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Introduction
This paper is mainly devoted to studying the regularity properties of global attractors for multivalued semiflows generated by strong solutions of reaction-diffusion equations.
The existence and properties of global attractors for dynamical systems generated by reaction-diffusion equations have been studied by many authors over the last thirty years. For equations generating a single-valued semigroup such results are well known since the 80s (see e.g. [5], [6], [7], [8], [24], [34]). Moreover, deep results concerning the structure of the attractors were proved for scalar equations (see [12], [13], [29], [30], [31]). Furthermore, for rather general parabolic equations boundedness of attractors in Sobolev spaces of higher orders were obtained as well (see e.g.
[3], [4]).
When uniqueness of the Cauchy problem is not satisfied we have to work with multivalued semiflows rather than semigroups. In this direction, existence and topological properties of global and trajectory attractors have been studied by several authors over the last years (see [2], [9], [10], [11], [14], [16], [20], [22], [23], [25], [26], [36]). However, concerning the structure of the attractor little is known so far in comparison with the single-valued case. Nevertheless, recently the global attractor has been characterised using the unstable manifold of the set of stationary points, generalizing in this way well-known results from the single-valued case (see [17], [18], [19]). In particular, such structure was proved to be true for the global attractor generated by strong solutions of reaction diffusion equations in which the nonlinear term satisfies a critical growth condition.
It is important to point out that there are two different approaches to the study of these equations. One method relies on the construction of weak solutions through Galerkin approximations, whereas the other one makes use of the properties of sectorial operators in order to obtain mild solutions, which are defined by the variation of constants formula. It seems that this has given rise to two separate groups of papers, whose paths have rarely crossed. However, we find it interesting to use the powerful technique of sectorial operators in order to improve the regularity of weak solutions and global attractors in the multivalued case. Indeed, in this paper we prove that under a suitable assumption mild and weak solutions are equivalent, and using this result we are able to improve the regularity of the global attractor generated by strong solutions which was obtained in [17].
This paper is split into three different parts.
In the first section, we prove that the concepts of weak and mild solutions are equivalent provided that an appropriate condition holds.
In the second section, we use the above equivalence in order to show that the global attractor generated by strong solutions of the reaction-diffusion equation is bounded in suitable Lr-spaces in the case where the nonlinear term satisfies a critical growth condition.
Finally, in the third section, considering a supercritical growth condition we define a multivalued semiflow by taking all strong solutions satisfying an energy inequality and then prove that a weak global attractor exists, that is, we construct an attractor which attracts bounded sets of the phase space with respect to the weak topology.
Equivalence of different definitions of solutions
Let Ω ⊂ ℝn, n ≥ 1, be an open bounded subset with sufficiently smooth boundary ∂Ω. We consider the following problem
for all (t, u) ∈ ℝ × ℝ, where C1, C2 are positive constants, p ≥ 2 and α > 0 if p > 2 but α ∈ ℝ if p = 2.
Denote
$\begin{array}{}
F(u)=\int_{0}^{u}f(s)ds.
\end{array}$
If p > 2 from (4)-(5) we obtain that
$\begin{array}{}
\displaystyle
\liminf\limits_{|u|\rightarrow\infty}\frac{f(u)}{u}=+\infty
\end{array}$
and that there exists D1, D2, δ > 0 such that
In the sequel we shall denote by H the space L2(Ω) endowed with the norm ∥⋅∥ and the scalar product (⋅, ⋅), and by V the space
$\begin{array}{}
\displaystyle
H_{0}^{1}
\end{array}$
(Ω) with the norm ∥⋅∥V and the scalar product ((⋅, ⋅)), whereas V′ = H−1(Ω) is the dual space of V with the norm ∥⋅∥V′. The pairing between the space V and V′ will be denoted by 〈⋅, ⋅〉. Also, putting
$\begin{array}{}
\displaystyle
\frac{1}{p}+\frac{1}{q}=1,
\end{array}$
the pairing between the spaces Lp(Ω) and Lq(Ω) will be given by 〈⋅, ⋅〉q,p.
