Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions

We investigate the long term behavior in terms of finite dimensional global 
attractors and (global) asymptotic stabilization to steady states, as time 
goes to infinity, of solutions to a non-local semilinear reaction-diffusion 
equation associated with the fractional Laplace operator on non-smooth 
domains subject to Dirichlet, fractional Neumann and Robin boundary 
conditions.


1.
Introduction. The main concerns in the present paper are to investigate the existence, the regularity and the long-time behavior of solutions to some non-local reaction-diffusion equations associated with the fractional Laplace operator with Dirichlet, fractional Neumann and fractional Robin type boundary conditions on non-smooth subsets of R N . In order to introduce the fractional Laplacian, let 0 < s < 1, Ω ⊂ R N an arbitrary open set and set provided that the limit exists. We notice that if 0 < s < 1/2 and u is smooth (for example, Lipschitz continuous), then the integral in (1.1) is in fact not singular near x. We also recall that in the whole space R N , using the Fourier transform, (−∆) s can be also defined as a pseudo-differential operator with symbol |ξ| 2s . If one wishes to consider the fractional Laplace operator (−∆) s on open subsets Ω of R N it cannot be used on Ω automatically due to its nonlocal character. In order to give a proper definition, we follow [28,29,30,48] in the following fashion. Let Ω ⊂ R N be an arbitrary open set. We restrict the integral kernel of the fractional Laplacian to the open set Ω. For u ∈ L 1 (Ω), x ∈ Ω and ε > 0, we let provided that the limit exists. As in the case R N , if s ∈ (0, 1/2) and u is smooth then the integral in (1.2) is not singular near x. We call the operator A s Ω the regional fractional Laplacian (cf. [28,29,30]). The regional fractional p-Laplace operator with p ∈ (1, ∞) has been also introduced in [49]. Let now u ∈ D(Ω), the space of infinitely continuously differentiable functions with compact support in Ω. Since u = 0 on R N \Ω, a simple calculation gives The operator A s Ω describes a particle jumping from one point x ∈ Ω to another y ∈ Ω with intensity proportional to |x − y| −N −2s . Parabolic problems associated with the operator A s Ω have been intensively studied in [6,11,29,30] and the references therein, employing probabilistic approaches, and in [48] by using the method of Dirichlet forms on non-smooth domains. Based on (1.4), we then view the fractional Laplacian (−∆) s with domain D(Ω) as a perturbation of the regional fractional operator A s Ω with the non-negative potential V Ω . Recently, various elliptic and parabolic equations associated with the fractional Laplace operator with Dirichlet boundary conditions were also investigated by Caffarelli et al. [8,9,10].
In this paper, we shall be concerned with non-local diffusion processes associated with the fractional Laplace operator with various boundary conditions. To be more precise, we consider diffusion processes described by the following systems in Ω, (1.5) and in Ω, (1.6) and in Ω. (1.7) In (1.5), (1.6) and (1.7), f = f (u) plays the role of nonlinear source, not necessarily monotone and d > 0 is a diffusion coefficient. In (1.5), the operator (−∆) s denotes the fractional Laplace operator defined in (1.1) and in (1.6) and (1.7), A s Ω is the regional fractional Laplacian given by (1.2). Finally in (1.7), N 2−2s u denotes the fractional normal derivative of the function u in direction of the outer normal vector (see Section 2 below), B N,s is a normalized constant (see (2.17) below) and γ ∈ L ∞ (∂Ω) is a non-negative function.
Our motivation for considering such problems is two-fold. First, the systems (1.5)-(1.7) and their stationary versions have been used recently to describe the motion of nonlinear deflects in crystalline materials in the field of dislocation dynamics (see, e.g., [26,36]). In the theory of phase-field and interfacial dynamics, these equations are usually referred as the fractional Allen-Cahn equation (see, e.g., [31,41]). Moreover, nonlocal reaction-diffusion equations have been also considered in the monograph [4] but the integral operators there are generally smooth or only mildly singular (i.e., the kernel is at least integrable over R N ). On the other hand, the linear parabolic equation ∂ t u + (−∆) s u = 0, s ∈ (0, 1), instead of the usual parabolic equation ∂ t u − ∆u = 0, is a much studied topic of anomalous diffusion in physics, probability and finance (see, e.g., [1,33,40,42]). We also refer the reader to an interesting tutorial in [47] which introduces the main concepts behind normal and anomalous diffusion. Second, our work is further motivated by the need to develop a complete dynamical theory for these problems where not much seems to be known about basic issues, such as global existence and regularity, uniqueness, blowup phenomena and longtime behavior of solutions, as time goes to infinity. This seems to be due to the fact that the parabolic structure of (1.5)-(1.7) has not been exploited before, an issue which is intimately connected with an L 2 -L ∞ smoothing result in (0, ∞) × Ω of solutions. This is essential to the study of the asymptotic behavior of these systems, in terms of global attractors and ω-limit sets. We also emphasize the generality of our results by assuming only minimal conditions on the regularity of Ω: we shall assume Ω to be simply an open subset of R N in the case of problems (1.5)-(1.6), while in the third case (1.7) it suffices to assume that ∂Ω is Lipschitz continuous. Our current contribution is also motivated by our recent work on parabolic equations with classical diffusion on rough domains and nonlocal boundary conditions (see [24]).
provided that f is a polynomial density function of arbitrary growth such that f (τ ) ≥ −c f , for any τ ∈ R, assuming f (0) < 0 as well. Here, the constants c 1 , c 2 > 0 depend on the shape of Ω and N only, but are independent of s. We recall that the dimension of the global attractor can be used in practice to indicate the number of degrees of freedom needed to simulate the given dynamical system since this dimension is usually associated with the temporal and spatial complexity of the long-time dynamics. Thus, the bound (1.9) indicates that the "permanent regime" for problem (1.5) is indeed more structurally complex when compared to that of the classical reaction-diffusion equation for the Laplace operator ∆. It is also worth noting that the foregoing bounds also stabilize as s → 1 − to the (classical) dimension bounds for the well-established reaction-diffusion equation associated with the operator ∆.
(III) The ω-limit sets of the problems (1.5)-(1.7) can exhibit a complicated structure if the function f is non-monotone. This can happen if the stationary problems associated with (1.5)-(1.7) possess a continuum of nonconstant solutions. We show the validity of the so-called Lojasiewicz-Simon inequality for our problems under the assumption that a certain elliptic boundary value problem has Hölder continuous (up to the boundary ∂Ω) solutions. To be more precise, we need only require that the bounded solution of has the property: w ∈ C 0,ν Ω , for some ν ∈ (0, 1). We also give an example when such a condition is met. Then, for any of these problems we prove the convergence of a given trajectory u = u(t; u 0 ), u 0 ∈ L 2 (Ω) , as time goes to infinity, to a single equilibrium which solves the corresponding stationary version associated with (1.5)-(1.7). More precisely, assuming also that f is a real analytic function over R, any weak solution u to problems (1.5)-(1.7) satisfies where ζ ∈ (0, 1) depends on u * ∈ L ∞ (Ω) ∩ W s,2 (Ω), such that u * solves the corresponding stationary problems.
