一类非经典扩散方程周期解及周期吸引子的存在性

considered in recent years [3, 5, 6], etc. , in this paper, we mainly consider the existence of
periodic solutions and periodic uniform forward attractors.
+
t))
=
0
uniformly with respect to σ ∈ R.
we will call Ωµ( ) in axiom(2) of the above definition the uniform forward attraction region of . In case Ωµ( ) = X, we simply call the global uniform forward attractor of P . Moreover, if A (σ) is T-periodic in t, i.e. , A (σ + T ) = A (σ), then we call = {A (σ)}σ∈R is periodic uniform forward attractor of P , simply say periodic attractor.
Definition 2.3 A family of nonempty compact subsets uniform forward attractor of P , if
= {A (σ)}σ∈R is said to be a
(1) is invariant under P in the following sense: P (t, σ)A (σ) = A (σ + t), ∀t ∈ R+, σ ∈ R.
andV2, respectively,
((u, v)) = ∇u · ∇vdx,
Ω
∀u, v ∈ V1,
[u, v] = ∆u · ∆vdx,
Ω
∀u, v ∈ V2.
Let · s be the corresponding norm of Vs (s=1,2), it is well known that the norm
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The Existence of Periodic Solution and Periodic Attractors for A Class of Nonclassical Diffusion Equation
Zhao xiang
Tianjin Petroleum Vocational and Technical College Tianjin tuanbowa 301607
1 Introduction
This paper is concerned with the dynamical behavior of the following nonclassical diffusion equation :
ut − ∆ut − ∆u + g(u) = f (t + θ), u(t, x) = 0, f or x ∈ ∂Ω
Assume that the nonlinear function g satisfies the following conditions: g is C2 function from R1 to R1, and
(G1) there exists L > 0, such that g (s) ≥ −L, ∀s ∈ R;
This paper is organized as follows. In Section 2, we give some basic notions and basic definition. In Sections 3, we establish the existence of periodic solution. In Section 4, we establish the existence of periodic uniform forward attractor.
2
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sup HX∗ (P (t, σ)B, A) → 0, as t → ∞.
σ∈R
LetM be a subset of X, denote by Ωµ(M ) be the set {x ∈ X| M uniformly attracts a neighborhood of x under P }
a∈A
where d(a, B) = infb∈B d(a, b).
Definition 2.1 A mappingP : R+ × R × X → X, is said to be a process on X, if it satisfies the following axioms:
(1) P (0, σ, x) = x, ∀σ ∈ R, x ∈ X; (2) P (t + s, σ, x) = P (t, s + σ, P (s, σ, x)), ∀t, s ∈ R+, σ ∈ R, x ∈ X.
in Ω × R+
(1.1)
u(0, x) = u0(x), ∀ x ∈ Ω.
when Ω is an open bounded set of Rn with sufficiently regular boundary ∂Ω, and f ∈ L∞(R; L2(Ω)), f (t) is T-periodic, parameter θ ∈ R.
This equation is a special form of the nonclassical diffusion equation used in non-Newtonian
flow, soil mechanics and heat conduction theory, see [1, 2, 3, 4]. This types equation has been
s 0
g(r)dr,
Then lim inf
|s|→∞
G(s) s2
≥ 0;
(G4) there exists k2 > 0 such that
lim inf
|s|→∞
sg(s)−k2G(s) s2
≥ 0;
1
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Infer from (G3) and (G4) we know that: for any δ > 0, there exist positive constants Cδ, Cδ such that (G5) G(s) + δs2 ≥ −Cδ, ∀s ∈ R. (G6) sg(s) − k2G(s) + δs2 ≥ −Cδ, ∀s ∈ R.
Let X be metric space with metric d(·, ·), and denote by HX∗ (·, ·) and Hausdorff semidistance, which is defined for any subsets A, B of X by
HX∗ (A, B) = sup d(a, B)
equivalent to the usual one of Vs, which is
· s is
u V1 = u · u + ∇u · ∇udx,
Ω
∀u ∈ V1,
u V2 = u · u + ∇u · ∇u + ∆u · ∆udx,
Ω
∀u ∈ V2.
Some basic definitions about attractor in non-autonomous systems [7].
If Ωµ(M ) is a neighborhood of M with M being the smallest uniformly attracting set of P in Ωµ(M ), then we say that M is a (local) uniform attractor of P . In case Ωµ(M ) = X, we say M is a global uniform attractor of P .
E-mail: zhaoxiang1225@163.com
Abstract This paper is concerned with the dynamical behavior of a class of nonclassical diffusion equations ut − ∆ut − ∆u + g(u) = f (t) in case f (t) is Tperiodic in t. Firstly, the existence of periodic solutions is established by using the Galerkin approximate method and Brouwer’s fixed point theorem. Secondly , the existence of periodic uniform forward attractor is investigated. Keywords: Non-autonomous, Nonclassical, Periodic, Galerkin approximate.
2 Basic notions and basic definition
Let H = L2(Ω), V1 = H01(Ω), and V2 = H2(Ω) ∩ H01(Ω). Denote by (·, ·) and | · | the inner
product and norm of H, respectively. Denote by ((·, ·)) and [·, ·] the inner products in V1
Clearly Ωµ(M ) is an open subset of X. Ωµ(M ) is said to be the uniform attraction region of M.
Definition 2.2 Let M be a nonempty closed subset of X. M is said to be a (local)uniformly attracting set of P , if Ωµ(M ) is a neighborhood of M . If Ωµ(M ) = X, we then say that M is a global uniformly attracting set.
(G2) k1 > 0, such that |g(s)| ≤ k1(1 + |s|γ+1), ∀s ∈ R;
with 0 ≤ γ < ∞, when n = 1, 2, 0 ≤ γ < 2/n − 2, when n ≥ 3;
(G3) denote by G(s) the primitive of g(s), G(s) =
(2) Ωµ( ) = Ωµ(A)is a neighborhood of A, where A = ∪σ∈RA (σ).
(3) uniformly forward attracts each bounded subset B of Ωµ( ), i.e. ,
lim
t→∞
HX∗
(P
(t,
Leabharlann Baidu
σ)B,
A
(σ
Let P be a process on X, we introduce a two-parameter family of mappings {P (t, σ) : X → X|P (t, σ)x = P (t, σ, x), ∀x ∈ X, (t, σ) ∈ R+ × R}.
We also say it definitions a process, and denote P (t, σ) by Pσ(t) when necessary. Now suppose that we are given a process P = P (t, σ) on X. For two subsets A and B of X, we say that A uniformly attracts B under P , if