Reaction-diffusion systems with supercritical nonlinearities revisited

We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.


Introduction
We study the following reaction-diffusion system in a bounded domain Ω ⊂ R d with smooth boundary: (1.1) ∂ t u = a∆ x u − f (u) + g, u t=0 = u 0 endowed with the Dirichlet boundary conditions. Here u = (u 1 , · · · , u k ) is an unknown vector-valued function, a is a given diffusion matrix and f (u) and g are given nonlinearity and external forces respectively. Equations of the form (1.1) model various classical phenomena in modern science (e.g., heat conduction, chemical kinetics, various quantum effects (Ginzburg-Landau equations), mathematical biology (Fitz-Hugh-Nagumo or Keller-Segel equations), etc.) and have been intensively studied from both mathematical and applied points of view, see [3,8,10,14,22,26,30,34,35] and references therein. In a sense, this is the most studied and somehow simplest model example of an evolutionary PDE which may generate nontrivial dynamics.
Since the analytic properties of the linear system (1.1) are completely understood, the analogous properties for the nonlinear equation depend strongly on whether or not we are able to treat the term f (u) as a perturbation. As usual, if we want to have global existence of a solution, we need to find the proper a priori estimates, usually with the help of energy functionals or some "wisely" chosen Lyapunov type functionals. This, in turn, requires some restrictions on the function f and matrix a (to prevent the finite-time blow up of solutions). Then, if the found a priori estimates are strong enough to treat the nonlinearity as a perturbation (the so-called subcritical case), the analytic properties of the nonlinear equation is usually the same as for the dominating linear one and more or less complete theory is available. In contrast to this, in the supercritical case, the nonlinearity is strong enough to destroy the nice properties of the underlying linear equation, for instance, to produce the finite-time blow up of initially smooth solutions (despite the fact that the "energy" remains bounded and dissipative, see [6] for such a phenomena in complex Ginzburg-Landau equation, [32] for chemical kinetics equations or [23] for chemotaxis models). Usually, the sub/super criticality of the considered equation is determined by the growth rate of the nonlinearity f (u) which depends on a priori estimates available (through the choice of the phase space for the problem) and the space dimension (through Sobolev embedding theorems). Thus, the typical picture for equation (1.1) is the following: we have the so-called critical growth exponent p = p crit > 1 and an extra condition (1.2) |f (u)| ≤ C(1 + |u| p ), u ∈ R k on the nonlinearity and the equation is subcritical if p < p crit , critical if p = p crit and supercritical if p > p crit , see [3,8,34,35] for more details. Unfortunately, the universal conditions on f and a which would allow to avoid the finite-time blow up and give the dissipativity in nice phase spaces are known in the scalar case k = 1 only, so many different classes of sufficient conditions are suggested for the case of systems strongly depending on the area of science where the considered system comes from. For instance, from the point of view of chemical kinetics, it is natural to assume that a is diagonal with non-negative entries and f (u) satisfies the balance law which mimics the acting mass law for the concentrations u i of reagents (which usually belong to the non-negative cone in R k ). The natural energy here is the L 1 -norm of the solution u(t) (the total mass is conserved or at least non-increasing), see [30,33] and the references therein for more details. We note that in the supercritical case the solutions may blow up in finite time despite the conservation of total mass, see [32].
Clearly, assumptions (1.3) are not appropriate for many other types of equations of the form (1.1), for instance, for complex Ginzburg Landau or Fitz-Hugh-Nagumo equations, so other types of assumptions should be used instead. The most widespread (especially in the literature related with the attractor theory, see [3,8,34,35]) is the following dissipativity condition: (1.4) f (u).u ≥ −C, u ∈ R k which is usually accompanied by the assumption that a has a positive symmetric part. These assumptions are related with the so-called L 2 -energy identity (1.5) 1 2 d dt u(t) 2 L 2 + (a∇ x u, ∇ x u(t)) + (f (u(t)), u(t)) = (g, u(t)) which can be formally obtained by multiplying equation (1.1) by u and integrating over x and which gives (due to these assumptions) the dissipative control of the L 2 -norm of u(t), see Lemma 3.1. However, the critical exponent which corresponds to this energy control (and the choice H = L 2 (Ω) as a phase space): p crit := 1 + 4 d is rather restrictive (the most natural cubic nonlinearity is supercritical in 3D case) and not much can be said in general about the supercritical case where the uniqueness of solutions may be lost and finite-time blow up of the L ∞ -norm may occur (see [6] for the numerical blow up evidence in 3D complex Ginzburg-Landau equation, see also [8,31] and references therein for study the long-time behavior of solutions without uniqueness using the multi-valued or trajectory approaches). We also mention here the so-called anisotropic dissipativity assumption: where l = (l 1 , · · · , l k ) is a sufficiently large vector, introduced in [18]. This restriction accompanied by the assumption that a is diagonal gives p crit = ∞ if l = l(d) is large enough.
A natural alternative is to use the so-called monotonicity assumption: which is also very widespread in the literature related with attractors. This assumption is naturally related with the H 1 -energy identity: which is obtained by formal multiplication of (1.1) by −∆ x u and integration over x. Together with (1.6) this gives the dissipative control of the H 1 -norm of the solution, see Lemma 3.2 for the details. The critical growth exponent associated with this H 1 -energy control is can be found in many works, see [3,8] and references therein. However, as pointed out in [38], the monotonicity assumption (1.6) gives for free the control of H 2 -norm of the solution u(t) together with the L 2 -norm of f (u(t)), namely, we have a priori estimates for the solutions in the nonlinear space (Ω)} due to the control of the L 2 -norm of ∂ t u(t), see Lemma 3.3 and Corollary 3.4 below, and this gives us much better value of the critical exponent: As far as we know, up to the moment, this is the best growth restriction which guarantees (of course, under the monotonicity assumption (1.6)) the global existence of smooth solutions and which is widely used nowadays not only for reaction-diffusion equations, but for many other related problems (like Cahn-Hilliard equations, see [31] and references therein; strongly damped wave equations, see [11,24] and reference therein, etc.). We also note that the monotonicity assumption (1.6) gives the uniqueness of weak solutions (= solutions in the energy phase space H = L 2 (Ω), see section 5 below) even in the supercritical case which, in turn, allows to get a lot of information about the solutions and their long-time behavior in the supercritical case as well. The theory of equations (1.1) in the critical or supercritical cases is of a great current interest, see for instance [8,9,10,31,39,40] and references therein. However, in most cases rather essential extra restrictions on the nonlinearity f are posed like the following two sided estimate: which really simplifies the situation, but automatically excludes some interesting new phenomena which may appear in a general case. The aim of the present paper (which can be considered as a continuation of our work [38]) is to give a comprehensive study of weak and strong solutions (=solutions in the phase space D) of problem (1.1) as well as their long-time behavior in the supercritical case p > p crit with dissipative (assumption (1.4) is fulfilled) and monotone (assumption (1.6) is satisfied) nonlinearities trying to avoid/minimize further restrictions on f .
Weak and strong solutions of problem (1.1) have been constructed in [38] (see also section 5). However, in contrast to the case of assumptions (1.10), for weak solutions the equation is understood only in a sense of variational inequalities since we cannot guarantee that f (u) ∈ L 1 and, therefore, cannot treat the equation in the sense of distributions. By this reason, even the parabolic smoothing property (whether or not a weak solution becomes strong at the next time moment) becomes non-trivial and has been posed in [38] as an open problem.