Finally, we can also mention that the present analysis can be exploited to extend and establish existence and existence of finite dimensional attractor results for systems of reaction-diffusion equations for a vector valued function − → u = (u 1 , ..., u k ) (k ≥ 2). For instance, our framework requires only minor modifications: the function spaces become product spaces, and the principal dissipation operators become block operators on these product spaces, typically with block diagonal form. The nonlinearities in these models can be treated in a similar way as in Section 3.
We also remark that one can also allow for time-dependent external forces h (t) , h ∈ C b R; L 2 (Ω) , acting on the right-hand side of these systems. In this case, one can generalize the notion of global attractor and replace it by the notion of pullback attractor, for example. One can still study the set of all complete bounded trajectories, that is, trajectories which are bounded for all t ∈ R + . We leave the details to the interested reader. The plan of the paper goes as follows. In Section 2, we introduce the functional analytic framework associated with (1.5)-(1.7). Then, in Section 3 (and corresponding subsections) we prove well-posedness and regularity results for any of the problems (1.5)-(1.7). In Section 4 we establish the existence of a compact semiflow in L 2 (Ω) and derive optimal estimates on the global attractor, while in the remaining Sections 5, 6 we deal with convergence of solutions and with blow-up phenomena, respectively.

2.
Preliminaries. In this section we introduce the function spaces needed to investigate our problems and we show some intermediate results. In particular we introduce the fractional normal derivative of a function u mentioned in the introduction. We also give the integration by parts formula for the regional fractional Laplacian and we introduce the fractional Neumann and Robin type boundary conditions associated with the operator A s Ω . Some generation of semigroup results, and the regularity of weak solutions of elliptic equations associated with these operators are also given. Ω Ω |u(x) − u(y)| 2 |x − y| N +2s dxdy < ∞} the fractional order Sobolev space endowed with the norm If Ω is a bounded open set with a Lipschitz continuous boundary, then by [16,Theorem 6.7], there exists a constant C > 0 such that for every u ∈ W s,2 (Ω), for all r satisfying If N < 2s, then For an arbitrary open set Ω ⊂ R N , we let .
2.2. The fractional Laplacian with Dirichlet boundary conditions. Let Ω ⊂ R N be an arbitrary bounded domain. Let 0 < s < 1 and let V Ω be the potential given in (1.3). Throughout the remainder of the article, if we write u ∈ W s,2 0 (Ω), we mean that u ∈ W s,2 (R N ) and u = 0 on R N \Ω.
Let E E be the bilinear symmetric closed form with domain D(E E ) = W s,2 0 (Ω) and defined for u, v ∈ W s,2 0 (Ω) by Let A E be the closed linear selfadjoint operator on L 2 (Ω) associated with E E in the sense that (2.6) We call A E a realization of the fractional Laplace operator (−∆) s on L 2 (Ω) with the Dirichlet boundary condition. We have the following more explicit description of the operator A E . Proof. Set D := {u ∈ W s,2 0 (Ω), (−∆) s u ∈ L 2 (Ω)}. Let u ∈ D(A E ). Then by definition, there exists a function v ∈ L 2 (Ω), such that for every ϕ ∈ W s,2 0 (Ω), We have shown that u ∈ D and A E u := v = (−∆) s u. Now, let u ∈ D and set v := (−∆) s u ∈ L 2 (Ω). Let ϕ ∈ W s,2 0 (Ω). Then We have shown that D ⊂ D(A E ) and the proof of (2.7) is complete.
Next, let E D be the bilinear symmetric closed form with domain D(E D ) = W s,2 0 (Ω) and defined for u, v ∈ W s,2 0 (Ω) by Let A D be the closed linear selfadjoint operator on L 2 (Ω) associated with E D in the sense that (2.8) We call A D a realization of the regional fractional Laplace operator A s Ω on L 2 (Ω) with the Dirichlet boundary condition. We have the following more explicit description of the operator A D .
Proposition 2.2. Let A D be the operator defined in (2.8). Then Proof. The proof follows as the proof of Proposition 2.1 by using also the integration by part formula Ω vA s We need not make any confusion between the operator A E and A D . They are different and coincide only if R N \Ω has capacity zero with respect to the capacity defined with the space W s,2 (R N ) (of course, this cannot be the case since Ω is bounded). On one hand, we have shown that A E u = A D u+V Ω u, for every u ∈ D(Ω). On the other hand, the potential V Ω is in general very difficult to describe. For example, if Ω has a Lipschitz continuous boundary then it has been shown in [27,Formula (1.3.2.12), p. 19] that there exist some constants 0 < C 1 ≤ C 2 such that for every x ∈ Ω, , where ρ(x) := dist(x, ∂Ω), x ∈ Ω. Instead of the fractional Laplace operator (−∆) s , whose definition is independent of the open set Ω, the regional fractional Laplace operator A s Ω depends on Ω and hence on the potential V Ω . But we have the following convergence result. Proposition 2.3. Let Ω ⊂ R N be an arbitrary bounded open set. Then for every u ∈ D(Ω) and v ∈ W 1,2 0 (Ω), we have that Proof. First, let u ∈ D(Ω). Then using [7], the fact that lim s↑1 (1 − s)Γ(1 − s) = 1 and the classical integration by part formula for the Laplace operator, we get that Proceeding as in (2.11), we also have that We have show (2.10) for u = v ∈ D(Ω). Replacing u by u + v in (2.11) and (2.12) for u, v ∈ D(Ω), we get (2.10) for every u, v ∈ D(Ω). Finally we get (2.10) for every u ∈ D(Ω) and v ∈ W 1,2 0 (Ω) by density and using that W 1,2 0 (Ω) is continuously embedded into W s,2 0 (Ω). The proof is finished.
2.3. The fractional normal derivative. In this subsection, we introduce the fractional normal derivative mentioned in the introduction. This will be used to define the fractional Neumann and Robin boundary conditions for the operator A s Ω . Throughout the remainder of this subsection, Ω ⊂ R N denotes a bounded open set of class C 1,1 and we will also use the following notations: ρ(x) = dist(x, ∂Ω) = inf{|y − x| : y ∈ ∂Ω}, ∀ x ∈ Ω, Ω δ = {x ∈ Ω : 0 < ρ(x) < δ}, δ > 0 is a real number, n(z) = the inner normal vector of ∂Ω at the point z ∈ ∂Ω, ν(z) = − n(z) the outer normal vector of ∂Ω at the point z ∈ ∂Ω.
The following definition is taken from [28, Definition 2.1] (see also [29,Definition 7.1] for the one-dimensional case).
β (Ω) be the representation of u. Let u 0 := f h β so that u = u 0 + g. Then the following assertions hold.
(a) If β ∈ (1, 2), then for z ∈ ∂Ω, Next, let and let the constant B N,s be such that We have the following fractional Green type formula for the regional fractional Laplace operator. Theorem 2.3. Let 1 2 < s < 1 and let A s Ω be the nonlocal operator defined in (1.2). Then, for every u := f h 2s + g = u 0 + g ∈ C 2 2s (Ω) and v ∈ W s,2 (Ω), where by u0 ρ 2s−1 (z) at the point z ∈ ∂Ω, we mean We mention that the first identity in Theorem 2.3 has been obtained in [28,Theorem 3.3] under the assumption that v also belongs to C 2 2s (Ω). Its validity for every v ∈ W s,2 (Ω) and the second and third identities have been proved in [48,Theorem 5.7].
