Existence , regularity and upper semicontinuity of pullback attractors for the evolution process associated to a neural field model

In this work we study the pullback dynamics of a class of nonlocal nonautonomous evolution equations for neural fields in a bounded smooth domain Ω in RN ∂tu(t, x) = −u(t, x) + ∫ RN J(x, y) f (t, u(t, y))dy, t ≥ τ, x ∈ Ω, u(τ, x) = uτ(x), x ∈ Ω, with u(t, x) = 0, t ≥ τ, x ∈ RN\Ω, where the integrable function J : RN ×RN → R is continuously differentiable, ∫ RN J(x, y)dy = ∫ RN J(x, y)dx = 1 and symmetric i.e., J(x, y) = J(y, x) for any x, y ∈ RN . Under suitable assumptions on the nonlinearity f : R2 → R, we prove existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to this problem.


Introduction
In this paper we study the pullback dynamics for a class of nonlocal non-autonomous evolution equations generated as continuum limits of computational models of neural fields theory.In short, neural field equations are tissue level models that describe the spatiotemporal evolution of coarse grained variables such as synaptic or firing rate activity in populations of neurons, see e.g.[1-3, 9, 20, 21, 24, 26, 28, 29].

Mathematical framework
To better present our results, we first introduce some terminology and notation.Let Ω ⊂ R N be a bounded smooth domain modelling the geometric configuration of the network, u : R × R N → R a function modelling the mean membrane potential, u(t, x) being the potential of a patch of tissue located at position x ∈ Ω at time t ∈ R and f : R × R → R a time dependent transfer function.Let also the integrable function J : R N × R N → R be the connection between locations, that is, J(x, y) is the strength, or weight, of the connections of neuronal activity at location y on the activity of the neuron at location x.The strength of the connection is supposed to be symmetric, that is J(x, y) = J(y, x), for any x, y ∈ R N .We also adopt a homogeneous and isotropic assumption for the layer so that, without loss of generality R N J(x, y)dy = R N J(x, y)dx = 1.
We say that a neuron at a point x is active at time t if f (t, u(t, x)) > 0.
We thus analyze the following non-autonomous theoretical model for networks of nerve cells with the "boundary" condition where the integral operator with symmetric kernel K is defined by for all v ∈ L 1 (R N ).Also we will assume that f : R 2 → R is a sufficiently smooth function (some growth conditions about f are also assumed, as presented along the Section 3).
We are interested in showing existence of the pullback attractor for the evolution process associated to Cauchy problem (1.1)-(1.2) in an appropriated Banach space, as well as some of its properties such as regularity and upper semicontinuity with respect to the functional parameter f .
Our model is a generalization of the one analyzed by many authors, (e.g.[1,9,20,25,27,28]), which takes the form where the strength of the connection depends only on the distance between cells, that is, J(x, y) = J(x − y) and the firing rate function is time-independent.

Outline of the paper
This paper is organized as follows.In Section 2 we recall some definitions from the theory of evolution process (or non-autonomous dynamical systems).
In Section 3, assuming the growth conditions (3.7), (3.8), (3.11) and (3.14), below for the nonlinearity f , we prove that (1.1)-(1.2) generates a C 1 flow in the phase space with the induced norm, satisfying the "variation of constants formula" In Section 4, we prove existence of the pullback attractor in X p and establish some regularity properties for it.
Finally, in section 5 we prove the upper semicontinuity of the pullback attractors with respect to the function f .

Functional setting and background results
In this section we recall some definitions from the theory of evolution processes (or infinitedimensional non-autonomous dynamical systems); following [7], where full proofs and more details can be found, (see also [8,15,16,22,23], and references therein).Definition 2.1.Let X be a complete metric space and d : X × X → R be its metric.An evolution process in X is a family of maps {S(t, τ); t ≥ τ, τ ∈ R} (or simply S(•, •)) from X into itself with the following properties: • S(t, t) = I, for all t ∈ R, where I : X → X is the identity map; • S(t, τ) = S(t, s)S(s, τ), for all t ≥ s ≥ τ;

