PULLBACK ATTRACTORS OF FITZHUGH-NAGUMO SYSTEM ON THE TIME-VARYING DOMAINS

. The existence and uniqueness of solutions satisfying energy equality is proved for non-autonomous FitzHugh-Nagumo system on a special time- varying domain which is a (possibly non-smooth) domain expanding with time. By constructing a suitable penalty function for the two cases respectively, we establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system on a special time-varying domain.


1.
Introduction. There are many papers devoted to studying the semilinear parabolic equation on a time-varying domain, we refer the readers to [1,2,5,6,7,8,16] and the references therein, most of them studied the nonlinear parablic equation on the time-varying domains. Recently, Kloeden, Marín-Rubio and Real in [7] followed the penalty function method developed by J.Lions in [9] to establish the existence of the pullback attractor for a semilinear heat equation with a homogeneous Dirichlet boundary condition defined on the domains expanding in time. Moreover, Kloeden, Real and Sun in [6] established the existence of a global pullback attractor for the same equation defined on general time-varying domain by using a C 2 -diffeomorphism transformation.
In this paper, we will study the dynamics of the non-autonomous FitzHugh-Nagumo system with homogeneous Dirichlet boundary condition defined on the time-varying domains Q τ by where the notations are defined later in section 2. We establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system. FitzHugh-Nagumo equations introduced by FitzHugh [4] and Nagumo, Arimoto and Yosimazawa [14] are intended to describe the signal transmission across axons. The long time behavior of FitzHugh-Nagumo equations on bounded and unbounded domains was presented by some authors in [10,11,12,13,17,18] and the references therein.
The main novelty of the present paper is to construct a new penalty function to show the uniqueness and existence of the variational solution for equations (1) on the time-varying domain Q τ expanding in time, which is motivated by the idea in [7] and [9], and establish the existence of a pullback attractor of non-autonomous FitzHugh-Nagumo system on the time-varying domain.
The rest of the paper is arranged as follows. In section 2, some notations and functions setting are introduced. In section 3, the existence of variational solution for FiztHugh-Nagumo system on the time-varying domain which increases with time are obtained by the penalty function method. In section 4, the existence of pullback attractor of FiztHugh-Nagumo system on the time-varying domain are obtained.

Preliminaries. Let {O t } t∈[τ,T ] be a family nonempty bounded open subset of
∂O t × {t} f or all T > τ, For any T > τ , the set Q τ,T is an open subset of R N +1 with boundary We consider the following auxiliary problem to equations (1) where τ ∈ R, u τ , v τ : O τ → R and f, g : Q τ → R are given, ν, λ, , γ are positive constants, h is a smooth nonlinear function such that for some positive constant α, β, l and p ≥ 2, For later observe, from (4), we have for some positive constantα,β −β +α 2 |s| p ≤ H(s) ≤β +α 1 |s| p , s ∈ R where H(x) := x 0 h(s)ds.