Definition 2.4. For 1 2 < s ≤ 1 and u ∈ C 2 2s (Ω), we call the function B N,s N 2−2s u the fractional normal derivative of the function u in direction of the outer normal vector.
We make some comments about the fractional normal derivative introduced above.
Remark 3. We mention that another definition of fractional normal derivative, called non-local normal derivative, has been introduced in [18,32] (see also [17]) for functions u defined on R N . More precisely, for 0 < α < 1 and u ∈ L 1 (R N ), the non-local normal derivative is defined by The definition of N α u in (2.18) requires that the function is defined on all R N . This is different from the fractional Normal derivative N α u given in (2.13) where the function u is defined only on Ω. Starting with a function defined only on Ω, it seems impossible to deal with N α u. For example if u ∈ W s,2 (Ω) and letting u ∈ W s,2 (R N ) be an extension to all R N , then the relation (2.18) can make sense but the definition cannot be independent of the extension, except in the case where there is only one such possible extension. This shows that the expression N α u cannot be used in our context since we consider functions defined a priori only on Ω. We recall that it has been shown in [17, Proposition 5.1] (see also [18,32]) that if Ω ⊂ R N is a bounded domain with Lipschitz continuous boundary ∂Ω, then for As we have seen in Remark 2, the fractional normal derivative N α u is continuous with respect to α, so that for every u ∈ C 2 2 (Ω) = C 1 (Ω) we have that N 1 u = ∂u ∂ν , i.e., the classical normal derivative of the function u in direction of the outer normal vector ν. Next, let B N,s be the constant given in (2.17). First, we notice that using a change of variable, we get that where B denotes the usual Beta function. Replacing this expression (2.19) in (2.17), we get that in fact B N,s = C s and hence, it is independent of N . Moreover, we have that lim s↑1 C s = 1. This shows that the integration by parts formula given in Theorem 2.3 is consistent with the well-known integration by part formula for the Laplace operator where there is no constant depending on the dimension in the boundary integral.
2.4. The fractional Neumann boundary conditions. Throughout this section, we assume that Ω ⊂ R N is a bounded domain with Lipschitz continuous boundary ∂Ω. We consider the bilinear symmetric closed form E N with domain D(E N ) = W s,2 (Ω) and given for u, v ∈ W s,2 (Ω) by Since W s,2 (Ω) = W s,2 0 (Ω) for all 0 < s ≤ 1/2 (by Remark 1), we have that E N = E D if 0 < s ≤ 1/2. Therefore, we assume that 1/2 < s < 1.
Let A N be the closed linear selfadjoint operator associated with E N in the sense that (2.20) We call A N a realization of the regional fractional Laplace operator A s Ω on L 2 (Ω) with the fractional Neumann type boundary conditions. In fact, we have the following more explicit description of the operator A N which has been proved in [48, Proposition 6.1] exploiting Theorem 2.3.
The following result shows in particular, that as s ↑ 1, the operator A N converges (in some sense) to the realization ∆ N in L 2 (Ω) of the Laplace operator with the classical Neumann boundary conditions. Proposition 2.5. For every u ∈ C 2 (Ω) and v ∈ W 1,2 (Ω) we have that Proof.
As we have mentioned in Remark 3, since our functions are a priori defined only on Ω, we have that N 2−2s u = 0 is the right fractional homogeneous Neumann type boundary conditions for the regional fractional Laplace operator.
2.5. The fractional Robin boundary conditions. Let Ω ⊂ R N be a bounded domain with Lipschitz continuous boundary ∂Ω and s ∈ (0, 1).
We assume that s ∈ (0, 1) is such that W s,2 0 (Ω) = W s,2 (Ω), that is, 1/2 < s < 1, otherwise we are in the situation of the Dirichlet boundary condition. It follows from [48,Theorem 6.4] that the form E R is closed on L 2 (Ω).
Let A R be the closed linear selfadjoint operator associated with the form E R in the sense that (2.24) We call A R a realization of the regional fractional Laplace operator A s Ω on L 2 (Ω) with the fractional Robin type boundary conditions. The following result has been proved in [48, Proposition 6.5] by using the integration by parts formula given in Theorem 2.3. Proposition 2.6. Let A R be the operator defined in (2.24). Assume also that Ω is a bounded open set of class C 1,1 . Then We refer to [28,29,48] for more details. We notice that it also follows from Proposition 2.5 that, as s ↑ 1, the operator A R converges (in some sense) to the realization ∆ R in L 2 (Ω) of the Laplace operator with the classical Robin boundary conditions. 2.6. Generation of semigroup. Let 0 < s < 1 and let A K , K ∈ {E, D, N , R} be the operators introduced above. We also let W s,2 We introduce the following assumption.
(H): If K = N or K = R, we assume that 1 2 < s < 1 and Ω ⊂ R N is a bounded domain with Lipschitz continuous boundary. If K = E or K = D, then Ω ⊂ R N is an arbitrary bounded open set.
Theorem 2.5. Let 0 < s < 1 and let assumption (H) be satisfied. Then the following assertions hold.
(a) The operator −A K generates a submarkovian semigroup (e −tA K ) t≥0 on L 2 (Ω) and hence, can be extended to contraction strongly continuous semigroups on L p (Ω) for every p ∈ [1, ∞), and to a contraction semigroup on L ∞ (Ω). (b) The operator A K has a compact resolvent, and hence has a discrete spectrum.
The spectrum of A K is an increasing sequence of real numbers 0 ≤ λ 1 < λ 2 < · · · < λ n < . . . that converges to +∞. Moreover, 0 is an eigenvalue of A N and is not an eigenvalue of A K for K ∈ {E, D, R}, and if u n is an eigenfunction associated with λ n , then Proof. Let 0 < s < 1 and let A K , K ∈ {E, D, N , R} be the operators introduced above. Assume the assumption (H).
(a) The proof of this part is contained in [48, Theorems 6.2 and 6.6]. We notice that in [48] the operator A K for K ∈ {N , R} has been considered. The proof of the corresponding result for A K , K ∈ {E, D} follows similarly.
(b) By [48, Theorems 6.2 and 6.6], the operator A K for K ∈ {N , R} has a compact resolvent. We have shown above that the embedding W s,2 0 (Ω) → L 2 (Ω) is compact. Hence, the operator A K for K ∈ {E, D} also has a compact resolvent. Since A K is a nonnegative self-adjoint operator and has a compact resolvent, then it has a discrete spectrum which is an increasing sequence of real numbers 0 ≤ λ 1 < λ 2 < · · · < λ n . . . , that converges to +∞. It is easy to see that 0 is an eigenvalue of A N and is not an eigenvalue of A K for K ∈ {E, D, R}. Next, let u n ∈ W s,2 K (Ω) be an eigenfunction associated with λ n . Then, A K u n = λ n u n . Let α > 0 be a real number. Since α ∈ ρ(−A K ), we have that αI + A K is invertible. From A K u n = λ n u n we have that By [48, Theorems 6.2 and 6.6], the semigroup (e −tA K ) t≥0 , for K ∈ {N , R}, is ultracontractive in the sense that it maps L 2 (Ω) into L ∞ (Ω). It also follows from (2.3) that the semigroup (e −tA K ) t≥0 , for K ∈ {E, D}, is ultracontractive. More precisely, there is a constant C > 0 such that for every f ∈ L p (Ω) and t > 0, and Since for every f ∈ L 2 (Ω) and α > 0, it follows from (2.25) and (2.26) and the fact that u n ∈ L p (Ω) for some p > N 2s that there exists a constant M > 0 such that This completes the proof of part (b).