Our first main result gives the positive answer on this question in the case where the nonlinearity has a polynomial growth rate.
Let the nonlinearity f satisfy the assumptions (1.2) (for some p > 0), (1.4) and (1.6), the diffusion matrix have positive symmetric part and g ∈ L 2 (Ω). Then any weak solution u(t) starting from u(0) ∈ H = L 2 (Ω) belongs to D for any t > 0. In other words, the instantaneous parabolic H to D smoothing property holds. In addition, the strong solutions of (1.1) are dissipative in D-norm as well.
The proof of this theorem is based on estimation of f (u) in Lebesgue spaces L q (Ω) with 0 < q < 1 and is given in section 6.
Our next result shows that the critical growth exponent can be slightly improved.
Theorem 1.2. Let the assumptions of Theorem 1.1 hold and let, in addition, the growth exponent p of the nonlinearity satisfy for some small positive ε = ε(a). Then any weak solution u(t) of problem (1.1) possesses an instantaneous H to L ∞ (Ω) smoothing property. In particular, finite-time blow up of smooth solutions is impossible and the actual regularity of a solution u(t) is restricted by the regularity of Ω, f and g only.
In the case where this data is C ∞ -smooth, the corresponding solution u(t) will be also C ∞ for any t > 0.
We now turn to the attractors. The existence of a global attractor for problem (1.1) in H has been verified in [38], however, the question about strong attraction in D has been remained open. Our next result gives a positive answer on this question under the extra restriction (1.11) |f ′ (u)| ≤ C(|f (u)| + 1 + |u|), u ∈ R k . Theorem 1.3. Let the assumptions of Theorem 1.1 hold and let, in addition, f satisfy (1.11). Then the solution semigroup S(t) associated with problem (1.1) possesses a compact global attractor A in the phase space D.
The proof of this theorem is given in section 7 and is based on the energy type arguments. We expect that assumption (1.11) is technical and can be removed, but it is strongly related with the validity of the integration by parts formula see the discussion in section 9 below.
Finally, we study the finite-dimensionality of the constructed global attractor A in D and the existence of the so-called exponential attractor (see [15,16,17,31] and also section 8 for more details). The finite-dimensionality of the global attractor A has been established in [38] under the similar assumptions using the so-called method of l-trajectories developed in [28,29]. In section 8 we suggest an alternative more transparent method for constructing of an exponential attractor which does not utilize l-trajectories and works directly in the phase space.
The paper is organized as follows. Notations and spaces which will be used throughout of the paper are introduced in section 2 and the standard a priori estimates for the solutions u(t) of problem (1.1) are recalled in section 3.
The existence of strong solutions for problem (1.1) is verified in section 4 based on special approximations of the nonlinearity f . The definition of a weak solution of problem (1.1) in the sense of variational inequalities as well as the proof of its global existence and uniqueness is given in section 5. Moreover, the existence of a global attractor A for the solution semigroup S(t) is also verified there.
The weak to strong instantaneous smoothing property, see Theorem 1.1, is verified in section 6. The further regularity of strong solutions is obtained in section 7. In particular, the proof of Theorems 1.2 and 1.3 are given there. Some results about the partial regularity of the elliptic problem associated with equations (1.1) which have an independent interest are obtained in Appendix A. The existence of an exponential attractor M, see Theorem 1.4, is given in section 8.
Finally, section 9 discusses natural extensions of the developed theory to other classes of dissipative PDEs, in particular, to fractional reactiondiffusion systems and (fractional) Cahn-Hilliard type equations. At the end of this section we also discuss some important (at least from our point of view) open problems for further investigation.

Assumptions and preliminaries
Throughout of the paper we consider the following reaction-diffusion system in a bounded domain Ω ⊂ R d : Here u = (u 1 , · · · , u k ) is an unknown vector valued function, a is a diffusion matrix satisfying (Ω) is a given external force and the nonlinearity f is assumed to satisfy the following conditions: where C and K are some fixed constants, u.v stands for the standard inner product in R k and f ′ (u) ≥ −K means f ′ (u)ξ.ξ ≥ −K|ξ| 2 for all ξ ∈ R k . For any l ∈ N and any 1 ≤ p ≤ ∞ we denote by W l,p (Ω) the Sobolev space of distributions u ∈ D ′ (Ω) such that u and all its partial derivatives up to order l inclusively belong to the Lebesgue space L p (Ω). As usual, for non-integer values of l, we define W l,p (Ω) := B l p,p (Ω) using real interpolation (B l p,p is a classical Besov space, see e.g., [36]). Moreover, the symbol W l,p 0 (Ω) stands for the closure of C ∞ 0 (Ω) in W l,p (Ω) and the space W −l,p (Ω) is defined as a dual space to W l,p 0 (Ω) with respect to the standard inner product in H = L 2 (Ω). To simplify the notations, we will write H l (Ω) instead of W l,2 (Ω).
In a sequel, we will also use the space L p (Ω) with 0 < p < 1 and use the standard notation simply ignoring the fact that it is not a norm. Recall that the topology in this space is defined by the metric d p (u, v) := u − v p L p . We say that the function u(t, x) is a strong solution of (2.1) if , L 2 (Ω)) and equation (2.1) is satisfied in the sense of distributions. In particular, for strong solutions we require that the initial data u 0 ∈ D, where Note that in general D is not a linear space and this causes a lot of extra difficulties in comparison with the case of linear phase space. We define the topology in the space D using the embedding In particular, the sequence u n → u strongly in D if u n → u in H 2 (Ω) and f (u n ) → f (u) in L 2 (Ω). Analogously, we say that u n ⇁ u weakly in D if u n → u weakly in H 2 (Ω) and f (u n ) → f (u) weakly in L 2 (Ω).

A priori estimates
In this section, we give a number of more or less standard estimates for strong solutions of problem (2.1) which will be justified later. We start with the dissipative estimate in the space H = L 2 (Ω).
Lemma 3.1. Let g ∈ H, assumptions (2.2)-(2.3) hold and let u be a sufficiently smooth solution of (2.1). Then, the following estimate is valid: , where the positive constants C and α are independent of t and u 0 .
Proof. We multiply equation (2.1) by u and integrate over x. This gives where (u, v) := Ω u(x).v(x) dx is a standard inner product in H. Using the dissipativity assumption f (u).u ≥ −C and positivity of the matrix a together with the Friedrichs inequality, we arrive at for some positive constants C and α. The Gronwall inequality applied to this relation gives (3.1) and finishes the proof of the lemma.
The next lemma gives the analogous dissipative estimate for the H 1 -norm of the solution.
Lemma 3.2. Let the assumptions of Lemma 3.1 hold and u be a sufficiently regular solution of (2.1). Then, the following estimate is valid: where the positive constants C and α are independent of t and u 0 .
Proof. We multiply equation (2.1) by −∆ x u and integrate over x to get Using the inequality f ′ (u) ≥ −K and positivity of matrix a again, we arrive at . Applying the Gronwall inequality to this relation and using (3.1) in order to control the integral of ∇ x u 2 L 2 , we end up with (3.2) and finish the proof of the lemma.
