Global attractors of impulsive parabolic inclusions

In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side \begin{document}$\varepsilon F(y)$\end{document} and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter \begin{document}$\varepsilon>0$\end{document} this system has a global attractor.


1.
Introduction. An autonomous evolution system is called impulsive (or discontinuous) dynamical system (DS) if its trajectories have jumps at moments of intersection with certain surface of the phase space. Unlike systems with impulses at fixed moments of time [23], the behavior of impulsive DS is far from complete understanding. Some aspects of the qualitative behavior of impulsive DS have been studied by many authors in recent years [23], [22], [18], [19], [7], [1], [3], [21], [9]. Focus of investigation was on stability of solutions and on properties of ω-limit sets for discontinuous DS generated by impulsive systems of ordinary differential equations. Recently in [4], [5] several concepts of global attractor were proposed and an application to a scalar reaction-diffusion equation with impulsive perturbation was given. However, it should be noted that in all papers the authors used detailed information about the nature of intersection of a given set by trajectories of the system. These conditions are formulated in an abstract form and cannot be effectively tested without explicit formulas of solutions. In this paper we propose a different concept of global attractor. This concept is based on the definition of uniform attractor for non-autonomous DS [6], [15]. In particular, this definition of attractor is used for systems with impulsive perturbation at fixed moments of time [11]. From this point of view we consider a multi-valued situation when there exists more than one solution for a given initial data. The theory of global attractors of multi-valued DS has many interesting results (see [26] and references there in). Its applications to parabolic inclusions and parabolic equations without uniqueness firstly appeared in the works of V.S. Melnik and his pupils [20], [12], [25], [13], [14], [17], [16].
Using the theory of global attractors we construct an abstract multi-valued impulsive DS and prove a result about existence of global attractors for it. We apply the obtained results to describe asymptotic dynamics of a wide class of dissipative infinite-dimensional multi-valued DS, generated by parabolic inclusion with impulsive perturbations at non-fixed moments of time. In particular, we give effective sufficient conditions for existence of global attractors.
The paper is organized in the following way. In Section 2 we give some abstract results concerning the existence and properties of global attractors of multi-valued impulsive dynamical systems. Main results are contained in Section 3 where impulsive parabolic inclusions are considered. The key moment is the presence of a small parameter in the right-hand part of the inclusion. This allows us to use the corresponding linear problem and, as a result, prove the theorem about existence of global attractor for the original nonlinear problem.

2.
Multi-valued impulsive dynamical systems. In this section we introduce a concept of impulsive multi-valued dynamical system (impulsive MDS for short). We also discuss a notion of global attractor for such systems.
Let (X, ρ) be a metric space, P (X) (β(X)) be a set of all nonempty ( nonempty bounded) subsets of X, for A, B ⊂ X dist X (A, B) := sup y∈A inf z∈B ρ(y, z).