Definition 2.2.
A globally-defined solution (or simply a global solution) of the evolution process {S(t, τ); t ≥ τ, τ ∈ R} is a function ξ : R → X such that for all t ≥ τ we have S(t, τ)ξ(τ) = ξ(t).A global solution ξ : R → X of the evolution process {S(t, τ); t ≥ τ, τ ∈ R} is backward-bounded if there is a τ ∈ R such that {ξ(t); t ≤ τ} is a bounded subset of X.The family {K(t); t ∈ R} of subsets of X pullback attracts bounded subsets of X under {S(t, τ); t ≥ τ, τ ∈ R} if K(t) pullback attracts bounded subsets of X at time t under the process {S(t, τ); t ≥ τ, τ ∈ R}, for each t ∈ R.
We observe that the Hausdorff semi-distance between A and B, dist H (A, B), measures how far the set A is from being contained in the set B. For example, dist H (A, B) = 0 if and only if A is contained in the closure of the set B. Now we remember the notion of an ω-limit for processes; we will build our pullback attractor as a union of ω-limit sets.
Definition 2.5.The pullback omega-limit set at time t of a subset B of X is defined by or equivalently, Definition 2.9.A family of continuous operators {S(t, τ); t ≥ τ, τ ∈ R} (which need not be a process) is called strongly compact if for each time t and each bounded B ⊂ X there exists a The following two results will be used to prove the existence of the pullback attractor for the evolution process generated by (1.1)-(1.2) in the Banach space X p (defined in (1.3)).
Theorem 2.10.Let X be a Banach space and | • | X : X → R be its norm.If an evolution process {S(t, τ); t ≥ τ, τ ∈ R} in X satisfies the properties where U(t, τ) is a strongly compact operator and there exists a non-increasing function k : [0, +∞) × [0, +∞) → R with k(σ, r) → 0 as σ → +∞, and for all τ ≤ t and z ∈ X with |z| X ≤ r, where {B(t); t ∈ R} is a family of bounded subsets of X which for each t ∈ R pullback attracts bounded subsets of X at time τ, for any τ ≤ t.
The pullback attractor of strongly bounded dissipative process however, is always bounded in the past.To be more precise, for every t ∈ R the union τ≤t A(τ) is bounded in X.