(c) Let p ∈ [1, ∞] and let A p,K be the generator of the semigroup on L p (Ω). Since A K = A 2,K has a compact resolvent and Ω is bounded, it follows from the ultracontractivity that each semigroup has a compact resolvent on L p (Ω) for p ∈ [1, ∞]. Now it follows from [15, Corollary 1.6.2] that the spectrum of A p,K is independent of p.
(d) Since I + A K is invertible we have that the L 2 -norm of (I + A K ) θ defines an equivalent norm on D A θ K . Besides, for every f ∈ L 2 (Ω), We shall prove the first claim in the case K ∈ {N } (the argument in the cases K ∈ {E, D, R} is similar). Using (2.26) for t ∈ (0, 1) and the contractivity of e −tA N for t > 1, for u ∈ D A θ K , we deduce The first integral is finite if and only if θ > N/4s. For the second claim, we begin by interpolating the inequality (2.26) with the L 2 (Ω)-contractivity of e −tA N to obtain that for every p ∈ (2, ∞). As above with α > 1−2/p ∈ (0, 1) , the L 2 -norm of (αI + A K ) θ defines an equivalent norm on D A θ K so that (2.27) for t ∈ (0, 1) and the contractivity of e −tA N for t > 1, for u ∈ D A θ K , allow us to deduce once again that provided that the first integral is finite, i.e., θ > N 1 − 2p −1 / (4s). The proof in the remaining cases K ∈ {E, D, R} is analogous and thus omitted.
We notice that the assumption 1 2 < s < 1 if K = N or K = R in (H) is not a restriction, since, otherwise Dirichlet, fractional Neumann and Robin boundary conditions coincide, that is, We conclude the section by the following regularity result taken from [44, Proposition 1.1].
Proposition 2.7. Let Ω ⊂ R N be a bounded domain with Lipschitz continuous boundary, f ∈ L ∞ (Ω) and let u ∈ W s,2 0 (Ω) be a weak solution of the elliptic problem Remark 4. We notice that even if (−∆) s and A s Ω are related by the relation (1.4), due to the effect of the potential V Ω one cannot immediately deduce from Proposition 2.7 a similar result for the elliptic problem associated with A s Ω . To have such a result, one needs to give a complete proof.
3. Well-posedness and regularity. Without loss of generality, we take d ≡ 1 in this section. We consider all problems (1.5)-(1.7) in unified form in Ω, (3.1) where A K , K ∈ {D, N , R, E} , is the self-adjoint operator associated with the regional fractional Laplacian, subject to Dirichlet, Neumann, Robin and the fractional Laplace operator associated with Dirichlet (i.e., u = 0 on R N \Ω) boundary conditions, respectively, as introduced in the previous section.
Our assumptions on the nonlinearity f that we will need in this section are as follows. (H1): for some appropriate positive constants C f , c f , C f , c f and some p > 1.
for some positive constant C f . Concerning regularity conditions for the domain Ω we assume the following. In what follows we shall use classical (linear/nonlinear semigroup) definitions of strong solutions to the unified problem (3.1). "Strong" solutions 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 "Laplacian" diffusion. Throughout the remainder of this article the solution of our system is a function that depends on both time and spatial variables but in our proofs we sometime omit the dependence in x. Definition 3.1. Let u 0 ∈ L 2 (Ω) be given and assume (H2) holds for some p > 1. The function u 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) . • Variational identity: for the weak solutions the following equality holds for all ξ ∈ W s,2 K (Ω) ∩ L p (Ω) , a.e. t ∈ (0, T ). Finally, we have, in the space L 2 (Ω) , u (0) = u 0 almost everywhere.
Finally, our notion of (global) strong solution is as follows.
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 and developed in [24]. Then exploiting a modified Moser iteration argument we show that the strong solution is actually a global solution. In the second part, we will show the existence of (globally-defined) weak solutions which satisfy the energy identity (3.6) and the variational form (3.5). Then combining the energy method with another refined iteration scheme we also show that any weak solution with initial data in L 2 (Ω) acquires additional smoothness in an infinitesimal time.
3.1. Weak and strong solutions. We first prove a Poincaré-type inequality in the space W s,2 (Ω), 0 < s < 1. Its application is crucial in the proof of strong solutions to semilinear parabolic equations with fractional diffusion. Then for all ∈ (0, 1) there is ζ > 0 such that for all u ∈ W s,2 K (Ω).

CIPRIAN G. GAL AND MAHAMADI WARMA
Proof. By a scaling argument it suffices to prove the inequality for u L 2 (Ω) = 1.
Suppose that there is no ζ > 0 such that the inequality holds for a given ∈ (0, 1). Then for any k ∈ N there is u k ∈ W s,2 The foregoing inequality implies that the resulting sequence (u k ) is bounded in W s,2 K (Ω). Since the identity operator is a compact map from W s,2 K (Ω) into L 2 (Ω) and into L 1 (Ω), respectively, we find a subsequence, again denoted by (u k ), that converges strongly in L 2 (Ω) and in L 1 (Ω) to some limit function u ∈ W s,2 K (Ω). By assumption we have u L 2 (Ω) = 1. On the other hand, the inequality shows that u k 2 L 1 (Ω) ≤ k for all k, such that u L 1 (Ω) = 0 and thus u = 0 a.e. in Ω. This is a contradiction which altogether completes the proof of the lemma.
We notice that it follows from Lemma 3.3 that for u ∈ W s,2 K (Ω), defines an equivalent norm on W s,2 K (Ω). In fact, it is clear that there exists a constant C > 0 such that (E K (u, u)) 1/2 + u L 1 (Ω) ≤ u W s,2 K (Ω) . Using Lemma 3.3 we get the converse inequality.
The next inequality is essential in comparing various energy forms. Proof. We prove the inequality by elementary analysis. Define the function g : Using the definition of E, we first notice that (3.8) is equivalent to showing that Indeed, assume this were true so that for u : Ω → R there holds (3.10) Then (3.8) is an immediate consequence of (3.10). We now prove our claim. First, we observe that g(z, t) = g(t, z), g(z, 0) ≥ 0, g(0, t) ≥ 0 and g(z, t) = g(−z, −t).
Therefore, without any restriction, we may assume that z ≥ t. A simple calculation shows that 2 p + 1 |z| Since the function ϕ : R → R given by ϕ(τ ) = |τ | p (p ≥ 2) is convex, then using the well-known Jensen inequality, it follows that We have shown the claim (3.9) and this completes the proof of lemma.
