Global Attractors for the Higher-Order Evolution Equation

We consider the following nonlinear evolution equation (1.1) ${\begin{matrix} u_{tt} + {(- Δ)}^{m} u + {(- Δ)}^{m} u_{t} + {(- Δ)}^{m} u_{tt} + g (x, u) = f (x), & (x, t) \in Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & x \in Ω, \\ \frac{\partial^{i} u (x, t)}{\partial^{i} v^{i}} = 0, i = 1, 2, \dots, m - 1, & (x, t) \in \partial Ω \times [0, \infty), \end{matrix}$ \left\{ {\begin{array}{*{20}{c}} {{u_{tt}} + {{( - \Delta )}^m}u + {{( - \Delta )}^m}{u_t} + {{( - \Delta )}^m}{u_{tt}} + g(x,u) = f(x),}&{(x,t) \in \Omega \times (0,\infty ),}\\ {u(x,0) = {u_0}(x),{u_t}(x,0) = {u_1}(x),}&{x \in \Omega ,}\\ {\frac{{{\partial ^i}u(x,t)}}{{{\partial ^i}{v^i}}} = 0,i = 1,2, \ldots ,m - 1,}&{(x,t) \in \partial \Omega \times [0,\infty ),} \end{array}} \right. where in a bounded domain Ω ⊂ Rⁿ with smooth boundary ∂Ω, the assumption on f, g, u₀ and u₁ will be made below.

When m = 1, the equation (1.1) is following form (1.2) $u_{tt} - Δ u - Δ u_{t} - Δ u_{tt} + g (x, u) = f (x) .$ {u_{tt}} - \Delta u - \Delta {u_t} - \Delta {u_{tt}} + g(x,u) = f(x). Chen and Wang [19] proved the existence of global attractor for the problem (1.2). Lately, Xie and Zhong in [8] studied the existence of global attractor of solution for the problem (1.1) with f = 0. Also, there are some authors studied the existence and nonexistence, asymptotic behavior of global solution for (1.2) (see [2, 3, 4, 5, 6, 7] for more details ). Nakao and Yang in [9] showed the global attractor of the Kirchhoff type wave equation.

In this paper, we improve our result by adopting and modifying the method of [19], we studied more general form of the equation.

This paper is organized as follows: In section 2, we give some assumptions and state the main results. In section 3, we prove the global existence of solution using the Faedo-Galerkin method. Also, we write some important estimates for the solution. In section 4, the existence of the global attractor is proved. In Section 5, the proof of decay property for solution is showed.

Preliminaries and main results

We write the Sobolev space H^k (Ω) = W^k,2 (Ω), $H_{0}^{k} (Ω) = W_{0}^{k, 2} (Ω)$ H_0^k(\Omega ) = W_0^{k,2}(\Omega ) . Furthermore, we show by (.,.) the inner product of L² (Ω), by ‖.‖_p the norm of L^p (Ω), p ≥ 1 and by ‖.‖_E the norm of any other Banach space E. As usual, we give u(t) instead of u(x,t), and u^′ (t) for u_t (t) and so on.

We write the following assumptions on f and g.

(A₁) Assume f (x) ∈ L² (Ω) and show F = ‖ f ‖₂;

(A₂) Suppose g(x,u) ∈ C¹ (Ω× R¹) and ∃k₁,k₂> 0, h₁ (x) ∈ L² (Ω), h₂ (x) ∈ L² (Ω) ∩ L^n/2 (Ω) such that (2.1) $g (x, u) u + h_{1} (x) | u | \geq k_{1} (G (x, u) + h_{1} (x) | u |) \geq 0, (x, u) \in Ω \times R^{1}$ g(x,u)u + {h_1}(x)|u| \ge {k_1}(G(x,u) + {h_1}(x)|u|) \ge 0,(x,u) \in \Omega \times {R^1} and the growth condition in u(2.2) $| g (x, u) | \leq k_{2} (| u |^{α} + h_{2} (x)), | g_{u} (x, u) | \leq k_{2} (| u |^{α - 1} + h_{2} (x)), (x, u) \in Ω \times R^{1}$ |g(x,u)| \le {k_2}(|u{|^{\rm{\alpha }}} + {h_2}(x)),|{g_u}(x,u)| \le {k_2}\left( {|u{|^{{\rm{\alpha }} - 1}} + {h_2}(x)} \right),(x,u) \in \Omega \times {R^1} with α ≥ 1, (n = 1,2), and $1 \leq α \leq \frac{n + 2}{n + 2}$ 1 \le {\rm{\alpha }} \le \frac{{n + 2}}{{n + 2}} , (n ≥ 3), $G (x, u) = \int_{0}^{u} g (x, s) ds$ G(x,u) = \int_0^u {g(x,s)ds} .