Definition 1
The function
$\begin{array}{}
\displaystyle
u\in L_{loc}^{p}(\tau,+\infty;L^{p}(\Omega))\cap L_{loc}
^{2}(\tau,+\infty;V)
\end{array}$
is called a weak solution to problem (1) on (τ, +∞) if for all T > τ, v ∈ V ∩ Lp(Ω) one has
For a weak solution let g(t) = −f(t, u(t)) + h(t) + Δ u(t). From
$\begin{array}{}
\displaystyle
|f(t,u)|\leq C_{1}(1+|u|^{p-1}),
\end{array}$
and condition (4) it is clear that
$\begin{array}{}
\displaystyle
g\in
L_{loc}^{q}(\tau,+\infty;L^{q}(\Omega))+L_{loc}^{2}(\tau,+\infty;V^{\prime}).
\end{array}$
Thus, we can rewrite (7) as
$$\begin{array}{}
\displaystyle
\frac{d}{dt}\left( u\left( t\right) ,v\right) =\left\langle
g(t),v\right\rangle _{V^{\prime}+L^{q}}\text{, for all }v\in V\cap
L^{p}(\Omega)\text{,}
\end{array}$$
where 〈⋅, ⋅〉V′+Lq denotes pairing between V′ + Lq(Ω) and V ∩ Lp(Ω).
It is well-known [10, p.284] that for any uτ ∈ H there exists at least one weak solution u(⋅). If, moreover, f(t, ⋅) ∈ C1(ℝ) and
$\begin{array}{}
\displaystyle
\dfrac{\partial f}{\partial u}
\end{array}$
(t, u) ≥ −C for any (t, u), then the solution is unique.
In order to consider an equivalent equality to (7) we recall the following well-known result.
Lemma 1
[33, p.250] LetXbe Banach space with dualX′, u, g ∈ L1(a, b; X). Then the following statements are equivalent:
$\begin{array}{}
u(t)=\xi+\int\limits_{a}^{t}g(s)ds,
\end{array}$ξ ∈ Xfor a.a. t ∈ (a, b);
$\begin{array}{}
\displaystyle
\frac{d}{dt}\langle u,\eta\rangle_{X^{\prime},X}=\langle g,\eta
\rangle_{X^{\prime},X}, \forall\ \eta\in X^{\prime},
\end{array}$in the sense of scalar distributions on (a, b), where 〈⋅, ⋅〉X′,Xdenotes pairing betweenXandX′.
If properties 1)-3) hold, then, moreover, u(⋅) is a.e. equivalent to a continuous function from [a, b] inX.
Applying this lemma with X = V′ + Lq(Ω) we obtain that (7) is equivalent to the equality
$\begin{array}{}
\displaystyle
\dfrac{du}{dt}=g
\end{array}$
in the sense of X-valued distributions on every interval [τ, T]. Hence,
Property (8) implies that u ∈ C([τ, +∞), H) and that the function t ↦ ∥u(t)∥2 is absolutely continuous on [τ, T] and
$\begin{array}{}
\displaystyle
\dfrac{d}{dt}\Vert u(t)\Vert^{2}=2\left\langle
\dfrac{du}{dt},u\right\rangle _{V^{\prime}+L^{q}}
\end{array}$
for a.a. t ∈ (τ, T) [10, p.285]. Hence, the initial condition u(τ) = uτ makes sense.
Also, since the space
$\begin{array}{}
\displaystyle
C_{0}^{\infty}
\end{array}$
((τ, T) × Ω) is dense in Lp(τ, T; Lp(Ω)) ∩ L2(τ, T; V), equality (9) is equivalent to
for any ϕ ∈
$\begin{array}{}
\displaystyle
C_{0}^{\infty}
\end{array}$
((τ, T) × Ω), that is, u(⋅) is a weak solution if and only if it is a solution in the sense of distributions.
