A Remark on the Global Attractors of the Nonlinear Evolution Equations

Chang Ya-Ya,Ma Qiao-Zhen

doi:10.1155/2013/372507

Abstract

We study the existence of global attractor of the nonlinear elastic rod oscillation equation when the forcing term belongs only to H−1(Ω); furthermore, we prove that the fractal dimension of global attractor is finite.

Highlights

Let Ω be an open bounded set of R3 with smooth boundary ∂Ω
We prove existence of global attractor and its fractal dimension for (1) under the condition that g(x) only satisfies the lower regularity
Let Ω ⊂ R3 be a bounded domain with smooth boundary, and one assumes that f satisfies (2) and (3), g ∈ H−1(Ω); the semigroup {S(t)}t≥0 possesses a global attractor A0 on H1

Summary

Introduction

Let Ω be an open bounded set of R3 with smooth boundary ∂Ω. We consider the following equation: utt − Δu − Δut − ωΔutt + f (u) = g (x) , (x, t) ∈ Ω × R+, (1)u |t=0 = u0, ut |t=0 = u1, ∀x ∈ Ω, u |∂Ω = 0, ∀t ≥ 0, where ω > 0 and g ∈ H−1(Ω). Let Ω be an open bounded set of R3 with smooth boundary ∂Ω. We prove existence of global attractor and its fractal dimension for (1) under the condition that g(x) only satisfies the lower regularity. Let A = −Δ and D(A) = H2(Ω) ∩ H01(Ω); we define D(A(s/2)); s ∈ R is Hilbert space family, and its inner product and norm are

Results

Conclusion