Remark 1.
In Definition of MDS we assume no conditions of continuity for the map x → G(t, x). If the MDS G also has global attractor in the classical sense [26], i.e. if there exists a compact uniformly attracting set Θ 1 ⊂ X and ∀t ≥ 0 The following result provides a criterion of existence of global attractors for dissipative MDS.
then the following conditions are equivalent: 1) MDS G has a global attractor Θ; 2) MDS G is asymtotically compact, i.e., ∀t n ∞ ∀B ∈ β(X) ∀ξ n ∈ G(t n , B) the sequence {ξ n } is precompact. (2) Moreover, under condition (1) Now we will define impulsive MDS. To do this we introduce a set K of continuous maps ϕ : [0, +∞) → X satisfying the following properties: We denote It is easy to prove that G is an MDS [16]. An impulsive MDSG consists of the set K, a closed set M ⊂ X and a multivalued map I : M → P (X). The set M is called the impulsive set and the map I is called the impulsive map.
For constructing impulsive trajectories we use arguments with slight modifications from [11], [13]. We assume the following conditions hold: It follows from the continuity of ϕ and (5) that for every ϕ ∈ K if M + (ϕ) = ∅, then there exists s := ∆(ϕ) > 0 such that Now we are ready to construct an impulsive trajectoryφ with initial point Arguing inductively we obtain an impulsive trajectoryφ with finite or infinite number of impulsive points {x + n } n≥1 ⊂ X, the corresponding moments of time {s n } n≥0 ⊂ (0, +∞) and the functions {ϕ n } n≥0 ⊂ K. We shall denote it bỹ We also put Ifφ has infinite number of impulses, then ∀n ≥ 0 ∀t ∈ [t n , t n+1 ] ByK x denote the set of all impulsive trajectories with initial point x ∈ X. We assume that all impulsive trajectories are defined on [0, +∞), i.e., the following global existence condition holds: The condition (6) means that for every impulsive trajectory the number of its impulsive points is either no more than finite or We define impulsive MDS as a mapG : Note that in the single-valued case (7) defines a classical semigroup [3].
Let t ∈ [t n , t n+1 ), n ≥ 0, where t n is defined as previously. We consider two cases: t + s < t n+1 or t + s ≥ t n+1 .
3. Application to impulsive parabolic inclusions. In this section we consider impulsive problem consisting of differential inclusion with a small parameter in the right-hand part and multi-valued impulsive perturbation. We prove that for sufficiently small value of parameter this impulsive problem generates an impulsive MDS which has a global attractor. Consider a triplet V ⊂ H ⊂ V * of Hilbert spaces with compact and dense embedding. Let ·, · be a canonical duality between V and V * . Denote by · and (·, ·), respectively, the norm and scalar product in H. Let · V be a norm in V and Consider a linear continuous self-adjoint operator A : V → V * such that Then there exists a complete orthonormal in H family Assume that a multi-valued perturbation F : H → P (H) satisfies the following conditions: ∀y ∈ H F (y) is closed and convex in H, Now we are ready to formulate our impulsive problem. The first part of this problem is the following parabolic inclusion dy dt + Ay ∈ ε · F (y), t > 0.
We are interested in mild solutions of (13) in the sense of the following definition. The first component y of the pair {y, f } is also called solution of (13).
The second part of our problem is an impulsive perturbation, which is characterized by the following parameters: I : M → P (H), As an example we consider the following simple situation: we have a standard parabolic inclusion of reaction-diffusion type [16]. But the proposed scheme can also be useful in other situations [2], [8].
The main result of the paper is the following theorem. Proof. If we denote by K ε the set of solutions of (13) then properties K1),K2) are satisfied. Moreover, for sufficiently small ε > 0 the corresponding (non-impulsive) has a global attractor [12]. We are going to prove that this property remains true under impulsive perturbation (14), (15).
Using (8), (9),(11), we get Then for ε ∈ (0, β αc1 ) we have Every mild solution {y, f } also satisfies the following equality: ∀i ≥ 1 ∀t ≥ 0 From the definition of the set M and the map I we immediately obtain (4). To verify (5) we take an arbitrary y 0 ∈ M , an arbitrary mild solution {y, f } of (13) with y(0) = y 0 and consider the function Then from (18) for sufficiently small ε > 0 we need to prove the following: Let us consider a function α i e −λit (y 0 , ψ i ).
Secondly, choosing every time impulsive points with c i = a(1+µ) αip , i = 1, p it is easy to see that if y ∈ K ε , y(0) = y 0 intersects M at least at one point then there is an impulsive trajectoryỹ,ỹ(0) = y 0 which intersects M infinitely many times.
As a result, we have a nonempty class of impulsive trajectories which intersects M and we have a nonempty class of impulsive trajectories which intersects M at an infinite number of points.
Let us prove the dissipativity condition for impulsive MDS (24). If y ∈K ε y0 , y 0 ≤ R does not intersect M then from (17) If for some τ > 0 y(t) / ∈ M ∀t ∈ (0, τ ), y(τ ) ∈ M then from (22) follows Thus it is enough to prove the following property for sufficiently small ε ∈ (0, ε 1 ): Without lost of generality in all further arguments we assume that if y 0 ∈ IM then y ∈K ε y0 has an infinite number of impulsive points. So for given y ∈K ε y0 with y 0 ∈ IM , y 0 ≤ R from (23) there are {s i } ∞ i=0 such that y(·) has jumps at the moments {s 0 , s 0 + s 1 , . . .} with impulsive points

Using inequality
After k steps we obtain Using (23), from (28), (29) we get Finally, let us prove thatG ε is asymptotically compact. Let {y (n) 0 } be an arbitrary bounded sequence of initial data, y (n) 0 ≤ R, ξ n ∈G ε (t n , y (n) 0 ), t n +∞. Then ξ n = y n (t n ), where y n ∈K ε y (n) 0 . If y n does not intersect M then y n ∈ K ε and from K2) z n (·) = y n (· + t n − 1) also belongs to K ε and does not intersect M . So ξ n = y n (t n ) = z n (1), z n (0) = y n (t n − 1).
From (17) we obtain Therefore from well-known regularity results [12], [10] the sequence {ξ n = z n (1)} is precompact in H. If the function y n intersects M at the first time at a point τ n then from (22) sequence τ n is bounded and {y n (τ n − 0)} is also bounded in H. So from the inequality it will be enough to prove the precompactness of the sequence {ξ n } ⊂ H, where ξ n ∈G ε (t n , z n ), t n ∞, z n ∈ IM, z n ≤ R.
For the sequence y n from (17) and (30) we deduce that there exists c = c(R) > 0 such that ∀t ≥ 0 ∀n ≥ 1 y n (t) ≤ c(R).