Later, we assume H₁ = ‖h₁‖₂, H₂ = max {‖h₂‖₂, ‖h₂‖_n/2}.

Clearly, the function g(x,u) = a(x)|u|^α−1u − b(x)|u|^β−1u(1 ≤ β < α) supplies (2.1) and (2.2) for some a(x), b(x).

Next, we show the definition and lemmas relating to the global attractor, (see [9, 11, 12]).

Definition 1

Suppose that E is Banach space and {S (t)}_t≥0a semigroup on E. A set A ⊂ E is said a (E,E)−global attractor if and only iff(1)

A is never changing (invariant), namely, S (t)A = A for whole t ≥ 0;

(2)

A is compact in E;

(3)

A is a bounded set in E and absorbs all bounded subset B in E relating with E topology, that is, for whichever bounded subset B ⊂ E, (2.3) ${dist}_{E} (S (t) B, A *) = \underset{y \in B^{x \in A *}}{sup inf} | | S (t) y - x | |_{E} \to 0 as t \to \infty .$ dis{t_E}(S(t)B,A*) = \mathop {\sup \inf }\limits_{y \in {B^{x \in A*}}} ||S(t)y - x|{|_E} \to 0\,as\,t \to \infty .

Lemma 2

Assume E is Banach space and {S (t)}_{t ≥0}is a semigroup of continuous operators on E. Then, there exists (E,E)−global attractor A if the following conditions are supplied:(1)

There exists a bounded absorbing set B₀in E, that is, for whichever bounded subset B ⊂ E, there is a T = T (B) such that S (t)B ⊂ B₀for any t ≥ T.

(2)

{S (t)}_t≥0as asymptotically compact in E, that is, for any bounded sequence {y_n} in E and t_n→ ∞ as n → ∞, ${S (t_{n}) y_{n}}_{n = 1}^{\infty}$ \left\{ {S\left( {t_n } \right)y_n } \right\}_{n = 1}^\inftyhas a convergent subsequence relating to E topology.

We show the basic results now.