Let now θ = ∂ t u. Then this function solves The next lemma gives the L 2 -estimate for the time derivative θ. . Then the following estimate is valid: where positive constants C and K 1 are independent of t and u 0 .
Proof. We multiply equation (3.4) by θ and use assumption f ′ (u) ≥ −K and positivity of matrix a to get Applying the Gronwall inequality to this relation, we get the desired estimate and finish the proof of the lemma.
As a corollary of this lemma, we get the key control for the norm of the solution in the space D.
Corollary 3.4. Let the assumptions of Lemma 3.1 hold and let u be a sufficiently regular solution of (2.1). Then the following estimate is valid: , where the positive constants C and K 1 are independent of T and u 0 .
Proof. We rewrite equation (2.1) as an elliptic problem for every fixed T . Multiplying then this equation by ∆ x u(T ) (without integration in time!) and using the control for θ(T ) obtained above (together with the elliptic regularity estimate for the Laplacian and the assumption f ′ (u) ≥ −K), we arrive at the estimate . Estimating the right-hand side of this inequality by (3.5) and (3.2) and using that we arrive at the desired estimate and finish the proof of the corollary.
Remark 3.5. Note that, in contrast to estimates for the L 2 and H 1 norms of the solution u(t), the obtained estimate for the D-norm of u(t) is not dissipative and even growth exponentially in time. We will remove this drawback later (under some extra assumptions on f ).
We conclude this section by establishing the global Lipschitz continuity with respect to the initial data which plays a crucial role in constructing weak solutions for (2.1).
Lemma 3.6. Let the assumptions of Lemma 3.1 hold and let u 1 (t) and u 2 (t) be two sufficiently regular solutions of equation (2.1). Then the following estimate is valid: where the positive constants C and K 1 are independent of T , u 1 and u 2 .
Then this function solves Multiplying this equation by v, using that, due to the monotonicity assump- and arguing as in the proof of Lemma 3.3, we arrive at the desired estimate.

Existence of strong solutions
Although the construction of a solution from a priori estimates obtained above is more or less standard, it is a bit delicate here since the phase space D is in general nonlinear and, particularly, it is not clear whether or not smooth functions are dense in D. By this reason, we sketch the proof here.
We expect that the existence result can be also obtained using the monotone operators theory (e.g., the standard Ioshida approximations), but we prefer to give an alternative, a bit more transparent proof. Our idea is to approximate the nonlinearity f by a sequence f n of functions of sub-linear growth rate without destroying assumptions (2.3). Then, on the one hand, the existence of solutions for such f n is well-known and, on the other hand, as not difficult to see, all estimates obtained above will be uniform with respect to n. Thus, it will only remain to pass to the limit n → ∞. We start with the approximation of f .
where the constant C n may depend on n.
We now introduce the approximating system for (2.1) where the functions f n are constructed in Lemma 4.1. However, the choice of the approximating initial data u n 0 requires some accuracy. Indeed, we cannot just fix u n 0 = u 0 since f n (u 0 ) L 2 will be not uniformly bounded and, as a result, we may lose the estimate of the D-norm of the limit solution. Instead, we define u n 0 as a solution of the following auxiliary elliptic problem: The next lemma gives useful properties of the solutions of this auxiliary problem.
Let the functions f n be as above and u 0 ∈ D. Then, for every fixed n, problem (4.5) has a unique solution v = u n 0 . Moreover, u n 0 H 2 and f n (u n 0 ) L 2 are uniformly bounded as n → ∞ and (4.6) Proof. Indeed, the existence and uniqueness of a solution for (4.5) is obvious since f n (u) are monotone and have sublinear growth rate. Let us prove uniform bounds. Indeed, multiplying (4.5) by ∆ x v = ∆ x u n 0 and using the monotonicity, we get the estimate , we see that f n (u n 0 ) are also uniformly bounded. Let us verify the convergence. Since u n 0 is uniformly bounded, passing to a subsequence if necessary, we may assume that u n 0 ⇁ w as n → ∞ and u n 0 → w strongly to w in H 1 . Then, we have the convergence u n 0 (x) → w(x) almost everywhere. Moreover, from this convergence and Lemma 4.1, we may conclude that f n (u n 0 (x)) → f (w(x)) almost everywhere. Since f n (u n 0 ) are uniformly bounded in L 2 , passing to a subsequence again, we infer that f n (u n 0 ) ⇁ f (w). Passing after that to the weak limit n → ∞ in equations (4.5), we see that the limit function w solves Finally, since the solution w ∈ D of equation (4.7) is unique (again due to the monotonicity of f (u) + Ku), we conclude that w = u 0 . The uniqueness also gives that passing to a subsequence was not necessary and the whole sequence u n 0 converges to u 0 . We are now ready to state and prove the main result of this section. Proof. We approximate the desired solution u by the approximate solutions u n (t) of problems (4.4), where f n and u n 0 are chosen as in Lemmas 4.1 and 4.2. Then, since f n has a sublinear growth, the existence and uniqueness of a solution u n (t) of (4.4) is straightforward. At the next step, we need to check that all estimates of section 3 are indeed uniform with respect to n (the justification of all these estimates for the case of sublinear growth rate is also obvious). This is obvious for Lemmas 3.1 and 3.2 (as well as for Lemma 3.6) since u n 0 → u 0 strongly in H 1 and f n satisfy (2.3) uniformly with respect to n. Thus, we only need to look on the estimates related with time differentiation and D norms. The key role in these estimates is played by the L 2 -norm of time derivative θ n (t) := ∂ t u n (t) and the L 2 -norm of it at time moment t is estimated by its L 2 -norm at time t = 0. But due to our construction Therefore, according to Lemma 4.2, By this reason, the analogue of estimate (3.6) on the level of approximations reads where positive constants C and K 1 are independent of t, u 0 and n. When the uniform estimates are obtained, we may pass to the limit n → ∞ in equations (4.4) and construct the desired strong solution u(t) of the limit problem (2.1) (the passage to the limit in the nonlinear term is done exactly as in Lemma 4.2. The uniqueness of a solution is an immediate corollary of Lemma 3.6 (which does not require justification on the level of strong solutions). Finally, passing to the limit in the corresponding estimates for u n , we prove that the limit solution u(t) satisfies indeed all of the estimates of section 3. The only non-immediate thing is the passage to the limit in the terms like (f ′ n (u n )∇ x u n , ∇ x u n ) (and in the analogous term containing ∂ t u n ) since we do not have any control of the integral norms of f ′ n (u n ). However, the passage to the limit could be performed here using x)) and the convexity arguments. Namely, under these conditions, we may establish that (see e.g [4], Theorem 5.4). Thus, the theorem is proved.

Weak solutions, dissipativity and attractors
In the previous section, we have proved the global existence and uniqueness of strong solutions of (2.1). Thus, the solution semigroup is well-defined. Moreover, according to Lemma 3.6, this semigroup is globally Lipschitz continuous in the L 2 -metric: Thus, we can extend this semigroup by continuity from D to its closure in L 2 which obviously coincides with the whole L 2 (since C ∞ ⊂ D). Thus, the semigroup is well-defined. Moreover, the limit in (5.3) can be considered in the space C(0, T ; L 2 (Ω)), so the trajectories u(t) := S(t)u 0 automatically belong to the space C(0, T ; L 2 (Ω)) for all T ≥ 0. Our next step is to understand in what sense the trajectory u(t) thus constructed satisfies the initial equation (2.1). Note that, for the general u 0 ∈ L 2 , we do not know whether or not f (u) ∈ L 1 , so we cannot treat it in the distributional sense. Indeed, the only control related with f (u) which we have up to now follows from estimate (3.1) and claims that f (u).u ∈ L 1 (Ω) (being pedantic, this is proved for the strong solutions only, but it can be easily extended to weak solutions using the Fatou lemma). Unfortunately, this is not enough to control the L 1 (Ω)-norm of the function f itself, so we cannot treat the term f (u) in a distributional sense.