Well-posedness of the problem
In this section we show the global well-posedness of the problem (1.1)-(1.2) in an appropriate Banach space, under suitable growth condition on the nonlinearity f .Consider, for any 1 ≤ p ≤ ∞, the subspace X p of L p (R N ) given by with the induced norm.The Banach space X p is canonically isometric to L p (Ω) and we usually identify the two spaces, without further comment.We also use the same notation for a function in R N and its restriction to Ω for simplicity, wherever we believe the intention is clear from the context.In order to obtain well-posedness of (1.1)-(1.2) in X p , we consider the Cauchy problem in the Banach space where the nonlinearity F : R × X p → X p is defined by where the map K given by is well defined as a bounded linear operator in various function spaces, depending on the properties assumed for J; for example, with J satisfying the hypotheses from introduction, K is well defined in X p as shown below.
The following simple estimates will be useful in the sequel.
Lemma 3.1.Let K be the map defined by (3.3) and J r : where 1 ≤ q ≤ ∞ is the conjugate exponent of p.Moreover, Proof.Estimate (3.4) follows easily from Hölder's inequality.Estimate (3.5) follows from the generalized Young's inequality (see [12]).The proof of (3.6) is similar to (3.5), but we include it here for the sake of completeness.Suppose 1 < p < ∞ and let q be its the conjugate exponent.Then, by Hölder's inequality .
Raising both sides to the p-th power and integrating, we obtain The inequality (3.6) then follows by taking p-th roots.
The case p = 1 is similar but easier, and the case p = ∞ is trivial.(3.8) then, for any 1 ≤ p < ∞ the function F is locally Lipschitz continuous in the second variable If p = ∞, this is true if f is locally Lipschitz in the second variable.
If f (t, x) is also continuous in t, then for any (t, u) ∈ R × X p we have for a small h ∈ R. From (3.7), the integrand is bounded by 2C(1 + |u(x)| p ), where C is a bound for C(t) in a neighborhood of t and goes to 0 as h → 0. Therefore, by Lebesgue's dominated convergence theorem, f (t, u) − f (t + h, u) L 1 (Ω) → 0 as h → 0. Thus which goes to 0 as h → 0, proving the continuity of F in t.Suppose now that for some 1 < p < ∞, where C 2 : R → R is a strictly positive function.Then, for u and v belonging to L p (Ω) we get where q is the conjugate exponent of p.Using (3.6) once again and the hypothesis on f , it follows that showing that F is Lipschitz in bounded sets of L p (Ω) as claimed.
If p = 1, the proof is similar, but simpler.Suppose finally that and this allows us to conclude that Thus, by (3.5) we have that and this completes the proof.
From Proposition 3.3, and well known results, it follows that the initial value problem (3.1) has a unique local solution for any initial condition in X p .For the global existence, we need the following result (see [18,Theorem 5.6.1]).Theorem 3.4.Let X be a Banach space, and suppose that G : [t 0 , +∞) × X → X is continuous and G(t, u) ≤ g(t, u ), for all (t, u) ∈ [t 0 , +∞) × X, where g : [t 0 , +∞) × [0, +∞) → [0, +∞) is continuous and g(t, r) is non decreasing in r ≥ 0, for each t ∈ [t 0 , +∞).Then, if the maximal solution r(t; t 0 , r 0 ) of the scalar initial value problem exists throughout [t 0 , +∞), the maximal interval of existence of any solution u(t; t 0 , y 0 ) of the initial value problem also contains [t 0 , +∞).
Corollary 3.5.Suppose, in addition to the hypotheses of Proposition 3.3, that f satisfies the dissipative condition for some constant k 1 ∈ R, independent of t.Then the problem (3.1) has a unique globally defined solution for any initial condition in X, which is given, for t ≥ τ, by the "variation of constants formula" (3.12) Proof.From Proposition 3.3, it follows that the right-hand-side of (3.1) is Lipschitz continuous in bounded sets of X and, therefore, the Cauchy problem (3.1) is well posed in X p , with a unique local solution u(t, x), given by (3.12) (see [10]).
From condition (3.11) it follows that where k 2 : R → R is a continuous and strictly positive function.If 1 ≤ p < ∞, we obtain from (3.5) and (3.13) the following estimate For p = ∞, we obtain by the same arguments (or by making p → ∞), that Now defining the function it follows that problem (3.1) satisfies the hypothesis of Theorem 3.4 and the global existence follows immediately.The variation of constants formula can be verified by direct derivation.
The result below can be found in [19].
•) is continuously Fréchet differentiable on X p with derivative given by Proof.From a simple computation, using the fact f is continuously differentiable in the second variable, it follows that the Gâteaux's derivative of F(t, •) is given by The operator D 2 F(t, u) is clearly a linear operator in X p .Suppose 1 ≤ p < ∞ and q is the conjugate exponent of p.Then, for u ∈ L p (Ω) we have that From Hölder's inequality and (3.15), it follows that Now from estimate (3.6) we concluded that showing that DF(t, u) is a bounded operator.In the case p = ∞, we have that and thus, from (3.5), we obtain showing the boundedness of DF(t, u) also in this case.Suppose now that u 1 and u 2 and v belong to L p (Ω), 1 ≤ p < ∞.From (3.6) and Hölder's inequality, it follows that Thus to prove continuity of the derivative, we only have to show that and a computation similar to (3.15) above shows that the right-hand side is integrable.The result then follows from Lebesgue's convergence theorem.
In the case p = ∞, we obtain from (3.5) and the continuity of DF follows from the continuity of ∂ 2 f (t, u).Therefore, it follows from Proposition 3.6 above that F(t, •) is Fréchet differentiable with continuous derivative in X p .Remark 3.8.Since, under the hypotheses of the Proposition 3.7 the right-hand side of (3.1) is continuous in t and C 1 in the second variable, the process generated by (3.1) in X p is C 1 with respect to initial conditions, (see [10] and [13]).
From the results above, we have that, for each t ∈ R and u τ ∈ X p , the unique solution of (3.1) with initial condition u τ exists for all t ≥ τ and this solution (t, τ, x) → u(t, x) = u(t; τ, x, u τ ) (defined by (3.12)) gives rise to a family of nonlinear C 1 flow on X p given by S(t, τ)u τ (x) := u(t, x), t ≥ τ ∈ R.
where q is the conjugate exponent of p. Hence, combining (4.2) with (4.3) we concluded that we have Therefore, while From this, the result follows easily for 1 ≤ p < ∞, and this complete the proof of the lemma.
Theorem 4.2.In addition to the conditions of Lemma 4.1, suppose that Then there exists a pullback attractor {A(t); t ∈ R} for the process {S(t, τ); t ≥ τ, τ ∈ R} generated by (3.1) in X = L p (Ω) and the 'section' A(t) of the pullback attractor A(•) of S(•, •) is contained in the ball centered at the origin with radius R δ (t) defined in (4.1), in L p (Ω), for any δ > 0, t ∈ R and 1 ≤ p < ∞.
Proof.We have proved that for each initial value u(τ, x) ∈ X and initial time τ ∈ R, (3.1) possesses a unique solution, which we now write as where from (3.12) we have that Now, using Theorem 2.10 (or Theorem 2.37, Chapter 2 in [7]), we prove that S(•, •) is pullback asymptotically compact.For this, suppose u ∈ B, where B is a bounded subset of X p .We may suppose that B is contained in the ball centered at the origin of radius r > 0. Then From (4.4), we have that u(t, •) L p (Ω) ≤ M, for t ≥ τ, where Hence, using (3.9), we obtain From estimate (3.6) (applied to J x in the place of J) it follows that Thus, we get Therefore, for t > τ and any u ∈ B, the value of ∂ x U(t, τ)u L p (Ω) is bounded by a constant (independent of u ∈ B).It follows that U(t, τ)u belongs to a ball of W 1,p (Ω) for all u ∈ B. From Sobolev's Embedding Theorem, it follows that U(t, τ) is a compact operator, for any t > τ.
Therefore it follows from Lemma 4.1 and Theorem 2.11 (or Theorem 2.23, Chapter 2 in [7]), that the pullback attractor {A(t); t ∈ R} exists and each 'section' A(t) of the pullback attractor A(•) is the pullback ω-limit set of any bounded subset of X p containing the ball centered at the origin with radius R δ , defined in (4.1), for any δ > 0. From this, since the ball centered at the origin with radius R δ pullback absorbs bounded subsets of X p , it also follows that the set A(t) is contained in the ball centered at the origin of radius Proof.Let T > τ and u n (t, x) = S n (t, τ)u τ (x) be the solution of the problem (5.2)-(5.3)for t ∈ [τ, T], given by (3.12).Then e −(t−s) K( f n (s, u n (s, x)) − f ∞ (s, u ∞ (s, x)))ds, x ∈ Ω, 0, x ∈ R N \Ω.For each value of the parameter n ∈ N we recall that S n (•, •) is the evolution process associated to problem (5.2)-(5.3).Now we prove the main result of this section.