Theorem 3

Suppose (A₁)–(A₂) satisfy and (u₀,u₁) ∈ X. Then, the problem (1.1) admits a unique weak solution u(t) in the class(2.4) $C^{1} ([0, \infty); H_{0}^{m} \cap C ([0, \infty); H^{2 m} \cap H_{0}^{m}) \cap W^{2, \infty} ([0, \infty); H_{0}^{m} \cap W^{1, \infty} (([0, \infty); H^{2 m})$ {C^1}([0,\infty );H_0^m \cap C([0,\infty );{H^{2m}} \cap H_0^m) \cap {W^{2,\infty }}([0,\infty );H_0^m \cap {W^{1,\infty }}(([0,\infty );{H^{2m}})holds.(2.5) ${‖ P^{\frac{1}{2}} u (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2} \leq C_{1} e^{- λ_{1} t} + C_{2}, t \geq 0$ \left\| {{P^{\frac{1}{2}}}u(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}(t)} \right\|_2^2 \le {C_1}{e^{ - {\lambda _1}t}} + {C_2},t \ge 0(2.6) ${‖ u_{tt} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{tt} (t) ‖}_{2}^{2} \leq C_{3} e^{- λ_{2} t} + C_{4}, t \geq 0$ \left\| {{u_{tt}}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_{tt}}(t)} \right\|_2^2 \le {C_3}{e^{ - {\lambda _2}t}} + {C_4},t \ge 0and(2.7) ${‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2} + {‖ Pu (t) ‖}_{2}^{2} + {‖ {Pu}_{t} (t) ‖}_{2}^{2} \leq C_{5} e^{- λ_{3} t} + C_{4}, t \geq 0$ \left\| {{P^{\frac{1}{2}}}{u_t}(t)} \right\|_2^2 + \left\| {Pu(t)} \right\|_2^2 + \left\| {P{u_t}(t)} \right\|_2^2 \le {C_5}{e^{ - {\lambda _3}t}} + {C_4},t \ge 0with some λ₁, λ₂,λ₃> 0. In this theorem $C_{1} = C_{1} ({‖ P^{\frac{1}{2}} u_{0} ‖}_{2}, {‖ P^{\frac{1}{2}} u_{1} ‖}_{2})$ {C_1} = {C_1}\left( {{{\left\| {{P^{\frac{1}{2}}}{u_0}} \right\|}_2},{{\left\| {{P^{\frac{1}{2}}}{u_1}} \right\|}_2}} \right) , C₂ = C₂(F,H₁), $C_{3} = C_{3} ({‖ P^{\frac{1}{2}} u_{0} ‖}_{2}, {‖ P^{\frac{1}{2}} u_{1} ‖}_{2}, F, H_{1}, H_{2})$ {C_3} = {C_3}\left( {{{\left\| {{P^{\frac{1}{2}}}{u_0}} \right\|}_2},{{\left\| {{P^{\frac{1}{2}}}{u_1}} \right\|}_2},F,{H_1},{H_2}} \right), C₄ = C₄ (F,H₁,H₂),C₅ = C₅ (‖Pu₀‖₂ ,‖Pu₁‖₂ ,F,H₁,H₂).

Show the solution in Theorem 1 by S (t)(u₀,u₁) = (u(t),u_t (t)). We are now in a position to prove some continuity of S (t) relating to the initial data (u₀,u₁), which will be needed for the proof of the existence of global attractor.

Theorem 4

Suppose whole conditions in Theorem 3. Assume S (t)(u_0k,u_1k) and S (t)(u₀,u₁) are the solutions of the problem (1.1) with the initial data (u_0k,u_1k) and (u₀,u₁). If (u_0k,u_1k) → (u₀,u₁) in X as k → ∞, then S (t)(u_0k,u_1k) → S (t)(u₀,u₁) in X as k → ∞.

Theorem 4 denotes that the semigroup S (t) : X → X is continuous on X.

Theorem 5

Assume every assumptions in Theorem 3 be provided. Then, the semigroup {S (t)}_t≥0related with the solution of the problem (1.1) accepts a (X,X)−global attractor A.

For the decay property of solution u(t) for the problem (1.1), we get

Theorem 6

Suppose u is a weak solution in Theorem 3 with f = 0 and g(x,u) = g(u). Besides, suppose 0 ≤ 2G(u) ≤ ug(u). Then, for whichever q > 0, there is $C_{1} = C_{1} ({‖ P^{\frac{1}{2}} u_{0} ‖}_{2}, {‖ P^{\frac{1}{2}} u_{1} ‖}_{2})$ {C_1} = {C_1}\left( {{{\left\| {{P^{\frac{1}{2}}}{u_0}} \right\|}_2},{{\left\| {{P^{\frac{1}{2}}}{u_1}} \right\|}_2}} \right)such that(2.8) $E (t) = \frac{1}{2} ({‖ u (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2}) + \int_{Ω} G (u (t)) dx \leq C_{1} {(1 + t)}^{- 1 / q} .$ E(t) = \frac{1}{2}\left( {\left\| {u(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}(t)} \right\|_2^2} \right) + \int_\Omega {G(} u(t))dx \le {C_1}{(1 + t)^{ - 1/q}}.

