The Existence Of The Uniformly Attractor For An Impulsive Reaction-Diffusion Equation

In this paper,we consider the long time behavior of an impulsive reaction-diffusion equation at fixed moments of time. LetΩdenote a bounded domain in Rⁿ with the smooth boundary (?)ΩThe unknown function u = u(t,x) which determined by the following initial boundary value condition:whereα> 0,τ≥0, g∈L²(Ω) and f∈C¹{R, R).The mathematical setting of this equations is classical. Let H = L²(Ω), and let (·,·) and |·| denote the inner product and the norm in H respectively. We can correctly pose the following impulsive problem: Assume that, at fixed moment of time {t_i}_i≥1. The solution of (1) in the phases space H has impulse effects of the formu(t_i + 0) - u(t_i) =ψ_i((u(t_i)),i≥1 (2)whereψ_i > 0 is a linear mapping, the constants t₁ > 0 and t_i+1 - t_i≥γ> 0, (?)i≥1.Our aim in the paper is to prove the existence of the uniform attractor of an impulsive reaction-diffusion equation at fixed moments of time in the phases space H. In the absence of pulse influence (2), problem (1) is autonomous and generates a semigroup. Since the shift of solution in time is not a solution of (1)—(2) because the pulse are fixed, problem (1)—(2) dose not generate a semigroup, and it seems natural to relate it to a family of semiprocesses. Problem (1)—(2) is nonautonomous, and so we should use the theory of the global attractor of infinite-dimensional nonautonomous dynamical systems to prove the existence of the uniform attractors for impulsive reaction-diffusion equations at fixed moments of time.