We will now state a well-known result for the non-homogeneous Cauchy problem  . Let ϕ K be the functional on L 2 (Ω) with domain D(ϕ K ) = W s,2 K (Ω) and defined by ϕ K (u) = 1 2 E K (u, u) for u ∈ W s,2 K (Ω). From Theorem 2.5 we know that −A K = −∂ϕ K generates a strongly continuous (linear) semigroup (e −tA K ) t≥0 of contraction operators on L 2 (Ω). Finally, e −tA K is non-expansive on L ∞ (Ω) , that is, and A K is strongly accretive only in the case K ∈ {D, R, E} whereas A N is only accretive on L 2 (Ω) (this follows from the definition of A N and from Theorem 2.5). Thus, the operator version of the original problem (3.1) reads and where we have set f K (τ ) = f (τ ) , K ∈ {D, E, R} and f N (τ ) = f (τ ) − χτ . Clearly, A N ,χ is also strongly accretive on L 2 (Ω) by construction and satisfies (3.12).
We adapt an argument we have developed in [24,Theorem 3.4]. We prove the existence of a (locally-defined) strong solution to (3.13), (3.14) by a fixed point argument. We shall focus on the case K ∈ {D, R, E} as the case K = N is similar. To this end, fix 0 < T * ≤ T , consider the space X T * ,R * ≡ u ∈ C ([0, T * ] ; L ∞ (Ω)) : u (t) L ∞ (Ω) ≤ R * and define the following mapping (3.15) Note that X T * ,R * , when endowed with the norm of C ([0, T * ] ; L ∞ (Ω)) , is a closed subset of C ([0, T * ] ; L ∞ (Ω)) , and since f is continuously differentiable, Σ (u) (t) is continuous on [0, T * ]. We will show that, by properly choosing T * , R * > 0, Σ : X T * ,R * → X T * ,R * is a contraction mapping with respect to the metric induced by the norm of C ([0, T * ] ; L ∞ (Ω)) . The appropriate choices for T * , R * > 0 will be specified below. First, we show that u ∈ X T * ,R * implies that Σ (u) ∈ X T * ,R * , that is, Σ maps X T * ,R * to itself. From (3.12) and the fact that f ∈ C 1 loc (R), we observe that the mapping Σ satisfies the following estimate for some positive continuous function Q f which depends only on the size of the nonlinearity f . Thus, provided that we set R * ≥ 2 u 0 L ∞ (Ω) , we can find a sufficiently small time T * > 0 such that in which case Σ (u (t)) ∈ X T * ,R * , for any u (t) ∈ X T * ,R * . Next, we show that by possibly choosing T * > 0 smaller, Σ : X T * ,R * → X T * ,R * is also a contraction. Indeed, for any u 1 , u 2 ∈ X T * ,R * , exploiting again (3.12), we estimate This shows that Σ is a contraction on X T * ,R * provided that T * < 1 is much smaller than the one determined by (3.16) and T * Q f (R * ) < 1. Therefore, owing to the contraction mapping principle, we conclude that problem (3.13) has a unique local solution u ∈ X T * ,R * . This solution can certainly be (uniquely) extended on a right maximal time interval [0, T max ), with T max > 0 depending on u 0 L ∞ (Ω) , such that, either T max = ∞ or T max < ∞, in which case lim t Tmax u (t) L ∞ (Ω) = ∞. Indeed, if T max < ∞ and the latter condition does not hold, we can find a sequence t n T max such that u (t n ) L ∞ (Ω) ≤ C. This would allow us to extend u as a solution to Equation (3.13) to an interval [0, t n + δ), for some δ > 0 independent of n. Hence u can be extended beyond T max which contradicts the construction of T max > 0. To conclude that the solution u belongs to the class in Definition 3.2, let us further set G (t) := −f (u (t)) , for u ∈ C ([0, T max ) ; L ∞ (Ω)) and notice that u is the "generalized" solution of such that u (0) = u 0 ∈ L ∞ (Ω) ⊂ L 2 (Ω) = D(A K ). By Theorem 3.5, the "generalized" solution u has the additional regularity ∂ t u ∈ L 2 (δ, T max ); L 2 (Ω) , which together with the facts that u is continuous on [0, T max ) and f ∈ C 1 loc (R), yield G ∈ W 1,2 (δ, T max ); L 2 (Ω) ∩ L ∞ (δ, T max ); L 2 (Ω) . Thus, we can apply Theorem 3.6 to deduce that such that the solution u is Lipschitz continuous on [δ, T max ), for every δ > 0. Thus, we have obtained a locally-defined strong solution in the sense of Definition 3.2. As to the variational equality in Definition 3.1, we note that this equality is satisfied even pointwise (in time t ∈ (0, T max )) by the strong solutions. Our final point is to show that T max = ∞, because of condition (H1). This ensures that the strong solution constructed above is also global.
Step 2. (Energy estimate) Let m ≥ 1 and consider the function . First, notice that E m is well-defined on (0, T max ) because u is bounded in Ω × (0, T max ) and because Ω has finite measure. Since u is a strong solution on (0, T max ) , see Definition 3.2 (or (3.20)), recall that u is continuous from [0, T max ) → L ∞ (Ω) and Lipschitz continuous on [δ, T max ) for every δ > 0. Thus, u (as function of t) is differentiable a.e., whence, 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 procedure yields the following standard energy identity: Assumption (H1) in the case K ∈ {D, R, E} implies that for some C f > 0 and for all τ ∈ R. This inequality allows us to estimate the nonlinear term in (3.21). We have (by using an equivalent norm in W s,2 K (Ω)) that 1 2 where |Ω| denotes the N -dimensional Lebesgue measure of Ω. In view of (3.3) and Gronwall's inequality, (3.23) gives the following estimate for t ∈ (0, T max ) , , for some constants ρ = ρ (N, Ω) > 0, C (f, |Ω|) > 0. The proof of the energy inequality in the case K = N is analogous (in this case, f N obeys (3.22)).
Step 3. (The iteration argument). In this step, 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 begin by showing that E m (t) satisfies a local recursive relation which can be used to perform an iterative argument. Testing the variational equation (3.5) for the strong solution with |u| m−1 u, m ≥ 1, gives on account of (3.22) and Lemma 3.4 the following inequality: , in all cases K ∈ {D, N , R, E}. Next, set m k + 1 = 2 k , k ∈ N, and define Our goal is to derive a recursive inequality for M k using (3.25). In order to do so, for q > 1 fixed that we will choose below, we define We aim to estimate the terms on the right-hand side of (3.25) in terms of the L 1+m k−1 (Ω)-norm of u. First, the Hölder inequality and the Sobolev inequality (i.e., W s,2 K (Ω) ⊂ L 2q (Ω), with q = q (N, s) ∈ (1, N/ (N − 2s)], if N > 2s and q ∈ (1, ∞) if N = 2s, see (2.3) and (2.1)) yield with s k = p k q ≡ q/ (2q − 1) ∈ (0, 1). Applying Young's inequality on the right-hand side of (3.27), we get for every > 0, for some α > 0 independent of k, since z k := q k / (1 − s k ) ≡ 2. In order to estimate the last term on the right-hand side of (3.25), we define two decreasing and increasing sequences (l k ) k∈N and (w k ) k∈N , respectively, such that and observe that they satisfy for all k ∈ N (in particular, w k → 2 as k → ∞). The application of Young's inequality in (3.28) yields again Hence, inserting (3.28), (3.29) into inequality (3.25), choosing a sufficiently small 0 < ≤ 0 := 1 4 , and simplifying, we obtain for t ∈ (0, for some positive constant δ > 0 independent of k. for some ζ > 0 independent of u, k. We can now combine (3.31) with (3.30) to for t ∈ (0, T max ) . Integrating (3.32) over (0, t), we infer from Gronwall-Bernoulli's inequality [13,Lemma 1.2.4] that there exists yet another constant c > 0, independent of k, such that On the other hand, let us observe that there exists a positive constant C ∞ = C ∞ ( u 0 L ∞ (Ω) ) ≥ 1, independent of k, such that u 0 L 2 k (Ω) ≤ C ∞ . Taking the 2 k -th root on both sides of (3.33), and defining X k := sup t∈(0,Tmax) u (t) L 2 k (Ω) , we easily arrive at It remains to notice that (3.35) together with the bound (3.24) shows that the norm u (t) L ∞ (Ω) is uniformly bounded for all times t > 0 with a bound, independent of T max , depending only on u 0 L ∞ (Ω) , the "size" of the domain and the non-linear function f. This gives T max = +∞ so that strong solutions are in fact global. This completes the proof of the theorem. for any T > δ > 0. This follows from the fact that the nonlinear function f is continuously differentiable. Note that the second regularity in (3.36) is a consequence of the first one, the time regularity in (3.7) (see Definition 3.2) and the variational identity (3.5).
