sense: (c) E δ attracts the images of all bounded subsets of L ∞ (∂Ω) at an exponential rate, namely, there exists two constants ρ > 0, C > 0 such that for every bounded subset B of L ∞ (∂Ω). (d) E δ has finite fractal dimension in L ∞ (∂Ω).
Proof. The proof follows the same scheme of the proof of Theorem 3.9 and requires only minor inessential modifications in the statements of Lemmas 3.13, 3.15 and 3.16. We leave the tedious details for the interested reader to check.

4.2.
A perturbation method. In this subsection, inspired by [14] we wish to consider a different approach to handle the well-posedness of our elliptic-parabolic system, of that in which (4.1) can be viewed as a singular perturbation of a sequence of fully parabolic problems, of the form where ε ∈ (0, 1] is a given relaxation parameter. Indeed, if we formally set ε = 0 in the first equation of (4.12), then we can easily deduce (4.1).
It turns out that (4.12) possesses a unique strong solution due to Theorem 3.5.
Our goal then is to prove the existence of at least one strong solution to (4.1) in the sense of Definition 4.1 by passing to the limit as ε → 0 in the parabolic system (4.12). However, this is not so straight-forward since when we collapse (4.12) into (4.1) by taking ε = 0, we lose the information on the initial datum u 0 in Ω. Indeed, (4.1) requires only knowledge of the initial value of u (t) | ∂Ω at the initial time t = 0. It is here where the linear elliptic problem (2.7) for the fractional regional Laplacian with inhomogeneous Dirichlet data enters into the picture. By virtue of Proposition 2.5, for a given g ∈ L ∞ (∂Ω) ∩ W s− 1 2 ,2 (∂Ω) the system has a unique weak solution u ∈ L ∞ (Ω) ∩ W s,2 (Ω). We denote the corresponding solving operator as the map We now set Z 0 := W s− 1 2 ,2 (∂Ω) and Z 1 = W s− 1 2 ,2 (∂Ω) ∩ W l,2 (∂Ω). Our main result in this subsection is the following.

(4.29)
Note that the second and third convergences of (4.29) imply that u ∈ C ([0, T ] ; L 2 (∂Ω) such that u (0) = v 0 a.e. on ∂Ω. Also by the classical Aubin-Lions-Simon compactness theorem we have u ε → u strongly in L 2 (0, T ); L 2 (∂Ω) , (4.30) which is enough to pass to the limit in the nonlinear boundary term. More precisely, using the fact that g ∈ C 1 , we have g (u ε ) → g (u) strongly in L 2 (0, T ); L 2 (∂Ω) , (4.31) thanks to (4.30), the first convergence of (4.29) and estimate (4.20). By means of the above convergence properties (4.29), (4.31), we can now pass to the limit in both equations of (4.12) to deduce a function U = (u, u| ∂Ω ) which solves the elliptic-parabolic system (4.1). Moreover, due to the arbitrariness of T > 0, passing to limit as ε → 0 in (4.28) and (4.20)-(4.21), and recalling (4.29), we also deduce that the limit (strong) solution U satisfies these inequalities with a constant C > 0 independent of ε > 0. The continuity of u in C ([0, T ] ; Z δ ∩ L ∞ (∂Ω)) is proved exactly in the same fashion as in the proof of Theorem 4.6. Finally, the uniqueness of the strong solution to the elliptic-parabolic problem (4.1) follows from (4.4). The proof is finished.

5.
Blow-up results. The goal in this section is to check that our assumptions on f, g in the previous sections, which ascertain the global well-posedness of strong solutions of both problems (3.1) and (4.1), respectively, are in fact optimal. We observe for instance that assumption (3.6) implies that if f and g are sources with a bad sign at infinity than they can only be of at most linear growth at infinity (i.e., if f (τ ) ∼ −c f |τ | p−2 τ , g (τ ) ∼ −c g |τ | q−2 τ, as |τ | → ∞ for some p, q > 2 then we necessarily must have c f < 0, c g < 0), exactly as in the classical case (see, however, Remark 5.4 below). Indeed, following the well-known concavity method of Levine and Payne we can easily show that as soon as both f and g have superlinear growth and a bad sign at infinity, then blowup in finite time of some strong solutions for (3.1) will occur. We establish a similar result for the elliptic-parabolic system (4.1).
We first recall the following local result which is a consequence of results proven in the previous sections.
In either case (a) or (b), the solution U must blow-up in finite time.
Proof. Our proof is inspired from [23]. We prove the claim in the case (a), the other is quite similar. Let U = (u, u| ∂Ω ) be a solution of (3.1) and consider the function for t > 0, where we have set µ Ω = |Ω| + σ (∂Ω). Integrating both equations of (3.1) over Ω and ∂Ω, respectively and adding the corresponding identities, we find The Green identity (2.18) together with the definition (2.24) gives for t > 0 and for as long the strong solution U exists. From (a) and the Jensen's inequality, we obtain for as along the strong solution exists. Since h > 0 this implies as usual that hence, by (5.1) it follows that and therefore if U (t) is a global solution for problem (3.1), (5.2) leads to a contradiction since the left-hand side of (5.2) goes to infinity as time goes to infinity. The proof is finished.
Remark 5.4. Note that this result does not say anything in the case when g is dissipative and of good sign at infinity, but f still has a bad sign at infinity. Indeed, for the standard reaction-diffusion system associated with the classical diffusion ∆, blow-up in finite time of some strong solutions actually holds (see [13,Section 3]). On the other hand, it was also proven in [13] for the same standard system that global boundedness of solutions still holds provided that g is of supercritical growth and of bad sign at infinity such that f is a "good" dissipative source whose role is to absorb the "bad" boundary reaction. Unfortunately, such questions remain open for the system (3.1).
The corresponding result for the elliptic-parabolic system (4.1) is proved in the same fashion.
Combining Theorem 5.5 with Theorem 4.6 yields the following.
Theorem 5.6. Assume that g (s) ∼ c g |s| r−2 s as |s| → ∞ for some c g ∈ R and r > 1. Then for all v 0 ∈ Z δ ∩ L ∞ (∂Ω) , problem (4.1) has a unique global strong solution if and only if either c g ≥ 0 or r ≤ 2.