PULLBACK ATTRACTORS OF FITZHUGH-NAGUMO SYSTEM 3693
Since is small, we can assume that Denote σ be a positive number given by Here, we consider the domains expanding with time, For each t < T , consider H 1 0 (O t ) as a closed subspace of H 1 0 (O T ) with the function belonging to H 1 0 (O t ) being trivially extended by zero. It follows from (9) that Similarly, we have (11) For the sake of simplicity, we denote and Denote by (·, ·) t , | · | t the usual inner product and associated norm in L 2 (O t ) or (L 2 (O t )) N , and by ((·, ·)) t , · t the usual inner product and associated norm in H 1 0 (O t ). Notice that (·, ·) t is also used to denote the duality product between L p/p−1 (O t ) and L p (O t ). More notations and properties about time-varying domains can be found in [6] and [7]. Now, we recall the basic concept of pullback attractor for non-autonomous dynamical systems. For each t ∈ R, and D 1 , D 2 nonempty subsets of L 2 (O t )×L 2 (O t ). Let us denote dist t (D 1 , D 2 ) the Hausdorff semi-distance defined as Let R σ be the set of all functions ρ : R → [0, ∞) such that is said to be a pullback D σ -attractor for the process Υ(t, τ ) : , A(t)) = 0 f or allD ∈ D σ and all t ∈ R; (3)Â is invariant, i.e.
Theorem 2.3. ( [7]) Suppose that the process Υ(·, ·) is L 2 -pullback D σ -asymptotically compact and thatD 0 is a family of L 2 -Pullback D σ -attracting set for Υ(·, ·). Then the familyÂ = {A(t), t ∈ R} defined by where for anyD ∈ D σ , is the unique pullback D σ -attractor for process Υ(t, τ ) belonging to D σ . In addition, A satisfies 3. Variational solution on domains expanding in time. In this section, we consider Fitzhugh-Nagumo system on bounded spatial domains (possibly nonsmooth) which are expanding in time. For each T > τ , denote be a variational solution of (3) on bounded spatial domains which are expanding in time and suppose that there exists a sequence {t n } ⊂ (τ, T ) of Lebesgue points of |u| 2 T and |v| 2 T such that t n → T , lim sup Then, (u, v) satisfies the energy equality a.e. in (τ, T ), Proof. The proof is similar to Lemma 10 in [7], and is omitted here.
Proof. The proof is similar to Lemma 11 in [7], and is omitted here.
Proposition 1 guarantees the uniqueness of the variational solutions for (3) on domain expending in time immediately.
3.1. Penalty method. In this subsection, we follow the penalty method in [7], which developed by [9] to establish the existence and uniqueness of the variational solution for Fitzhugh-Nagumo system on the domain expending in time.

Lemma 3.3. For any integers
Lemma 3.4. For any integers 1 ≤ h ≤ k, any t ≥ τ , and every φ, For each integer k ≥ 1 and each t ∈ [τ, T ], we consider the following symmetric bilinear operators and We have the following estimates and Now, let (u τ , v τ ) ∈ (L 2 (O T )) 2 be given and k ≥ 1. we consider the problem and fixed k ≥ 1, there exists a unique variational solution (u k , v k ) of (28). Moreover, for a.e. t ∈ (τ, T ), and Proof. The existence of solution of(28) can be obtained by the Galerkin method( [3,9,15]   (v τ , e j ) T e j , and j ∈ N. Moreover, the solution (u km , v km ) satisfies the energy equality It follows from (5), (26) and (27) that Due to (8), (34) and (35), we obtain Since {u τm } and {v τm } are bounded in L 2 (O T ), then Multiplying the equation by a km,j yields (u km (t), e j ) T + A k (t)(u km (t), e j ) + (h(u km ), e j ) T + λ(u km , e j ) T + (v km , e j ) T = (f (t), e j ) T a.e. t ∈ (τ, T ).
Summing the above equations from k = 1 to k = m implies It follows from [7] that Thus, we have T τ |u km | 2 T dr + ν u km (t) 2 + k((P k (t)u km (t), u km (t))) T + 2α 1 u km (t) p and Define the functional Φ : Then Φ is continuous and convex and

ZHEN ZHANG, JIANHUA HUANG AND XUEKE PU
Similarly, we have the following estimates By using the Hölder inequality and the Cauchy inequality, we have for a.e. t ∈ (τ, T ). Hence, we have {v km } is bounded in L 2 (Q τ,T ).
Since v τ ∈ L 2 (O T ), (33) indicates that there exists a sequence v τm converging to v τ in L 2 (O T ). Due to (46), extracting a subsequence if necessary, we assume that {v km } converges weakly-star to {v k } in L ∞ (τ, T ; L 2 (O T )) and {v km } converges weakly to {v k } in L 2 (Q τ,T ), and a.e. t ∈ (τ, T ), Define the functional Ψ : Then Ψ is continuous and convex, and it holds Thus, the proof is completed.
Next, we will establish the existence of variational solutions satisfying the energy equations.
Then, combining it with Lemma 4.1, Lemma 4.3 and (71), it is easy to see that there exists a compact D−attracting K for the non-autonomous process Υ defined above, which attracts bounded subsets of {(L 2 (O t )) 2 } t∈R . Thus, using Theorem 4.1, we can obtain a unique non-autonomous pullback attractor in {(L 2 (O t )) 2 } t∈R .