Instead, we use the ideas from the monotone operator theory and variational inequalities, see e.g., [5]. Namely, following [20], we take an arbitrary test function Then, integrating by parts and using that The advantage of this approach is that the variational inequality (5.5) has a sense for all u ∈ C(0, T ; L 2 (Ω)) and, therefore, can be used to define a weak solution u(t) of problem (2.1).
We are now ready to state the key result of this section.
Theorem 5.2. Let g ∈ L 2 (Ω), the diffusion matrix a be such that a+a * > 0 and f satisfy assumptions (2.3). Then, for every u 0 ∈ L 2 (Ω), there exists a unique weak solution u(t) of problem (2.1) and this solution has the form Proof. Indeed, any strong solution is a weak solution of (2.1) (since all manipulations used in the derivation of the variational inequality (5.5) are obviously justified on the level of strong solutions). Let now u(t) = lim n→∞ u n (t), where u n (t) are the strong solutions u n (t) := S(t)u n 0 , u n 0 ∈ D and u n 0 → u 0 in L 2 . The variational inequality for u n reads Using that u n → u in C(0, T ; L 2 (Ω)) and passing to the limit n → ∞ in (5.6), we see that u(t) satisfies (5.5) and, therefore, u(t) := S(t)u 0 is the desired weak solution of (2.1). Vice versa, letū(t) be a weak solution of (2.1) and letū(t) := S(t)ū(0). Then, there exists a sequence u n 0 ∈ D such that u n 0 →ū(0) and the sequence of strong solutions u n (t) = S(t)u n 0 which converges as n → ∞ to the weak solution u(t). We need to show that u(t) =ū(t).
Indeed, by the definition of a strong solution, u n (t) satisfies the assumptions of (5.4) and therefore can be used as a test function v = u n in the variational inequality (5.5) for the weak solutionū. Taking v = u n in it and using that ∂ t u n = a∆ x u n − f (u n ) + g, we get Passing to the limit n → ∞ in this inequality, we get and, since T is arbitrary, the Gronwall inequality gives thatū(t) = u(t) for all t. Thus, the theorem is proved.
There is an alternative possibility to relate a weak solution should be satisfied in a sense of distributions, see [38]. It is not difficult to verify that, indeed, any weak solution u(t) should satisfy this identity. The drawback of this approach is that it is unclear whether or not (5.7) is enough for the uniqueness.
Using identity (5.7) it is not difficult to show that under the additional restriction which guarantees that f (u) ∈ L 1 ([0, T ] × Ω), any weak solution satisfies equation (2.1) in a sense of distributions. However, in contrast to the scalar case k = 1, in the case of systems (5.8) is an extra restriction which we prefer to avoid.
As a next step, we note that the weak solutions are dissipative. Indeed, passing to the limit in the estimate of Lemma 3.1 for strong solutions, we derive that which is a standard dissipative estimate for the semigroup S(t). Analogously, passing to the limit in the estimate of Lemma 3.2, we get the dissipative estimate in H 1 for weak solutions In addition, estimates (5.9) and (5.10) give in a standard way the L 2 -H 1 smoothing property for the semigroup S(t), namely, the following estimate holds: These estimates ensure us that the ball B R := {u ∈ H 1 0 , u H 1 ≤ R} will be a compact (in L 2 ) absorbing ball for the semigroup S(t) if R = R( g L 2 ) is large enough. Remind that the latter means that for every bounded set is the set of all complete bounded trajectories of the semigroup S(t): Proof. According to the abstract attractor existence theorem, see e.g., [3], we need to verify two assumptions: 1) The operators S(t) : H → H are continuous for every fixed t; 2) The semigroup S(t) possesses a compact absorbing set in H.
The first assumption is guaranteed by Lemma 3.6 and the second one follows from estimate (5.11). Thus, the global attractor A exists. The fact that A is a bounded subset of H 1 follows from the fact that the attractor is always a subset of an absorbing set and the representation formula (5.12) is a standard corollary of the attractor existence theorem. Thus, the theorem is proved.

Weak to strong smoothing property
In this section, we establish that any weak solution u(t) of problem (2.1) becomes strong for t > 0. The main difficulty here is the fact that we cannot in general estimate |f (u)| through f (u).u and, by this reason, we do not know whether or not f (u) and ∂ t u are distributions. This makes the situation with the parabolic smoothing property a bit more delicate than usual. We overcome this difficulty under the extra assumption that for some p > 1 by using the L q -spaces with q < 1. Namely, we will use the fact that where, for q < 1, we denote by v L q exactly the same expression as for the case q ≥ 1 (simply ignoring the fact that it is no more a norm). Thus, at least on the level of approximations, we may expect that, for θ = ∂ t u, 3) θ(t) L 2 (0,1;L 2 (Ω))+L ∞ (0,1;L 2/p (Ω)) ≤ C( u 0 p H 1 + g p L 2 + 1) and this can be used in order to establish the smoothing property for θ. Namely, the following theorem holds.
Theorem 6.1. Let d ≥ 3, the assumptions of Theorem 5.2 and (6.1) hold, and let u 0 ∈ H 1 and u(t) be the corresponding weak solution of equation (2.1). Then, u(t) ∈ D for all t > 0 and the following estimate is valid: ) , t ∈ (0, 1], where the exponent N = N (p), positive constant C and the monotone increasing function Q are independent of u 0 and t.
Proof. We first note that it is enough to verify (6.4) for u 0 ∈ D only when u(t) is a strong solution. Moreover, it is enough to obtain the estimate for ∂ t u only since then the estimate for u(t) D will follow from the elliptic problem (3.7). Second, we approximate the strong solution u(t) by the solutions u n (t) of auxiliary problems (4.4). Finally, analyzing the proof of Lemma 4.1, we see that assumption (6.1) allows us to pose the extra assumption for some p 1 > p > 1 and constant C independent of n (e.g., by taking Ψ(u) := |u| p 1 +1 ). Let θ n (t) := ∂ t u n (t). Then, this function solves the equation (6.6) ∂ t θ n = a∆ x θ n − f ′ n (u n )θ n . Multiplying this equation by t N θ n (t) where N is a sufficiently big number, we end up with We need to estimate the integral in the right-hand side. To this end, we fix sufficiently small s > 0 which will be specified below and write where we have used equation (4.4) in order to express θ n through u n . Let us estimate every term in the RHS separately. Applying the Hölder and Young inequalities to the first term, we get This gives us a good estimate if N is chosen in such a way that 2 N −1 2−s ≥ N . The second term in the RHS of (6.8) can be estimated analogously to have (6.10) t Let us now estimate the most complicated third term. To this end, we use the embedding theorem H 1 ⊂ L r where 1 r = 1 2 − 1 d together with the Hölder inequality with exponents q 1 and q 2 , 1 Moreover, due to assumptions (6.5), we have where C is independent of n. We may also fix sq 1 = 2 p 1 to get (6.11) (|f n (u n )| s , |θ n | 2−s ) ≤ C( u 0 L 2 + 1 + g L 2 ) sp 1 θ n 2−s H 1 and end up with the following system for the exponents q 1 , q 2 and s Solving this system, we get and we see that 0 < s < 2 and 1 < q 1 < ∞, so all of the exponents are in the prescribed range and (6.11) holds indeed. Applying the Young inequality, we arrive at where ε > 0 is arbitrary small and the monotone function Q ε is independent of u 0 . Combining estimates (6.8), (6.9), (6.10) and (6.12) for estimating the RHS of (6.7) and fixing ε > 0 to be small enough and N satisfying 2 N −1 2−s ≥ N , we arrive at , t ≤ 1. Applying the Gronwall inequality to this relation and using (5.10) for estimating the integral of the H 2 -norm of the solution, we end up with (6.14) t N θ n (t) 2 L 2 ≤ Q( u n 0 H 1 ) + Q( g L 2 ), t ∈ (0, 1] and passing to the limit n → ∞, we derive the desired estimate for ∂ t u(t). Thus, the theorem is proved.