Definition 2.3. The
subset B of X pullback absorbs bounded subsets of X at time t ∈ R under {S(t, τ); t ≥ τ, τ ∈ R} if there exists τ 0 = τ 0 (t, D) withS(t, τ)D ⊂ B for any τ ≤ τ 0 ≤ t.The family {B(t); t ∈ R} of subsets of X pullback absorbs bounded sets if B(t) pullback absorbs bounded sets in X at time t, for each t ∈

R. Definition 2.4. The
subset K of X pullback attracts bounded subsets of X under {S(t, τ); t ≥ τ, τ ∈ R} at time t if, for each bounded subset C of X

Definition 2.8. An evolution process
and {x k } in B, such that y = lim k→∞ S(t, τ k )x k .
{S(t, τ); t ≥ τ, τ ∈ R} in a Banach space X is pullback asymptotically compact if, for each t ∈ R, each sequence Definition 3.2.If E is a normed space, and I ⊂ R is an interval, we say that a function F : I × E → E is locally Lipschitz continuous (or simply locally Lipschitz) in the second variable if, for any (t 0 , x 0 ) ∈ I × E, there exists a constant C and arectangle R = {(t, x) ∈ I × E : |t − t 0 | < b 1 , x − x 0 < b 2 } such that, if (t,x) and (t, y) belong to R, then Suppose, in addition to the hypotheses of Lemma 3.1, that the function f satisfies the growth condition | f (t, x)| ≤ C 1 (t)(1 + |x| p ), for any (t, x) ∈ R × R N , (3.7) with 1 ≤ p < ∞, where C 1 : R → R is a locally bounded function.Then the function F given by (3.2) is well defined in R × X p .If f (t, •) is locally bounded for any t ∈ R, F is well defined in R × L ∞ (Ω).Additionally, if f is continuous in the first variable,then F is also continuous in the first variable.If there exists a strictly positive function C 2 : R → R such that F(t, x) − F(t, y) ≤ C x − y .Now we prove that the map F, given in (3.2), is well defined under appropriate growth conditions on f and is locally Lipschitz continuous (see Proposition 3.3).

4 Existence and regularity of the pullback attractor for 1 ≤ p < ∞
We prove the existence of a pullback attractor {A(t); t ∈ R} in X p for the evolution process {S(t, τ); t ≥ τ, τ ∈ R} when 1 ≤ p < ∞.Suppose that the hypotheses of Proposition 3.7 hold with the constant k 1 in (3.11) satisfying k 1 < 1.Then the ball of L p (Ω), 1 ≤ p < ∞, centered at the origin with radius R δ (t) R δ (t)), where k 1 and k 2 are derived from (3.13) and δ is any positive constant, pullback absorbs bounded subsets of X p at time t ∈ R with respect to the process S(•, •) generated by (3.1).