The Proof of Theorem 3

In this section, we suppose that all assumptions in Theorem 3 are supplied. Firstly, we establish the global existence of a solution to problem (1.1) with Fadeo-Galerkin method as in [16, 17].

Assume ω_j (x) (j = 1,2,...) is the complete set of properly normalized eigenfunctions for the operator (−Δ)^m in $H_{0}^{m} (Ω)$ H_0^m(\Omega ) . Then, the family {ω₁,ω₂...,ω_k,...} holds an orthogonal basis for both $H_{0}^{m} (Ω)$ H_0^m(\Omega ) and L² (Ω), see [16, 17]. For each positive integer k, show V_k = span{ω₁,ω₂...,ω_k,...}. We search for an approximation solution u_k (t) to the problem (1.1) in the form $u_{k} (t) = \sum_{j = 1}^{k} d_{jk} (t) ω_{j}$ {u_k}(t) = \sum\limits_{j = 1}^k {{d_{jk}}(t){\omega _j}} where djk (t) are the solution of the nonlinear ordinary differential equation (ODE) system in the variant t: (3.1) $(u_{k}^{″}, ω_{j}) - ({Pu}_{k}, ω_{j}) - (P u_{k}^{'}, ω_{j}) - (P u_{k}^{″}, ω_{j}) + (g, ω_{j}) = (f, ω_{j}), j = 1, 2, \dots, k,$ (u_k^{\prime\prime},{\omega _j}) - (P{u_k},{\omega _j}) - (Pu_k^\prime,{\omega _j}) - (Pu_k^{\prime\prime},{\omega _j}) + (g,{\omega _j}) = (f,{\omega _j}),j = 1,2, \ldots ,k, with the initial conditions (3.2) $d_{jk} (0) = (u_{0 k}, ω_{j}), d_{jk}^{'} (0) = (u_{1 k}, ω_{j})$ {d_{jk}}(0) = ({u_{0k}},{\omega _j}),d_{jk}^\prime(0) = ({u_{1k}},{\omega _j}) where u_0k and u_1k are chosen in V_k so that (3.3) $u_{0 k} \to u_{0}, u_{1 k} \to u_{1} in H^{2 m} (Ω) \cap H_{0}^{m} (Ω) as k \to \infty .$ {u_{0k}} \to {u_0},{u_{1k}} \to {u_1}\,{\rm{in}}\,{H^{2m}}(\Omega ) \cap H_0^m(\Omega )\,{\rm{as}}\,k\, \to \infty . Here (.,.) shows the inner product in L² (Ω). Then, Sobolev imbedding theorem means that ∃c₀> 0, such that (3.4) ${‖ u_{k} (0) ‖}_{H_{0}^{m}}^{2} \leq c_{0} {‖ P^{\frac{1}{2}} u_{0} ‖}_{2}^{2}, {‖ u_{k}^{'} (0) ‖}_{H_{0}^{m}}^{2} \leq c_{0} {‖ P^{\frac{1}{2}} u_{1} ‖}_{2}^{2} \forall k = 1, 2, \dots,$ \left\| {{u_k}(0)} \right\|_{H_0^m}^2 \le {c_0}\left\| {{P^{\frac{1}{2}}}{u_0}} \right\|_2^2,\left\| {u_k^\prime(0)} \right\|_{H_0^m}^2 \le {c_0}\left\| {{P^{\frac{1}{2}}}{u_1}} \right\|_2^2\forall k = 1,2, \ldots , and (3.1) shows that for any v ∈ V_k, (3.5) $(u_{k}^{″}, v) - ({Pu}_{k}, v) - (P u_{k}^{'}, v) - (P u_{k}^{″}, v) + (g, v) = (f, v), \forall v \in V_{k} .$ \left( {u_k^{\prime\prime},v} \right) - \left( {P{u_k},v} \right) - \left( {Pu_k^\prime,v} \right) - \left( {Pu_k^{\prime\prime},v} \right) + (g,v) = (f,v),\forall v \in {V_k}. We know, the system (3.1) and (3.2) accept a unique solution u_k (t) on the interval [0,T ] for any T > 0. Such a solution can be expanded to the overall interval [0,∞). We show by C_i (i = 1,2,...) the constants that are independent of k and t ≥ 0, by C₀ the constant depending on k₁, k₂ in (A₂) and Sobolev imbedding constant c₀ in (3.4). These constants may be different from line to line.