The following result is immediate. given by

37)
where u is the (unique) strong solution in the sense of Definition 3.2.
In the final part of this section, we aim to prove the existence of weak solutions in the sense of Definition 3.1. Proof. We divide the proof into three steps. For practical purposes, C will denote a positive constant that is independent of time, T , > 0 and initial data, but which only depends on the other structural parameters. Such a constant may vary even from line to line.
Step 1. (Approximation scheme). First, we consider a sequence u 0 ∈ L ∞ (Ω)∩ W s,2 K (Ω) such that u 0 → u 0 = u (0) in L 2 (Ω) . Next, for each K ∈ {D, N , R, E} let u (t) be a strong solution, in the sense of Definition 3.2, of the system in Ω. (3.38) Note that such a smooth solution exists since every function that satisfies (H2) also obeys (3.22). Testing the weak formulation associated with problem (3.38), cf.
(3.45) By refining in (3.45), u converges to u a.e. in Ω × (0, T ). Then, by means of known results in measure theory [12], the continuity of f and the convergence of (3.45) imply that f (u ) converges weakly to f (u) in L p ((0, T ) × Ω), while from (3.41)-(3.42) and the linearity of A K , we further see that (Ω)). (3.46) We can now pass to the limit as → 0 in the weak form (3.5) for u to deduce the desired weak solution u, satisfying the variational identity (3.5) and the regularity properties (3.4 (Ω) + L p (Ω)) can be represented as These spaces are precisely the dual of the space L 2 ((0, T ); W s,2 K (Ω)), and the space L p ((0, T ); L p (Ω)), respectively. In particular, we obtain that every weak solution u ∈ C [0, T ] ; L 2 (Ω) , and that the map t → u (t) 2 2,Ω is absolutely continuous on [0, T ], such that u satisfies the energy identity (3.6).
Step 2. (Uniqueness and continuous dependence). As usual, consider any two weak solutions u 1 , u 2 , and set u (t) = u 1 (t) − u 2 (t). According to the energy identity (3.6) and assumption (H3), we obtain d dt Upon integration over (0, t), we infer This yields the desired continuous dependance result with respect to the initial data. The proof of the theorem is finished.
Consequently, problem (3.1) defines a dynamical system in the classical sense.

49)
where u is the (unique) weak solution in the sense of Definition 3.1.

3.2.
Regularity of weak solutions. The main result of this section is concerned with proving that any weak solution with initial condition in L 2 (Ω) acquires additional smoothness in an infinitesimal time; more precisely, it becomes a strong solution in the sense of Definition 3.2. Moreover, the same result also establishes the existence of an absorbing ball for the semigroup S K in the space W s,2 K (Ω) ∩ L ∞ (Ω). The latter is an essential property in the theory of attractors (see the next section). Fix now K ∈ {D, N , R, E} .
Step 1. (The bound in L ∞ (Ω)). In this case, as in the proof of Theorem 3.8, we can use strong solutions in order to provide sufficient regularity to justify all the calculations performed in the proof below. At the very end one can pass to the limit and obtain the estimate even for the weak solutions. Let now τ > τ > 0 and fix µ := τ − τ . We claim that there exists a positive constant C = C (µ) ∼ µ −η (for some η > 0), independent of t and the initial data, such that sup The argument leading to (3.51) follows exactly as in [21, Theorem 2.3] (cf. also [22,24]). It is based on the following recursive inequality for E m k (t), which is a consequence of (3.32) and (3.27)-(3.30): 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, l > 0 are independent of k and that C (µ) is uniformly bounded in µ if µ ≥ 1 (see [21,Theorem 2.3]). We can iterate in (3.52) with respect to k ≥ 1 and obtain that sup Therefore, we can take the 2 k -th root on both sides of (3.53) and let k → +∞. Using the facts that ζ : i 2 i < ∞, we easily deduce (3.51). From the inequality (3.51) together with the energy estimate (3.40), which is also satisfied by the weak solution u (t) with initial datum u 0 ∈ L 2 (Ω), we deduce with C ρ ∼ ρ −η , for some η > 0, for each ρ > 0. This yields the first part in (3.50).
Step 2. (The bound in W s,2 K (Ω)). The argument relies on using the test function ξ = ∂ t u (t) into the variational equation (3.5). However, in order to further justify this choice in (3.5) we actually need to require more regularity of the strong solution, in particular we need to have u ∈ W 1,q loc ((0, ∞); W s,2 K (Ω)), for some q > 1. (3.55) Due to the non-smooth nature of the domain Ω and its boundary ∂Ω, one generally lacks any further information on both weak and strong solutions than the one provided by Definitions 3.1 and 3.2. In order to overcome this difficulty, we need to further truncate the strong solutions resulting in approximate solutions which will now have the desired regularity (3.55). The latter is provided by a proper basis associated with the nonnegative self-adjoint operator A K on L 2 (Ω). Indeed, as a basis for L 2 (Ω) we can choose the complete system of eigenfunctions ξ K i i∈N for , which is a key regularity provided by the statement of Theorem 2.5. According to the general spectral theory, the eigenvalues λ i can be increasingly ordered and counted according to their multiplicities in order to form a real divergent sequence. Moreover, the respective eigenvectors ξ K i turn out to form an orthogonal basis in W s,2 K (Ω) and L 2 (Ω) , respectively. The eigenvectors ξ K i may be assumed to be normalized in L 2 (Ω). Let P n : W s,2 K (Ω) → span {ξ 1 , ξ 2 , ..., ξ n } be the usual orthogonal projector and consider a Galerkin truncation of u in the form solving the problem ∂ t u ,n = −P n (A K u ε,n + f (u ,n )) such that u ,n (0) = P n u (0). We recall that the latter problem is uniquely solvable by the Cauchy-Lipschitz theorem since f ∈ C 1 , and that the solution u ,n has the desired regularity (3.55). Thus, they key choice of the test function ξ = ∂ t u n, into the variational formulation (3.5) is now allowed for these truncated solutions u ,n . We infer d dt for all t ≥ 0. As usual, F denotes the primitive of f , i.e., F (σ) = σ 0 f (y) dy. Multiply the foregoing equation by t ≥ ρ > 0 and integrate over (0, t) to get t 1 2 E K (u ,n (t) , u ,n (t)) + (F (u ,n (t)) , 1) for all t ≥ ρ. Recalling that, due to (H2)-(H3), F is bounded from below, independently of n, and |F (σ)| ≤ C (1 + |σ| p ), we infer from (3.40) (which is also satisfied by u ,n ) and (3.54), for some constant c > 0 independent of t, n, . On the basis of standard lowersemicontinuity arguments, we can now pass to the limit, first with respect to n → ∞ and then as → 0 + , to obtain the desired inequality (3.50), owing once more to estimates (3.40)-(3.54) and uniqueness (cf. Theorem 3.8). The proof is finished.