Let us consider now another concept of solution. Namely, we will define a mild solution to (1).
It is known that the operator A = Δ:D(A) = H2(Ω) ∩
$\begin{array}{}
\displaystyle
H_{0}^{1}
\end{array}$
(Ω) → H is the generator of a strongly continuous semigroup of contractions T : ℝ+ × H → H. Moreover, it follows from well-known results [27] that for any x ∈ D(A) the function u(t) = T(t) x is the unique classical solution to the problem
If g ∈
$\begin{array}{}
\displaystyle
L_{loc}^{1}
\end{array}$
([τ, +∞)), then u ∈ C([τ, +∞), H) is called a mild solution if (12) is satisfied on every interval [τ, T].
It is called a classical solution on [τ, T] if u(⋅) is continuously differentiable on (τ, T), u(t) ∈ D(A) for any t ∈ (τ, T), u(τ) = x and the equality in (11) is satisfied for every t ∈ (τ, T).
It follows readily from this definition that problem (11) possesses a unique mild solution for every x ∈ H. Also, if g is continuously differentiable on [τ, T] and x ∈ D(A), then the mild solution is in fact the unique classical solution [27, p.107].
Coming back to our problem (1), let us introduce the concept of mild solution for it.
Definition 3
The function
$\begin{array}{}
\displaystyle
u\in L_{loc}^{p}(\tau,+\infty;L^{p}(\Omega))\cap L_{loc}
^{2}(\tau,+\infty;V)\cap C([\tau,+\infty),H)
\end{array}$
is called a mild solution to problem (1) on (τ, +∞) if for all T > τ the function g(⋅) = h(⋅) − f(⋅, u(⋅)) belongs to L1(τ, T; H) and the equality (12) holds true.
Our aim now is to show that under an additional assumption the concepts of weak and mild solutions coincide.
Lemma 2
Assume thatu(⋅) is a weak solution to problem (1) with initial datumuτ ∈ H which satisfies
Let us define the function g(t) = − f(t, u(t)) + h(t), which belongs to L2(τ, T; H) for every τ < T due to (13). We need to prove that the equality (12) holds. For an arbitrary fixed interval [τ, T] we take sequences
$\begin{array}{}
\displaystyle
u_{0}^{n}\in H^{2}(\Omega)\cap
H_{0}^{1}(\Omega)
\end{array}$
, gn(⋅) ∈ C1([τ, T], H), T < T, such that
$\begin{array}{}
\displaystyle
u_{0}
^{n}\rightarrow u_{0}
\end{array}$
in H, gn → g in L2(τ, T; H). Let un be the unique classical solution to the problem
possesses a unique weak solution v(⋅). However, arguing in the same way as in Lemma 2 we deduce that v(⋅) is a mild solution to problem (11) with x = uτ. Therefore, by uniqueness of the mild solution we have u(⋅) = v(⋅), so our statement follows.
Regularity of the strong global attractor
Let us consider problem (1) in the autonomous case, that is,
and we assume that conditions (2)-(5) are satisfied. In particular, this means that h ∈ H.
In [17] the existence of a global attractor for the multivalued semiflow generated by the strong solutions to (14) was proved assuming a critical growth condition on the nonlinear function f. In this section, our aim is to prove that this attractor is bounded in suitable Lr-spaces.
Definition 4
A weak solution u(⋅) to problem (1) is called a strong solution if, additionally, it satisfies
As u ∈ L∞(0, T; V ∩ Lp(Ω)) ∩ C([0, T]; H), we deduce that
$\begin{array}{}
\displaystyle
u\in C([0,T];H_{0w}^{1}(\Omega)\cap L_{w}^{p}(\Omega))
\end{array}$
for any T > 0, where
$\begin{array}{}
\displaystyle
H_{0w}^{1}(\Omega),\, L_{w}^{p}(\Omega)
\end{array}$
are respectively the spaces
$\begin{array}{}
\displaystyle
H_{0}^{1}
\end{array}$
(Ω), Lp(Ω) with the weak topology [33, Lemma 1.4, p.263].
if N ≥ 3, that is, f satisfies a critical growth condition. No assumption is imposed if N ≤2.