Combining this result with the L 2 to H 1 smoothing property (5.11), we get the following result. Corollary 6.2. Under the assumptions of Theorem 6.1, the weak solution semigroup S(t) possesses the following smoothing property: ) , t ∈ (0, 1], where the positive constant C and monotone function Q are independent of t and u 0 ∈ L 2 (Ω).
Thus, under the assumption (6.1), any weak solution indeed becomes strong for t > 0 (the extra assumption d ≥ 3 is not essential since for d ≤ 2 the equation is subcritical and the smoothing property is obvious). This, in particular, gives the following result on the regularity of the global attractor.
for some monotone increasing function Q.
Indeed, this assertion is an immediate corollary of (6.15) and the strict invariance of the global attractor.

Further regularity and strong attraction
In this section we discuss the possibility to get more regular than u ∈ D solutions. We start with some partial result on the regularity of ∂ t u which does not require any extra assumptions on f and g.
Proposition 7.1. Let the assumptions of Theorem 4.3 hold. Then, there exists a positive number r > 0 depending only on the matrix a such that, for any strong solution u(t) ∈ D, the following estimate holds: where 0 ≤ t ≤ 1 and the constant C is independent of t and u.
Proof. Let θ := ∂ t u. Then this function satisfies equation (3.4). Let us multiply this equation by θ|θ| r and integrate over x. This gives Integrating by parts in the second term, we get for some positive α. Fixing now r > 0 small enough and estimating the term containing f using f ′ (u) ≥ −K, we arrive at Multiplying this estimate by t and integrating in time, we arrive at To estimate the right-hand side of this inequality, we use estimate (3.5) and Sobolev embedding theorem which gives that, for sufficiently small r > 0, and finishes the proof of the proposition.
for some monotone function Q. Then every trajectory u(t), t ∈ R belonging to the kernel K possesses the following extra regularity of time derivative: for some r > 0 and monotone function Q which is independent of t and u.
This extra regularity in time can be transformed to extra regularity in space assuming that right-hand side g is slightly more regular. for some q > 2. Then, there exists r = r(a, q) > 0 such that for 0 ≤ t ≤ 1. In particular, if the attractor A is a bounded set in D then it is also bounded in W 1, d(r+2) d−2 (Ω).
Indeed, due to Proposition 7.1, we control the L r+2 -norm of ∂ t u. Rewriting problem (2.1) as an elliptic boundary value problem we get also the control for the L r+2 -norm ofg(t) (point-wisely in time). Applying the elliptic regularity result proved in Appendix (see Theorem A.1) to this equation, we arrive at the desired estimate (7.6).
The obtained partial regularity results allow us to establish the crucial L ∞ -estimates for critical and slightly supercritical growth rate of the nonlinearity f . Namely, the following result holds. Theorem 7.4. Let the assumptions of Proposition 7.1 hold and let, in addition, the nonlinearity f satisfy (6.1) with the exponent p restricted by the assumption if d ≥ 4 and the external forces g satisfy (7.5) for some q > d 2 . Then any weak solution u(t) of problem (2.1) possesses the following smoothing property: for some monotone function Q t depending on t, but independent of u and g.
Proof. Note that, due to estimate (6.15) we may assume from the very beginning that u 0 ∈ D and work with strong solutions only. The derivation of (7.8) can be done by the standard bootstrapping arguments by iterating the classical interior regularity result for the linear parabolic equation namely, where κ > 0 is arbitrarily small, 1 < s < ∞ and T > 0. This estimate, in turn, can be easily deduced from the fact that this linear equation generates an analytic semigroup in L s (Ω) or from the maximal L q (t, t+1; L s )-regularity estimate for parabolic equations (see e.g., [36]), so we left the details to the reader). From this smoothing property and Sobolev embedding theorem, we derive the iterative estimate (7.11) u L ∞ (T k+1 ,1;L q k+1 ) ≤ C k u L 2 (0,1; where T k+1 > T k and . In our situation h(t) = g − f (u(t)) and, due to our growth restrictions, we have h L s k ≤ C( g L q + 1 + u p L q k ) where s k := min{q, q k p −1 }. Thus, in order to prove the theorem, it is sufficient to verify that the sequence q k defined via , κ ≪ 1 will become large than q > d 2 in finitely many steps (we have used here estimate (7.6) and the embedding W 1, d(r+2) d−2 ⊂ L q 0 to initialize the iterations and the embedding W 2−κ,q ⊂ L ∞ which holds for sufficiently small κ due to the condition q > d 2 ).
Obviously this sequence will be monotone increasing if (and only if) Then it must converge to +∞, so we only need to verify the last inequality.
Using assumption (7.7) and the explicit formula for q 0 , we only need the inequality It remains to note that the last inequality is satisfied if κ ≪ 1 and ε < ε 0 = ε 0 (r) for some positive ε 0 if r > 0. This finishes the proof of the theorem.
Remark 7.5. The growth rate of the nonlinearity is no more important if the L ∞ -estimate for the solutions is obtained, so further regularity can be obtained by bootstrapping exactly as in the subcritical case. Thus, under the growth restriction (7.7), the actual regularity of a solution is determined by the smoothness of Ω, f and g only (if all of them are C ∞ -smooth, the solutions will be also C ∞ -smooth). In other words, we may say that the critical growth exponent for f in our problem (2.1) is slightly larger than p crit = 1+ 4 d−4 . We also note that the value ε = ε(a) somehow measures how far the matrix a is from the scalar matrix. It is easy to show that ε(a) = ∞ if a is scalar.
We now turn to the question of whether or not the attraction to A holds in the space D. Since in this case we at least need the dissipativity of our semigroup in D, we assume that f has a polynomial growth rate (i.e., that (6.1) is satisfied for some p ∈ R + ). Of course, the most interesting here is the supercritical case when the assumption (7.7) is not satisfied. Unfortunately, we do not know the answer on this question in general and have to pose some extra restrictions which however look natural. Namely, we assume that the nonlinearity also satisfies Then, the following result holds. Proof. We only need to prove the compactness ofŜ(1)B R , the rest is a corollary of the standard attractor's existence theorem. The fact that this set is closed is also standard and we left it to the reader. So, we will only check pre-compactness below.