4.
Finite dimensional attractors. The first main result of this section is the following. As before, we fix K ∈ {D, N , R, E} .
Theorem 4.1. Let the assumptions of Theorems 3.8 be satisfied for some f ∈ C 2 (R). There exists a compact attractor A K L 2 (Ω) for the parabolic system (3.1) which attracts the bounded sets of L 2 (Ω). Moreover, A K is the maximal bounded attractor in D (A K ) ∩ L ∞ (Ω) and has finite fractal dimension, that is, Proof. Step 1. (Global attractor). By the proof of Theorem 3.8, (3.40), there is a ball B K in L 2 (Ω) which is absorbing in L 2 (Ω) , meaning that for any bounded set U ⊂ L 2 (Ω) there exists t 0 = t 0 ( U L 2 (Ω) ) > 0 such that S K (t)U ⊂ B K for all t ≥ t 0 . Moreover, by Theorem 3.9, (3.50) and (3.54), we infer the existence of a new time t 1 ≥ 1 such that for some positive constant C independent of time and the initial data. We also observe that the Galerkin truncated solutions u ,n satisfy (in the weak sense of Definition 3.1) the following "time-differentiated" version of the original problem where we have set u ,n = ∂ t u ,n . In particular, testing the aforementioned equation with 2tu ,n we deduce upon integrating over (0, t) that where in the last line we have used assumption (H3). Exploiting (3.58) we obtain in the limit as ( , n) → (0, ∞) that The usual comparison argument in equation (3.38) together with the uniform bound (4.1) yields u ∈ L ∞ ((t 1 , ∞); D (A K )) uniformly with respect to time. Thus, for any bounded set U ⊂ L 2 (Ω) , we have that ∪ t≥t1 S K (t)U is relatively compact in L 2 (Ω), when endowed with the metric topology of L 2 (Ω). Finally, applying [46, Theorem I.1.1] we have that the set is a compact attractor for S K , and the rest of the result is immediate.
Step 2. (Uniform differentiability on A K ). We show that the bound obtained in (4.1) is sufficient to show the uniform differentiability of S K on the attractor A K with Υ (t; u 0 ) ξ := S K (t) ξ as a solution of To this end, consider two solutions u 1 , u 2 of problem (3.1) with initial conditions u i (0) ∈ A K , i = 1, 2 and let U be the solution of (4.2) with ξ = u 1 (0) − u 2 (0). Then the function ω (t) = u 1 (t) − u 2 (t) − U (t) satisfies the equation with g := f (u 1 ) − f (u 2 ) − f (u) (u 1 − u 2 ). Next, by Taylor's theorem and the fact that u i ∈ L ∞ (Ω) (as both u 1 , u 2 lie on the attractor A K ⊂ L ∞ (Ω)), we infer that |g (x)| ≤ C |u 1 (x) − u 2 (x)| 2 , for some C > 0. Let r be the conjugate exponent 1308 CIPRIAN G. GAL AND MAHAMADI WARMA to 2 * from the Sobolev embedding inequality (2.1) such that W s,2 K (Ω) ⊂ L 2 * (Ω). Therefore, if we write g (t) = g (u (x, t)) it follows that , owing to Hölder's inequality and the L ∞ (Ω) bound on u 1 , u 2 . Choosing now δ = 2 − r (1 + ε) , for some ε ∈ (0, (2 − r) /r), we easily deduce from (3.48) that which upon integrating over (0, t) gives L 2 (Ω) , for some function C (t) > 0. Thus, the flow S K (t) is indeed differentiable on A K and the derivative S K (t) is given by the solution of the linearized equation (4.2). Finally, we also observe that for ξ ∈ L 2 (Ω) the set Υ (t; u 0 ) ξ is relatively bounded in W s,2 K (Ω) ∩ L ∞ (Ω), whence the mapping Υ (t; u 0 ) is also compact in L 2 (Ω) for each t > 0. The desired finite dimensionality of the global attractor A K follows from standard results in the theory of infinite dimensional dynamical systems (see, e.g., [12,46]). The proof of the theorem is now complete.
The following lemma states other basic properties of the dynamical system associated with problem (3.1). In particular, it shows that S K (t) , L 2 (Ω) is a "gradient" system, namely, we have the following.
has along the strong solutions of (3.1), the derivative d dt , a.e. t > 0. In other words, the functional L K is decreasing, and becomes stationary exactly on equilibria u * , which are solutions of the system: Proof. The proof is a consequence of the calculation (3.57) and the fact that strong solutions are smooth enough, see Definition 3.2 and Remark 6.
The foregoing Lemma 4.2 can now be used to study the asymptotic behavior of the solutions of (3.1) by means of the LaSalle's invariance principle. To this end, to any (weak) trajectory of (3.1) we associate the respective ω-limit set ω L 2 : ω L 2 := y ∈ L 2 (Ω) : ∃t n → ∞, z n ∈ L 2 (Ω) such that S K (t n ) z n → y in L 2 -topology . The following lemma states some basic properties of the ω-limit sets associated with the dynamical system S K (t) , L 2 (Ω) . Lemma 4.3. (i) Any ω-limit set ω L 2 is nonempty, compact and connected.
(ii) The trajectory approaches its own limit set in the norm of L 2 (Ω), i.e., (iii) Any ω-limit set is invariant: new trajectories which start at some point in ω L 2 remain in ω L 2 for all times t > 0.
Proof. The proof is immediate owing to the continuity properties of the strong solution and the compactness of the embedding W s,2 K (Ω) ⊂ L 2 (Ω) . The second main result is concerned with the parabolic problem in Ω, (4.5) in the case K = E (recall that u = 0 in R N \Ω, N ≥ 1), with d > 0 playing the role of a diffusion coefficient. Recall that 0 < s < 1.
as either C f → ∞ or d → 0 + , where c * depends on the shape of Ω and N only.
Proof. In order to deduce (4.6), it is sufficient (see, e.g., [12,Chapter III,Definition 4.1]) to estimate the j-trace of the operator L (t, U (t)) : where the set of real-valued functions ϕ j ∈ L 2 (Ω) ∩ W s,2 E (Ω) is an orthonormal basis in Q m (L 2 (Ω)). Since the family ϕ j is orthonormal in Q m L 2 (Ω) , using assumption (H3) on f (i.e., f (σ) ≥ −C f , for all σ ∈ R), we find We now consider the eigenvalue problem (−∆) s ϕ = λϕ in Ω and ϕ = 0 in R N \Ω, which is equivalent to the eigenvalue problem A E ϕ = λϕ, ϕ ∈ D (A E ). By [5], the eigenvalues λ j obey the following Weyl asymptotic formula: N +1 + C f m, for some c 0 > 0 which only depends on the shape of Ω and N . Let us define the function on the right-hand side as ρ (m). The function ρ is concave and the non-zero root of the equation ρ (m) = 0 is Thus, we can apply [12, Corollary 4.2 and Remark 4.1] to deduce that from which (4.6) follows.