From (4), (17) and the imbeddings H1(Ω) ⊂
$\begin{array}{}
\displaystyle
L^{\frac{2N}{N-2}}
\end{array}$
(Ω), if N ≥ 3, H1(Ω) ⊂ Lq(Ω), for any q ≥ 1 if N ≤2, we get
It is known [17] that for any u0 ∈ V there exists at least one strong solution u(⋅) such that u(0) = u0. Moreover, every strong solution satisfies the energy inequality
$$\begin{array}{}
\displaystyle
E\left( u\left( t\right) \right) +2\int_{s}^{t}\left\Vert u_{r}\right\Vert
^{2}dr=E\left( u\left( s\right) \right) \text{, for all }t\geq s\geq0,
\end{array}$$
We recall now some results proved in [17] about the existence and structure of a global attractor for G. It is worth pointing out that, although in that paper such theorems were proved in the particular three-dimensional case (i.e. N = 3), the proofs in the general N-dimensional setting are identical.
First, we note that G is a strict multivalued semiflow, that is, G(0, x) = x and G(t + s, x) = G(t, G(s, x)) for any t, s ∈ ℝ+, x ∈ V.
Moreover, G possesses a global compact invariant attractor 𝓐, that is, 𝓐 is compact in V, it is invariant (i.e. 𝓐 = G(t, 𝓐) for any t ≥ 0) and attracts every bounded set in V, that is,
$$\begin{array}{}
\displaystyle
dist(G(t,B),\mathscr{A)}\rightarrow0\text{ as }t\rightarrow+\infty\text{, }
\end{array}$$
for any B set bounded in V, where dist refers to the Hausdorff semidistance between sets given by dist(A, B) = supx∈A infy∈B ∥x − y∥V.
The map γ : ℝ → V is called a complete trajectory of
$\begin{array}{}
\displaystyle
K_{s}^{+}
\end{array}$
if γ(⋅ + h)|[0,+∞) ∈
$\begin{array}{}
\displaystyle
K_{s}^{+}
\end{array}$
for any h ∈ ℝ, and this is equivalent to γ being continuous and satisfying
$$\begin{array}{}
\displaystyle
\gamma\left( t+s\right) \in G_{s}\left( t,\gamma\left( s\right) \right)
\text{ for all }s\in\mathbb{R}\text{ and }t\geq0.
\end{array}$$
The set of all complete trajectories of
$\begin{array}{}
\displaystyle
K_{s}^{+}
\end{array}$
will be denoted by 𝔽s. Let 𝕂s be the set of all complete trajectories which are bounded in V.
The attractor 𝓐 is characterised by the union of all points lying in a bounded complete trajectory, that is,
A point z ∈ X is a fixed point of
$\begin{array}{}
\displaystyle
K_{s}^{+}
\end{array}$
if φ(t) ≡ z ∈
$\begin{array}{}
\displaystyle
K_{s}^{+}
\end{array}$
, whereas it is called a fixed point of G if z ∈ G(t, z) for all t ≥ 0. In our case these two concepts are equivalent, so we will simply call them fixed points. Moreover, z is a fixed point if and only if z ∈ V ∩ H2(Ω) and
$$\begin{array}{}
\displaystyle
-\Delta z+f(z)=h\text{ in }H.
\end{array}$$
The set of all fixed points will be denoted by ℜ.
Finally, in [17] it is proved that the strong global attractor coincides with the unstable manifold of the set of stationary points, and also with the stable one when we consider only bounded complete solutions. This means that
Concerning boundedness of the attractor in stronger Lr-spaces than
$\begin{array}{}
\displaystyle
L^{\frac{2N}{N-2}}
\end{array}$
(Ω) (which follows from the embedding V ⊂
$\begin{array}{}
\displaystyle
L^{\frac{2N}{N-2}}
\end{array}$
(Ω)) in [17] and [18] an estimate in the space L∞(Ω) is shown to be true if h ∈ L∞(Ω).