The proof of this fact is a combination of parabolic regularity estimates which gives the pre-compactness of the set in L 2 (Ω) and energy type estimates for the elliptic equation which then give the desired compactness in H 2 (Ω).
Step 1. B R is compact in L 2 (Ω). We already know that B R is a bounded set in L 2+r (Ω), due to Proposition 7.1 and Corollary 6.2. In order to get the desired compactness we will use the standard interpolation embedding: see [36]. This embedding together with the compactness of the embedding H ε ⊂ L 2 will give the desired result if we prove boundedness of B R in W 1−κ,1 for some 0 < κ < 1. To this end, we note that according to Corollary 6.2, u(t) ∈ D for t ∈ [1/2, 1] and is uniformly bounded there if u 0 ∈ B R . Thus, according to assumption (7.12), f ′ (u(t)) L 2 is uniformly bounded. Since the L 2+r -norm of ∂ t u(t) is also bounded due to Proposition 7.1, we have the estimate Thus, applying the L s interior estimate (with s > 1) to equation we arrive at (7.14) θ(1) W 2(1− 1 s ),s ≤ C( ∂ t θ L s (3/4,1;L s ) + ∆ x θ L s (3/4,1;L s ) ) ≤ ≤ C( f ′ (u)∂ t u L s (1/2,1;L s ) + ∂ t u L 2 (1/2;1;L 2 ) ) ≤ C ′ R . This estimate gives the desired boundedness of B R in W 1−κ,1 (Ω) and completes the first step of the proof.
Step 2. Compactness in H 2 . Let us consider a sequence of solutions u n (t), u n (0) ∈ B R and find a subsequence which is convergent strongly in H 2 to some solution u(t). Due to the result of Step 1, we may assume without loss of generality that u n (1) → u(1) weakly in H 2 and ∂ t u n (1) → ∂ t u(1) strongly in L 2 . In other words, we need to pass to the limit n → ∞ in the semilinear elliptic equation Without loss of generality we may assume also that f ′ (u) ≥ 0. We will utilize the so-called energy method. Assume at this moment that we are able to integrate by parts and get this formula will be verified later at the end of the proof. Then, multiplying (7.15) by ∆ x u n and integrating over x, we get the energy identity Our aim here is to pass to the limit n → ∞ in this equality and compare it with the energy equality for the limit solution. Indeed, using the convexity arguments (similarly to (4.11), we get and due to the strong convergence h n → h, we have Then, the comparison with the limit energy identity shows that we must have lim n→∞ (a∆ x u n (1), ∆ x u n (1)) = (a∆ x u(1), ∆ x u(1)).
Together with the weak convergence ∆ x u n (1) → ∆ x u(1) this gives the strong convergence ∆ x u n (1) → ∆ x u(1) in L 2 and, therefore, the strong convergence u n (1) → u(1) in H 2 . From the equation (7.15) we finally establish that f (u n (1)) → f (u(1)) also strongly. Thus, the compactness ofŜ(1)B R in D is proved. Thus, the theorem is proved by modulo of the integration by parts formula (7.16) which we prove in the following lemma.
Lemma 7.7. Let the nonlinearity f satisfy the assumptions of Theorem 7.6. Then integration by parts (7.16) is valid for every u ∈ D.
Proof of the lemma. We first establish the identity for all vector-fields W ∈ C ∞ (Ω). Due to our assumption (7.12) both parts of this equality make sense. The identity may be proved by approximating the function f by "good" functions f n as in Lemma 4.1. Since f has a polynomial growth, we may take, say, and this allows us to keep also assumption (7.12) uniformly in n. Let u n be the corresponding approximating functions for u constructed as in (4.5). Then, we first verify the integration by parts for f n and u n (which is trivial since everything is smooth) and after that pass to the limit n → ∞ (which is also straightforward since as in Lemma 4.2, we have weak convergence f n (u n ) → f (u) in L 2 and, due to our assumption (7.12), we also have weak convergence f ′ n (u n )∂ t u n to f ′ (u)∇ x u in L 1+ε for small positive ε. Thus, the integration by parts (7.19) is verified for smooth vector fields W .
Note that the C ∞ smoothness assumption on the vector field W can be relaxed till W ∈ H 1 (Ω) ∩ L ∞ (Ω) by density arguments.
We now construct a sequence of Lipschitz continuous cut-off functions (7.20) ϕ n (z) = Then, the sequence ϕ n (z) monotone increasing in n and is convergent pointwise to one. Moreover, the following estimate holds: (there are no problems to construct similar smooth sequence, but we prefer to give relatively simple explicit expression). Then, we define a special vector-field W = W n as follows: Then, as simple calculation shows, W ∈ L ∞ (Ω) and, due to condition (7.21), where the constant C is independent of n. Thus, we may conclude that div W n → ∆ x u weakly in L 2 (Ω). Moreover, we may put W n to the integration by parts formula (7.19) and get It only remains to pass to the limit n → ∞ here. Passing to the limit in the left-hand side is immediate and to pass to the limit n → ∞ in the right-hand side, it is enough to note that f (u)∇ x u.∇ x u is non-negative and belongs to L 1 (Ω). The monotonicity of ϕ n in n and its point-wise convergence to one allow us to apply the Levy monotone convergence theorem and get the desired result. Thus, the lemma is proved and the theorem is also proved.
Remark 7.8. We expect that the integration by parts formula (7.16) holds without the extra assumption (7.12), however, it is not clear how to verify it. Key difficulty here is that D is a nonlinear set and it is not easy to construct good smooth approximations for functions u ∈ D.
The first step in the proof of Theorem 7.6 can be also done using the energy type arguments. To this end one just need to verify the energy identity which can be verified similarly to the proof of Lemma 7.7. In the case of reaction diffusion system (2.1) this is not necessary since Proposition 7.1 gives a simpler way to verify the compactness. However, it may be useful in the case of higher order equations where the technique of Proposition 7.1 may not work.

Finite dimensionality and exponential attractors
In this section we discuss the finite-dimensionality of the global attractor for problem (2.1) and the existence of the so-called exponential attractor. We recall that a set M ⊂ H is called an exponential attractor of the semigroup It is well-known that the exponential attractor if exists always contains a global attractor, so the existence of M automatically implies the finitedimensionality of a global attractor. In contrast to global attractors, exponential attractors usually more robust with respect to perturbations and allow us to control the rate of attraction in terms of physical parameters of the system considered, but as a price to pay for that, an exponential attractor is not unique, see [15,17,31] for more details.
The existence of an exponential attractor is usually verified using the following abstract result for discrete semigroupsŜ(n) :=Ŝ n : H → H generated by the mapŜ : H → H.
Then the corresponding discrete semigroupŜ(n) : B → B possesses an exponential attractor M ⊂ B.
For the proof of this proposition, see [16,17]. In applications usually B is an absorbing ball of the considered continuous semigroupŜ(t) : H → H, S := S(T ) for some properly chosen T and (8.1) is verified using the proper parabolic smoothing property for the equation on differences of two solutions. If the existence of a discrete exponential attractor M d is established, the exponential attractor for the continuous semigroup can be constructed by the standard formula: and in order to get its finite-dimensionality, we need to assume in addition that the semigroup is also Hölder continuous in time: for some α ∈ (0, 1] and all t i ∈ [T, 2T ] and ξ i ∈ B, see [17] for the details.
