### Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions

Matheus C. Bortolan, ,Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis - Brazil, José Manuel Uzal, ,Departamento de Estatística, Análise Matemática e Optimización & Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela - Spain
2020 Discrete and Continuous Dynamical Systems. Series A
We define the notions of impulsive evolution processes and their pullback attractors, and exhibit conditions under which a given impulsive evolution process has a pullback attractor. We apply our results to a nonautonomous ordinary differential equation describing an integrate-and-fire model of neuron membrane, as well as to a heat equation with nonautonomous impulse and a nonautonomous 2D Navier-Stokes equation. Finally, we introduce the notion of tube conditions to impulsive evolution
more » ... e evolution processes, and use them as an alternative way to obtain pullback attractors. 2010 Mathematics Subject Classification. Primary: 37L05, 37L30. Secondary: 34H05. 2791 2792 MATHEUS C. BORTOLAN AND JOSÉ MANUEL UZAL IMPULSIVE EVOLUTION PROCESSES AND THEIR ATTRACTORS 2793 an invariance property for all impulsive ω-limit sets, and not only for the pullback attractor. 2. Impulsive evolution processes and their pullback attractors. We begin by fixing a metric space (Z, d) and denoting by C(Z) the set of all continuous maps from Z to itself. Also we denote P = {(t, s) ∈ R 2 : t s}. Definition 2.1. An evolution process in Z is a two-parameter family of maps U = {U (t, s) : (t, s) ∈ P} ⊂ C(Z) which satisfies the following conditions: is nonempty, open, closed, compact in Z, respectively, for each t ∈ R. Also ifD 1 andD 2 are families in Z, we will denoteD 1 ⊂D 2 if D 1 (t) ⊂ D 2 (t) for each t ∈ R. The fundamental properties of a pullback attractor, as we will see, are the invariance and the pullback attraction. We will define these concepts precisely. Definition 2.2. Given an evolution process U, a nonempty familyD = Definition 2.3. Given an evolution process U and nonempty familiesÂ andB, we say thatÂ pullback U-attractsB if Now we are able to define the concept of a pullback attractor for a given evolution process. Definition 2.4. Let D be a collection of families in Z. A familyÂ is called the pullback D-attractor for an evolution process U if it satisfies the following: (iv)Â is the minimal closed family satisfying (iii), that is, ifĈ is a closed family that pullback U-attracts each familyD ∈ D, thenÂ ⊂Ĉ. Condition (iv) is used to ensure the uniqueness of the pullback D-attractor. However, without condition (iv), we would still have uniqueness with the additional requirement thatÂ ∈ D. Since ifÂ 1 ,Â 2 ∈ D satisfy (i), (ii) and (iii), we would have d H (A 1 (t), A 2 (t)) = d H (U (t, s)A 1 (s), A 2 (t)) → 0, as s → −∞, and hence A 1 (t) ⊂ A 2 (t). Interchanging A 1 and A 2 we obtain A 2 (t) ⊂ A 1 (t), and since t ∈ R is arbitrary, we haveÂ 1 =Â 2 . MATHEUS C. BORTOLAN AND JOSÉ MANUEL UZAL 2.1. Impulsive evolution processes. Now we can present the theory of impulsive evolution processes, and to that end, let U be an evolution process in a metric space (Z, d). For each s ∈ R, r 0 and familyD we define F (D, r, s) = {z ∈ Z : U (r + s, s)z ∈ D(r + s)}. ( 2.2) When D(t) = D for all t ∈ R (D is a fixed subset of Z), then we simplify the notation setting F (D, r, s) := F (D, r, s). Moreover, we denote F (z, r, s) := F ({z}, r, s). Definition 2.5. A familyD = {D(t)} t∈R will be called collectively closed if for t n → t, x n ∈ D(t n ) with x n → x, then x ∈ D(t). Also, a familyD will be called collectively compact if given t n → t and x n ∈ D(t n ), then {x n } has a convergent subsequence to a point in D(t). Note that, in particular, a collectively closed family is also closed. With these definitions and notations, we are able to present the impulsive evolution processes: Definition 2.6. An impulsive evolution process, denoted bỹ U = (U, Z,M , I), consists of an evolution process U in the metric space Z, a collectively closed familŷ M such that for every z ∈ M (s) there exists = (z, s) > 0 such that and a collectively continuous family of functions I = {I t : M (t) → Z} t∈R , that is, given t n → t and x n ∈ M (t n ) with x n → x then I tn (x n ) → I t (x). When all the elements in this definition are well understood, we will denote the impulsive evolution process simply byŨ. Some remarks are necessary to a better understanding of this definition, before we can continue. Remark 2.7. 1. Conditions (2.3) are, in some sense, a transversality property of the evolution process U with respect to the familyM . Roughly speaking, if z ∈ M (s), then the flow defined by U has to effectively cross M (s) when passing through z. 2. The collectively closedness condition forM is already required for the definition of the collectively continuity of I, since if t n → t, x n ∈ M (t n ) and x n → x, then x ∈ M (t), and hence we can compute I t (x). Nevertheless, it will also be required in the proof of some results throughout the text. 3. Regarding the family of functions I, its role is not yet explicit in this definitions, and it will become clear in the definition of the impulsive semitrajectories. 4. The familyM is often called impulsive family and each M (t) is called the impulsive set at time t, and the family of functionsÎ is called the impulse function and each individual function I t is called impulse function at time t. We define {U (r + s, s)z} ∩ M (r + s) for each z ∈ Z and s ∈ R. (2.4) IMPULSIVE EVOLUTION PROCESSES AND THEIR ATTRACTORS 2795 Our first result is the following: Proposition 2.8. If M(z, s) = ∅ then there exists a unique τ > 0 such that U (τ + s, s)z ∈ M (τ + s) and U (r + s, s)z / ∈ M (r + s) for 0 < r < τ . Proof. Indeed, let τ = inf{r > 0 : U (r + s, s)z ∈ M (r + s)}. If z ∈ M (s) and τ = 0 then there exists a sequence r n → 0 + such that U (r n + s, s)z ∈ M (r n + s) for each n ∈ N, which contradicts (2.3). Now if z / ∈ M (s), sinceM is collectively closed and U is an evolution process, and hence it satisfies property (iii) of Definition 2.1, there exists ε > 0 such that U (r + s, s)z / ∈ M (r + s) for 0 r < , which implies that τ > 0. In any case τ has the required properties and the result is complete. With this proposition, we are able to present the next definition. Definition 2.9. Define φ : Z × R → (0, ∞] by φ(z, s) = ∞ if M(z, s) = ∅ and if M(z, s) = ∅ we define φ(z, s) = τ , where τ > 0 is given in Proposition 2.8. The function φ is called the impact time map of U. An important straightforward property of φ is the following: if r > 0 and U (r + s, s)z ∈ M (r + s) then φ(z, s) r. Now we can define the impulsive semitrajectories, and the role of the impulsive function I will become clear.