Now, using the equivalence between weak and mild solutions from the first section we are able to obtain suitable estimates in Lr-spaces under much weaker assumptions on the function h.
Theorem 4
The global attractor 𝓐 is bounded inLr(Ω), wherer ∈ [1, +∞] ifN ≤3, r ∈ [1, +∞) ifN = 4 and 1 ≤ r <
$\begin{array}{}
\displaystyle
\frac{2N}{N-4}
\end{array}$ifN ≥ 5.
IfN ≥ 4 andh ∈ Lq(Ω), for someq >
$\begin{array}{}
\displaystyle
\frac{N}{2}
\end{array}$
, then 𝓐 is bounded inL∞(Ω).
Proof
The semigroup T(t) generated by the solutions of problem (10) satisfies the following well-known estimate [32, p.2988]:
Take an arbitrary y ∈ 𝓐. In view of (23), there exists a complete trajectory ψ such that ψ(0) = y and ψ(t) ∈ 𝓐 for all t ∈ ℝ. Inequality (18), combined with (15) and Lemma 2, implies that u(⋅) = ψ(⋅ + h) is a mild solution to problem (14) for any h ∈ ℝ. We choose h = −1 and then by the variation of constants formula we have
$$\begin{array}{}
\displaystyle
y=u(1)=T\left( 1\right) u\left( 0\right) +\int_{0}^{1}T(1-s)\left(
h-f(u(s)\right) ds.
\end{array}$$
for any r ≥ 2. Take an arbitrary value r satisfying r ∈ [2, +∞] if N ≤3, r ∈ [2, +∞) if N = 4 and 2 ≤ r <
$\begin{array}{}
\displaystyle
\frac{2N}{N-4}
\end{array}$
if N ≥ 5. The inequality ∥f(u(s)∥2 ≤ R1(1 +
$\begin{array}{}
\displaystyle
\left\Vert u\left( t\right) \right\Vert _{V}^{2p-2}
\end{array}$
), together with the boundedness of the attractor in the space V, gives the existence of a constant R2 such that
where we have used that due to the conditions imposed on the parameter r, it follows that
$\begin{array}{}
\displaystyle
\frac{N}{2}\left( \frac{1}{r}-\frac{1}{2}\right) +1 \gt 0
\end{array}$.
Thus the first statement of the theorem is proved.
In order to prove the second one, we first will prove that if the global attractor is bounded in Ls(Ω), where s satisfies
Therefore, applying again formula (26) with r = ∞ and
$\begin{array}{}
\displaystyle
q=\min\{\overline{q},s\frac{N-2}{N}\}
\end{array}$
and arguing as before for any y ∈ 𝓐 we have
where we have used that
$\begin{array}{}
\displaystyle
-\frac{N}{2q}+1 \gt 0.
\end{array}$
Thus, the result follows.
Observe that if N = 4, then the attractor is bounded in Ls(Ω) for an arbitrary s ∈ [2, ∞). Hence, we can pick s such that (27) holds. On the other hand, since we have proved that the attractor is bounded in Ls(Ω) for any s <
$\begin{array}{}
\displaystyle
\frac{2N}{N-4}
\end{array}$
, then (27) is also satisfied if N = 5, 6.
Further, for N ≥ 7 we will apply the above procedure iteratively so as to achieve (27).
Assume that the attractor is bounded in Ls(Ω) for some
$\begin{array}{}
\displaystyle
s\geq
\frac{2N}{N-2}\,\, \text{such that}\,\, s\frac{N-2}{N} \lt \frac{N}{2}.