The main result of this section is the following theorem.
Let the nonlinearity f satisfy assumptions (2.3), (6.1) for some p ∈ R + , (7.12) and the following convexity property: there exist a convex function Ψ : R k → R such that for some positive constants C 1 and C 2 . Let also g ∈ L 2 (Ω) and a satisfy (2.2). Then problem (2.1) possesses an exponential attractor M in the space H := L 2 (Ω) which is a compact set in D.
Proof. According to Corollary 6. We recall that, due to the assumption f ′ (u) ≥ −K, multiplication of this equation on θ gives the basic Lipschitz continuity estimate: see Lemma 3.6. Moreover, multiplying (8.4) by θ|θ| r and arguing exactly as in the proof of Proposition 7.1, we get the estimate for some sufficiently small positive r depending only on the matrix a.
In order to get smoothing estimate for θ, we argue as in Step 1 of the proof of Theorem 7.6. Namely, from (7.12) and (8.3), we conclude that and, therefore, since f (u(t)) is uniformly bounded in L 2 -norm for our solutions u 1 and u 2 , we have This estimate, in turn, implies (together with (8.5), Sobolev embedding theorem and Hölder inequality) that for some 1 < s < 2. Applying now the L s interior regularity estimate to equation (8.4) and arguing as in the proof of Theorem 7.6, we get θ(T ) W 2(1− 1 s ),s ≤ C T θ(0) L 2 which together with the embedding (7.13) gives for some positive exponent ε. Setting finally V = H ε (Ω) we get the desired smoothing property (8.1) and finish the proof of the theorem.
Remark 8.3. The finite-dimensionality of the global attractor A has been established under similar assumptions on f in [38] using the so-called method of l-trajectories, see also [28,29]. In the present work we suggest the simplified version of the proof which is based on multiplication of equation (8.4) on the quantities like θ|θ| r . Although the proof becomes more transparent, it is slightly less general than the one suggested in [38] since this multiplication is suitable for reaction-diffusion systems and may not work for more general (e.g., higher order equations). In such cases one should return back to the method of l-trajectories.

Generalizations and concluding remarks
In this concluding section we briefly consider other types of equations for which the technique developed above works (with some minor changes which we will discuss) and state some interesting open problems. We start with the case of fractional Laplacians and the corresponding reaction-diffusion equations which are becoming more and more popular nowadays, see [1,2,21,27] and references therein for more details. 9.1. Fractional reaction-diffusion systems. Let us define A := (−∆ x ) α , 0 < α < 1, in the domain Ω endowed with Dirichlet boundary conditions and consider the following fractional reaction-diffusion system: where the function f and the matrix a satisfy assumptions (2.3) and (2.2) respectively. In this case, the definition of the phase space D should be reduced as follows: . All of the estimates and results stated above for the case α = 1 can be extended in a straightforward way to a general case 0 < α < 1. The only non-trivial place is the estimates of the terms like ((−∆ x ) α u, f (u)) or ((−∆ x ) α u, u|u| r ). In the case when Ω = R d (or in the case of periodic BC), we have a nice explicit formula for such inner products which trivializes the required estimates (see e.g., [36]), namely, In particular, it gives the positivity of (Au, f (u)) if f ′ (u) ≥ 0. Fortunately, there is an extension of this formula to the case of bounded domains (see [7]), namely, for some non-negative functions K Ω,α and B Ω,α . This formula allows us to get the same type of estimates as for the local case α = 1. For the convenience of the reader, we state below the analogues of two main results for the fractional case.
Theorem 9.1. Let the matrix a and the nonlinearity f satisfy (2.2) and (2.3) respectively and let, in addition, the nonlinearity f satisfy (6.1) with the exponent p restricted by the assumption if d ≥ 4α and the external forces g ∈ L q (Ω) for some q > d 2α . Then any weak solution u(t) of problem (9.1) starting from u(0) ∈ H possesses the following smoothing property: for some monotone function Q t depending on t, but independent of u and g.
Remark 9.2. Note that this result is not very helpful if 0 < α < 1 2 since the direct H 1 -estimate which is obtained by multiplication of the equation by −∆ x u gives the control of the H 1 -norm (of course, assuming in addition that g ∈ H 1−α ) which is better than H 2α -control finally obtained from u ∈ D α . However, it is useful for α ≥ 1 2 . In particular, in the case 0 < α < 3 4 , we may have the supercritical growth rate in the case of physical dimension d = 3 as well. So, main results become applicable for d = 3 as well. Note also that many of the results of our paper may be extended also to the case α > 1 (e.g., to the Swift-Hohenberg type equations where α = 2), but in this case we will be not able to multiply the equation by Au since the term (Au, f (u)) will be out of control, so we may multiply it only on ∆ x u and this gives the control of the H 1+α 2 -norm of u(t) (not H 2α as before). Moreover, in this case ε(a) = 0 since multiplication on u|u| r is no more available.
We now state the key result about exponential attractors for the supercritical case. Theorem 9.3. Let the nonlinearity f satisfy assumptions (2.3), (6.1) for some p ∈ R + , (7.12) and the following convexity property: there exist a convex function Ψ : R k → R such that for some positive constants C 1 and C 2 . Let also 0 < α < 1, g ∈ L 2 (Ω) and a satisfy (2.2). Then problem (9.1) possesses an exponential attractor M in the space H := L 2 (Ω) which is a compact set in D α .

9.2.
Cahn-Hilliard type systems. Let us consider the following fractional Cahn-Hilliard system in Ω ⊂ R d : endowed by the Dirichlet boundary conditions. We assume here that 0 < β ≤ 1, 0 < α ≤ 1. Note that α = β = 1 corresponds to the classical Cahn-Hilliard system and β = 0, α = 1 to the reaction-diffusion system considered above. See [1,31,35] and references therein for more details concerning classical and fractional CH-equations. It is natural to take D α as the phase space for this problem and rewrite it in the following form: Then we may utilize the monotonicity of the function f and apply the developed above theory to this equation (see also [31] for the case α = β = 1). In this case, weak solutions are naturally defined in the space H := H −β (Ω) and strong solutions live in D α .
The key result on the existence of exponential attractors now reads.
Theorem 9.4. Let the nonlinearity f satisfy assumptions (2.3), (6.1) for some p ∈ R + , (7.12) and the following convexity property: there exist a convex function Ψ : R k → R such that (9.10) for some positive constants C 1 and C 2 . Let also g ∈ L 2 (Ω) and a satisfy (2.2). Then problem (9.8) possesses an exponential attractor M in the space We leave the rigorous proof of this theorem to the reader. 9.3. Open problems. We conclude this section by a discussion of some open questions and possible further improvements of the above developed theory. Problem 1. We start with the already posed question about the validity of the integration by parts formula for every u ∈ D. We know that both parts of this equality are well-defined for any u ∈ D. However, since we do not know the density of smooth functions in D, we cannot verify the identity in a standard way, so we need to use something else. We have proved this identity under the extra assumption (7.12) which allows us to control the Lebesgue norm of f ′ (u)∇ x u and simplifies the situation. Clarifying the situation with this integration by parts in general would be very useful for establishing energy equalities for many other equations containing monotone nonlinearities which, in turn, may give compactness of the corresponding global attractors. We were sure that (7.12) is technical, but surprisingly are unable to remove it (or find the proper reference). Problem 2. Next problem is related with smoothness of weak/strong solutions of problem (2.1). We have established that under the assumption (6.1) that f has a polynomial growth rate, the problem possesses an instantaneous smoothing H to D smoothing property. It would be interesting to understand whether or not this polynomial growth restriction is really necessary for the smoothing (ideally, to construct a non-smoothing weak solution for problem (2.1)), say, with exponential or stronger nonlinearities. A natural idea here is to extend the proof of Theorem 6.1 to the case where ∂ t u belongs to some weaker spaces than L p (Ω) with p ≪ 1 using the technique of Orlich spaces. But more detailed analysis shows that this does not work already when ln(1 + |f (u)|) ∈ L 1 , so we may expect the existence of such exotic non-smoothing weak solutions for fast growing nonlinearities.