Remark 7.
It is worth emphasizing that when s = 1, in (4.7) we obtain the classical Weyl's formula for the Dirichlet Laplacian eigenvalue problem −∆ϕ = λϕ in Ω, ϕ = 0 on ∂Ω. Moreover, the upper bound in (4.6) also stabilizes as s → 1 to the corresponding upper bound for the fractal dimension of the parabolic equation ∂ t u − ∆u + f (u) = 0 in Ω × (0, ∞) , and u = 0 on ∂Ω × (0, ∞), see, for instance, [46, Chapter VI], [12]. We conjecture that a similar bound on the dimension of A K also holds in the remaining cases K ∈ {D, N , R} where at the moment Weyl asymptotic formulas are not yet available.
Remark 8. One can also provide a lower bound on the dimension of A E : for some c > 0 which depends only on the shape of Ω and N , if f (0) < 0 is sufficiently large or d > 0 is sufficiently small. This estimate is obtained in the same spirit of [46, Chapter VII] (see also [22]) and relies on the fact that, owing to the boundedness of u ∈ L ∞ (Ω), the semigroup S E (t) is uniformly differentiable with derivative of Hölder class C α , α ∈ (0, 1) (in fact, in our case α = 1). More precisely, there exists a smooth manifold W loc (u * ) (of class C 1,α ) localized in an open neighborhood of the hyperbolic equlibrium u * = 0, with finite instability dimension dim X * (u * ) < ∞. Here, X * (u * ) is the unstable subspace of −A K − f (u * ) which is tangent to W loc (u * ) at the point u * and we recall that the global attractor always contains localized unstable manifolds.

5.
Asymptotic stabilization to single equilibria. Let u be a weak solution of (3.1) according to Theorem 3.8. We show that any such weak solution converges (in a certain sense) to a single steady state as time tends to infinity. Recall that any weak solution of (3.1) regularizes in finite time to a strong solution by Theorem 3.9. Moreover, observe that, by virtue of (3.50) and the proof of Theorem 4.1, all stationary solutions u * ∈ ω L 2 (u) of problem A K u * + f (u * ) = 0 in Ω, are bounded in L ∞ (Ω) ∩ D (A K ). Setting now, for each K ∈ {D, E, N , R} , it is easy to see that L K ∈ C 1 L 2 (Ω) , R but L K / ∈ C 2 L 2 (Ω) , R no matter how smooth F is due to the nature of the nonlocal term E K (cf. also [19,23,39], where the same issue occurs for other nonlocal problems).
Consequently, we shall employ a generalized version of the Lojasiewicz-Simon theorem which is well-suited for our nonlocal problem (3.1). As usual, we fix K ∈ {D, E, N , R} and set J (r) := |r| −N −2s . Recall that the Lojasiewicz-Simon result applies in principle to functionals which can be written as a maximal monotone operator plus a linear compact perturbation. The version that applies to our cases K ∈ {D, E, N , R} is formulated in the subsequent lemma requiring the following condition: (H-er): Let w be a bounded solution of the elliptic boundary value problem A K w = h in Ω, for some h ∈ L ∞ (Ω). Then, w ∈ C 0,ν Ω for some ν ∈ (0, 1).
Lemma 5.1. Let F ∈ C 2 be a real analytic function satisfying (H1), (H3) and let Ω obey condition (H4). Assume condition (H-er). Then, there exist constants θ ∈ (0, 1 2 ], C > 0, ε > 0 such that the following inequality holds: We employ an abstract version of the Lojasiewicz-Simon inequality provided by [20,Theorem 6] and which has been specifically tailored to our needs. We need the following assumptions: A1: Let (V, · V ) and (W, · W ) be Banach spaces densely and continuously embedded into the Hilbert space (H, · H ) and its dual (H * , · H * ), respectively. We assume that the restriction j |V of the duality map j ∈ L (H, H * ) to V is an isomorphism from V onto W = j (V ). A2: Let T ∈ L (H, H * ) be a self-adjoint and completely continuous operator such that its restriction T |V to V is a completely continuous operator in L (V, W ). For fixed π ∈ W and d ∈ R consider the quadratic functional Ψ : H → R given by Ψ (u) = T u, u + π, u + d, u ∈ H.
A3: Let U be an open subset of V and Φ : U → R be a Fréchet differentiable function. Additionally, assume that the Fréchet derivative DΦ : U → R is a real analytic operator which satisfies for all u, v ∈ U, for some constants c 1 , c 2 > 0. Moreover, assume that the second Fréchet derivative D 2 Φ (u) ∈ L (V, W ) is an isomorphism for all u ∈ U.
We make some comments on the assumption (H-er).
Remark 9. We notice that by Proposition 2.7, if Ω is a bounded domain with Lipschitz continuous boundary, then the assumption (H-er) is satisfied for the operator A E . It is also satisfied by the operator A K , K ∈ {D, N , R} for bounded domains in dimension N = 1 and if s > 1/2 (cf. Section 2). We think that assumption (H-er) is also satisfied for A K , K ∈ {D, N , R} for bounded domains with Lipschitz boundary, in any space dimension and for any s ∈ (0, 1). Such a result is not yet available in the literature, and is interesting on its own independently of the application given in the present paper, and also necessitates a careful study. Since this is not the main objective of the paper, we will not go into details.
Remark 10. Exploiting the L 2 (Ω) → (L ∞ (Ω) ∩ D (A K )) smoothing property of the weak and stationary solutions together with a similar argument from Theorem 3.9, and the convergence rate (5.3) it is possible to show the convergence rate: for some positive constant κ = κ (θ, u * ) ∈ (0, 1) . Indeed, we can also prove (3.51) for the difference u − u * owing to the boundedness of u, u * ∈ L ∞ (Ω) .
6. Some blow-up results. Our goal in this section is to show that assumption (H1) is in fact quite optimal for global well-posedness of strong solutions of the problem (3.1). We recall that this condition implies in particular that if f is a source with a bad sign at infinity then it can only be of at most linear growth at infinity. Indeed, we will show below by the concavity method of Levine-Payne [37] that as soon as f has superlinear growth and a bad sign at infinity as |σ| → ∞, as provided by the example f (σ) = − |σ| p−1 σ, σ ∈ R, with p > 1, then blowup in finite time of some strong solutions occurs. Theorem 6.1. Let Ω satisfy condition (H4) and suppose that f ∈ C 1 loc (R) obeys and u 0 ∈ L ∞ (Ω) ∩ W s,2 K (Ω) is an initial datum such that it follows for any ε > 0, that For α > 0, combining these estimates together yields provided that ε and α are small enough such that (1 + α) (1 + ε) ≤ 1 and A > 0 is large enough (since Q K (0) > 0, by assumption). The foregoing inequality implies as usual that which yields that the quantity V K (t) cannot remain finite for all time t > 0. The proof is finished.
Remark 12. The same blow-up result also holds for some initial datum for which E K (u 0 , u 0 ) > 0, see [25].