\end{array}$
Using (28) and applying again (26) with
$\begin{array}{}
\displaystyle
q=s\frac{N-2}{N}\,\, \text{and}\,\, \overline
{s} \lt \frac{s(N-2)N}{N^{2}-2s(N-2)}
\end{array}$
for any y ∈ 𝓐 we obtain
and thus proceeding iteratively we obtain in k steps that the global attractor is bounded in Ls(Ω), where s satisfies (27).
Weak attractor for strong solutions in the supercritical case
In this section, our aim is to prove the existence of a weak global attractor for the multivalued semiflow generated by strong solutions to problem (14) satisfying a suitable energy inequality without imposing the assumption (17). In this case, we do not know whether strong solutions belong to the space of continuous functions with values in V, and therefore we are still not able to prove the existence of a strong attractor. Instead, we have to consider a weaker attractor in which the attracting property is satisfied with respect to the weak topology of the space V ∩ Lp(Ω).
Lemma 5
Letuτ ∈ V ∩ Lp(Ω). Then there exists at least one strong solutionuof (1) such thatu(τ) = uτ. Moreover, the energy inequality
holds for allt ≥ s, a.a. s > 0 ands = 0, where$\begin{array}{}
\displaystyle
E(u(t))=\left\Vert
u\left( t\right) \right\Vert _{V}^{2}+\left( F(u(t)),1\right) -2\left(
h,u\left( t\right) \right) .
\end{array}$
Proof
As usual, we take the Galerkin approximations using the basis of eigenfunctions {wj(x), j ∈ ℕ} of −Δ with Dirichlet boundary conditions. Let Xm = span{w1, …, wm} and let Pm be the orthogonal projector from H onto Xm. Then
$\begin{array}{}
u_{m}\left( t,x\right) =\sum_{j=i}^{m}a_{j,m}\left( t\right) w_{j}\left(
x\right)
\end{array}$
will be a solution of the system of ordinary differential equations
It is proved in [10, p.281] that passing to a subsequence um converges to a weak solution u of (1) weakly star in L∞(0, T; H), weakly in Lp(0, T; Lp(Ω)) and weakly in L2(0, T; V) for all T > 0. Also, u(τ) = uτ and
$\begin{array}{}
\displaystyle
\dfrac{du_{m}}
{dt}\rightarrow\dfrac{du}{dt}
\end{array}$
weakly in Lq(0, T; H−s(Ω)) for some s > 0.
Multiplying (30) by
$\begin{array}{}
\displaystyle
\dfrac{du_{m}}{dt}
\end{array}$
we get
By the choise of the special basis we have that um(0) → u0 in
$\begin{array}{}
\displaystyle
H_{0}^{1}
\end{array}$
(Ω) ∩ Lp(Ω).
Hence, um → u weakly star in L∞(0, T; V ∩ Lp (Ω)) and
$\begin{array}{}
\displaystyle
\dfrac{du_{m}}{dt}\rightarrow\dfrac{du}{dt}
\end{array}$
weakly in L2(0, T; L2(Ω)). Thus from the Ascoli-Arzelà theorem {um} is pre-compact in C([0, T]; H) and then um → u in C([0, T]; H). Moreover, it follows that um (t) → u(t) in
$\begin{array}{}
\displaystyle
H_{0w}^{1}(\Omega)\cap L_{w}^{p}(\Omega))
\end{array}$
for any t ∈ [0, T].
Finally, we must check the validity of the energy inequality (29). It is clear from (31) that um satisfy
$$\begin{array}{}
\displaystyle
u_{m}\rightarrow u\, \text{strongly in }L^{2}(r,T;V)\cap L^{p}(r,T;L^{p}
(\Omega)),
\end{array}$$
so
$$\begin{array}{}
\displaystyle
u_{m}(t)\rightarrow u\left( t\right) \text{ in }V\cap L^{p}(\Omega)\text{
for a.a. }t\in\left( 0,T\right) .