The phenomenon of delayed regularization is well-known in the class of nonlinear diffusion problems, see [37] and reference therein. For example, the equation ∂ t u|∂ t u| p = ∆ x u, u ∂Ω = 0, p ≥ 0 is well-posed in a natural energy phase space Φ = W 1,2 0 (Ω). However, the solutions of this equation do not possess an instantaneous smoothing if, say, p > 4 and d = 3. Indeed, the energy identity for this equation reads so if u(0) / ∈ L p+2 (Ω), we have u(T ) / ∈ L p+2 (Ω) for any finite T > 0. However, if we start from more regular phase space Ψ := W 1,2 0 (Ω) ∩ L ∞ (Ω), we will have instantaneous further regularization, see [19]. The open question is whether or not something similar happens in the case of system (1.1) of reaction-diffusion equations with fast growing nonlinearity f satisfying (2.3).
Another related question is about generating singularities in finite time in equations like (2.1). It is known that general reaction-diffusion systems may generate singularities in higher norms even if the natural energy norm remains finite and dissipative, see e.g., [32] for RDS satisfying balance law (=action mass law), [23] for the case of reaction-diffusion with chemotaxis or [6] for Ginzburg-Landau equations in R 3 (see also references therein). However, to the best of our knowledge, there are no such examples in the class of equation (2.1) with nonlinearities satisfying f ′ (u) ≥ −K. As we know, in this case the H 2 -norm cannot blow up, so this is the question of possible blow up of higher norms and high space dimension d > 4.
Problem 3. Finally, about the finite-dimensionality of global attractors. The most popular scheme for proving this result is related with volume contraction technique, see [3,35] and references therein. Using this technique, we need to estimate l dimensional traces Tr l L u , where is the linearized operator on the trajectory u(t) of the equation (2.1) lying on the attractor. Formal estimates of this quantity depend only on K (if the assumption f ′ (u) ≥ −K is posed) and are independent on the norm of u(t) and any norms of f (u).
However, to justify this method we need to verify the differentiability of the semigroupŜ(T ) with respect to the initial data (at least the socalled uniform quasi-differentiability on the attractor, see [35]) and such a differentiability usually does not hold in supercritical cases.
This was the main reason to use the alternative scheme based on Proposition 8.1 for verifying the finite-dimensionality. In this scheme the differentiability is not required, but as the price to pay, we get essentially worse estimates than expected since now the norm of |f ′ (u)| is involved into all dimension estimates.
It would be interesting to remove this drawback and remove the dependence on |f ′ (u)| from these estimates, e.g., by finding a "clever" choice of spaces H and V in Proposition 8.1. Up to the moment we know how to do this in a scalar case only, due to the possibility to multiply (8.4) by sgn v and using the Kato inequality. This in turn gives the estimate of the L 1norm of l(t)θ through quantities depending only on K. To the best of our knowledge, nothing similar is known for the vector case.

Appendix A. Nonlinear localization and elliptic regularity
In this appendix we consider the following semi-linear elliptic problem: where the matrix a satisfies assumption (2. 2) and f enjoys assumptions (2.3). Then, arguing as before, we get the H 2 -elliptic regularity The question addressed here concerns an additional regularity under the extra assumption A partial answer on this question is given in the following theorem.
Theorem A.1. Let the above assumptions hold. Then there exists κ = κ(a) > 0 such that if d > 2, q < d − d(d−2) κ+d and r = d(q−2) d−q . Here and below D 2 x u stands for the collection of all second derivatives of the function u.
Proof. We give below only the formal derivation of estimate (A.4) which can be justified in a standard way (e.g., by cutting off the nonlinearity as explained in section 4, mollifying g and using the corresponding smooth solutions of the cut off equation to approximate the initial solution u).
Step 1. We start with the simplest case of periodic boundary conditions where no difficult terms related with the boundary arise and we may integrate by parts freely. Without loss of generality, we may also assume that f ′ (u) ≥ 0 and f (0) = 0.
Finally, using the Hölder inequality with the exponents q, 2 and 2d(r+2) r(d−2) , we get (A.9) |(g, |D 2 x u| 2 |∇ x u| r/2 |∇ x u| r/2 )| ≤ ≤ α 2 (|D 2 x u| 2 , |∇ x u| r ) + ∇ x u r+2 L d(r+2) d−2 and finish the proof of the theorem in the case of periodic BC. The case of Dirichlet BC is more delicate since hardly controllable boundary terms will appear if we try to integrate by parts in the first term of (A.5).
To avoid them, we will use the nonlinear localization technique suggested in [24] (see also [25]) for more details.
Step 3. Boundary estimates: tangential derivatives. To treat the neighbourhood of the boundary, we introduce an x-depending smooth orthonormal base (τ 1 (x), · · · , τ d−1 (x), n(x)) near the boundary such that, when x ∈ ∂Ω, τ i correspond to tangential directions and n(x) is outer normal. We also assume that these vector fields are cut-off outside of small neighbourhood of the boundary similarly to Step 2. Note also that in general such smooth vector fields exist near the boundary only locally, but we will ignore this fact assuming that they exist globally (one more localization is necessary in general). We also define the corresponding directional derivatives ∂ τ i := τ i .∇ x , ∂ n := n.∇ x .
Step 5. Combining all together. Combining the interior estimates obtained at Step 2 with the tangential and normal boundary estimates (A.14) and (A.15) (e.g., with the help of the proper partition of unity), we finally arrive at (|D 2 x u| 2 , |∇ x u| r ) ≤ C ε ( g r+2 L r+2 + ∇ x u r+2 L r+2 ) + Cε(|D 2 x u|, |∇ x u| r ) and after fixing ε > 0 small enough, we end up with (|D 2 x u| 2 , |∇ x u| r ) ≤ C( g r+2 L r+2 + ∇ x u r+2 L r+2 ). So, it only remains to estimate the L r+2 -norm of the gradient ∇ x u. To this end, it is enough to use (A.7) together with the obvious estimate and estimate (A.2). This completes the proof of the theorem.
Remark A.2. As we see, the nonlinear localization uses the general strategy of the classical (linear) localization technique. However, it is more delicate since we need also to treat the nonlinear term f (u) which is now not subordinated to the linear ones, so we can multiply the equation only on the terms which can be estimated using the monotonicity assumption f ′ (u) ≥ 0.
Fortunately, the amount of such multipliers is enough to get the estimates similar to the linear case.