\end{array}$$
Applying then the dominated convergence theorem we deduce that F(um(t)) → F(u(t)) in L1(Ω) for a.a. t ∈ (0, T). On top of that, in view of um → u in C([0, T], H), for any t ∈ [0, T] it follows that F(um(t, x)) → F(u(t, x)) for a.a. x ∈ Ω. Hence, by the inequality
$$\begin{array}{}
\displaystyle
F(u_{m}(t,x)) \geq-D_{2}\text{ if }p \gt 2,\\
F(u_{m}(t,x)) \geq\delta\left\vert u_{m}(t,x)\right\vert ^{2}-D_{2}\text{
if }p=2,
\end{array}$$
For any u0 ∈ V ∩ Lp(Ω) let 𝓡(u0) be the set of strong solutions to problem (14) which satisfy the energy inequality (29) for all t ≥ s and a.a. s > 0. This set is non-empty for every initial datum u0 ∈ V ∩ Lp(Ω) due to Lemma 5. Denoting X = V ∩ Lp(Ω) we define now the multivalued mapping G : ℝ+ × X → P(X) by
It is straightforward to check that G(0, x) = x and G(t + s, x) ⊂ G(t, G(s, x)) for any x ∈ X and t, s ∈ ℝ+, that is, G is a multivalued semiflow.
In the sequel,
$\begin{array}{}
\displaystyle
X_{w}=H_{0w}^{1}\left( \Omega\right) \cap L_{w}^{p}(\Omega).
\end{array}$
Definition 5
The set 𝓐 is called a weak global attractor for the multivalued semiflow G if the following properties hold:
𝓐 is bounded in X and compact in Xw.
𝓐 is weakly attracting, that is, for any bounded set B in X and any neighborhood 𝓞 of 𝓐 in Xw there exists T(𝓞) such that
$$\begin{array}{}
\displaystyle
G(t,B)\subset\mathscr{O}\text{ for all }t\geq T.
\end{array}$$
𝓐 is negatively semi-invariant, that is,
$$\begin{array}{}
\displaystyle
\mathscr{A\subset}G(t,\mathscr{A)}\text{ for any }t\geq0.
\end{array}$$
It follows from this definition that if K is a weakly closed set which is weakly attracting, then for any neighborhood 𝓞 of K in Xw there exists T(𝓞) such that
$$\begin{array}{}
\displaystyle
\mathscr{A\subset}G(t,\mathscr{A)\subset O}\text{ for all }t\geq T.
\end{array}$$
Hence, 𝓐 ⊂ 𝓚 as Xw is a Hausdorff topological space. This means that 𝓐 is the minimal weakly closed set which is weakly attracting.
We recall that the set B0 is said to be absorbing if for any bounded set B in X there exists T(B) such that
$$\begin{array}{}
\displaystyle
G(t,B)\subset B_{0}\text{ for any }t\geq T.
\end{array}$$
Lemma 6
The multivalued semiflowGpossesses a bounded absorbing set$\begin{array}{}
\displaystyle
B_{0}=\{v\in X:\left\Vert v\right\Vert _{V}^{2}+\left\Vert v\right\Vert
_{L^{p}(\Omega)}^{p}\leq R\},
\end{array}$whereRis a positive constant.
Remark 1
The set B0 is weakly closed.
Proof
The function E(u(t)) satisfies (29) and multiplying the equation in (14) by u we have
The theory of global attractors in topological spaces for multivalued semiflows and processes has been developed for example in [1, 15, 21]. However, since the abstract conditions imposed in those papers are not met in our particular situation, we will prove the existence of the weak attractor from scratch.
Theorem 7
The multivalued semiflowGpossesses a weak global attractor 𝓐.
Proof
Take an arbitrary bounded set B. We recall that the ω − limit set of B is given by
Since the space Xw satisfies the first axiom of countability, an equivalent definition for ω(B) is the following
$$\begin{array}{}
\displaystyle
\omega(B)=\left\{
\begin{array}
[c]{c}
y\in X:\text{ there exist sequences }y_{n}\in G(t_{n},x_{n})\text{, }
t_{n}\rightarrow+\infty\text{, }x_{n}\in X\\
\text{such that }y_{n}\rightarrow y\text{ in }X_{w}
\end{array}
\right\} .
\end{array}$$
Therefore, as X is a reflexive Banach space and by Lemma 6 the set ∪t≥ T(B)G(t, B) is bounded for some T(B), any sequence yn ∈ G(tn, B) with tn → +∞ has a weakly convergent subsequence. Thus, ω(B) is non-empty. It is obvious that ω(B) belongs to the absorbing set B0, so it is bounded in X, and that it is weakly closed, so it is compact in Xw.
Further, let us check that ω(B) weakly attracts B. Assuming the opposite, there exists a neighborhood 𝓞 of ω(B) and a sequence yn ∈ G(tn, B), where tn → +∞, such that yn ∉ 𝓞. But this leads to a contradiction as from {yn} we can extract a converging subsequence whose limit belongs to ω(B).
It remains to prove that ω(B) is negatively semi-invariant.
First of all, let us consider a sequence of strong solutions un(⋅) ∈ 𝓡(
$\begin{array}{}
\displaystyle
u_{0}^{n}
\end{array}$
) such that un(t) ∈ B0 for any t ≥ 0. In view of well-known results (see [20, Lemma 15] and [16, Theorem 3.11]), there exists a weak solution u(⋅) to problem (14) and a subsequence unk(⋅) such that unk → u strongly in C([0, T], H), for all T > 0 (among other convergences). But un(t) are uniformly bounded in X, so
$$\begin{array}{}
\displaystyle
u^{n_{k}}\rightarrow u\text{ weakly star in }L^{\infty}(0,T;V\cap L^{p}
(\Omega))\text{ for all }T \gt 0,\\\displaystyle\qquad\qquad\quad
u^{n_{k}}(t)\rightarrow u\left( t\right) \text{ in }X_{w}\text{ for any
}t\geq0.
\end{array}$$
Also, inequalities (29), (6) and E(un(s)) ≤ C, for any s ≥ 0 and n, imply that
$\begin{array}{}
\displaystyle
\dfrac{du^{n}}{dt}
\end{array}$
are bounded in L2(0, T; H) and then
$$\begin{array}{}
\displaystyle
\frac{du^{n_{k}}}{dt}\rightarrow\frac{du}{dt}\text{ weakly in }L^{2}(0,T;H).
\end{array}$$
Therefore, u(⋅) is in fact a strong solution to problem (14). On top of that, arguing in the same way as in Lemma 5 we obtain that
$$\begin{array}{}
\displaystyle
u^{n_{k}}\rightarrow u\, \text{strongly in }L^{2}(r,T;V)\cap L^{p}
(r,T;L^{p}(\Omega)),\text{ for any }0 \lt r \lt T,
\end{array}$$
and also that (29) is satisfied for all t ≥ s and a.a. s > 0. Thereby, u(⋅) ∈ 𝓡(u0).
Consider now an arbitrary element y ∈ ω(B) and t > 0. Then there is a sequence yn ∈ G(tn, xn) with tn → +∞, xn ∈ B such that yn → y in Xw. In addition, we take N(B) for which G(s, B) ⊂ B0 if s ≥ tn − t and n ≥ N. Since G(tn, xn) ⊂ G(t, G(tn − t, xn)), there are zn ∈ G(tn − t, xn), un(⋅) ∈ 𝓡(zn) satisfying that un(s) ∈ B0, for any s ≥ 0, and yn = un(t). Passing to a subsequence zn converges in the space Xw to some z ∈ ω(B). In light of the previous arguments there exists u(⋅) ∈ 𝓡(z) such that u(t) = y. Thus, y ∈ G(t, z) ⊂ G(t, ω(B)), which proves that ω(B) is negatively semi-invariant.
Finally, let 𝓐 = ω(B0). Obviously, 𝓐 is bounded in X, compact in Xw and negatively semi-invariant. The weakly attracting property follows from the chain of inclusions