Multiplying (3.1) by d^′_jk (t) and summing the resulting equations over j, we obtain (3.6) $E_{1}^{'} (t) + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} = 0, \forall t \geq 0$ E_1^\prime(t) + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2 = 0,\forall t \ge 0 where (3.7) $E_{1} (t) = \frac{1}{2} ({‖ u_{k}^{'} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2}) + \int_{Ω} G (x, u_{k} (t)) dx - \int_{Ω} f (x) u_{k} (t) dx .$ {E_1}(t) = \frac{1}{2}\left( {\left\| {u_k^\prime(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2} \right) + \int_\Omega {G(x,{u_k}(t))dx - } \int_\Omega {f(x){u_k}(t)dx.} Also, multiplying (3.1) by djk (t), we get (3.8) $E_{2}^{1} (t) + {‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2} + \int_{Ω} g (x, u_{k}) u_{k} (t) dx = {‖ u_{k}^{'} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} + \int_{Ω} f (x) u_{k} (t) dx$ E_2^1(t) + \left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^2 + \int_\Omega {g(x,{u_k}){u_k}(t)dx = \left\| {u_k^\prime(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2 + } \int_\Omega {f(x){u_k}(t)dx} where (3.9) $E_{2} (t) = \frac{1}{2} {‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2} + \int_{Ω} u_{k} (t) u_{k}^{1} (t) dx + \int_{Ω} P^{\frac{1}{2}} u_{k} (t) P^{\frac{1}{2}} u_{k}^{'} (t) dx .$ {E_2}(t) = \frac{1}{2}\left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^2 + \int_\Omega {{u_k}(t)u_k^1(t)dx + } \int_\Omega {{P^{\frac{1}{2}}}{u_k}(t){P^{\frac{1}{2}}}u_k^\prime(t)dx.} If we take sufficient large k₁> 0 and use the assumption (A₂), we get (3.10) $ψ_{k}^{'} (t) + λ_{1} ψ_{k} (t) \leq C_{0} (F^{2} + H_{1}^{2 m}), ψ_{k} (t) = k_{1} E_{1} (t) + E_{2} (t)$ \psi _k^\prime(t) + {\lambda _1}{\psi _k}(t) \le {C_0}({F^2} + H_1^{2m}),{\psi _k}(t) = {k_1}{E_1}(t) + {E_2}(t) with some positive λ 1, relating to the indicated constants in (A₂).

We note that (3.11) $ψ_{k} (t) \leq c_{0} (k_{1} + c_{0}) ({‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2}) + k_{1}^{2} (F^{2} + H_{1}^{2 m}) + k_{1} \int_{Ω} (G + h_{1} | u_{k} (t) |) dx$ {\psi _k}(t) \le {c_0}({k_1} + {c_0})\left( {\left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^2} \right) + k_1^2\left( {{F^2} + H_1^{2m}} \right) + {k_1}\int_\Omega {\left( {G + {h_1}\left| {{u_k}(t)} \right|} \right)dx} and (3.12) $2 ψ_{k} (t) \geq (k_{1} - 1) ({‖ u_{k}^{'} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{1} (t) ‖}_{2}^{2}) + (k_{1} - 5 c_{0}) {‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2} + k_{1} \int_{Ω} (G + h_{1} | u_{k} (t) |) dx - k_{1}^{2} (F^{2} + H_{1}^{2 m})$ 2{\psi _k}(t) \ge ({k_1} - 1)\left( {\left\| {u_k^\prime(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^1(t)} \right\|_2^2} \right) + \left( {{k_1} - 5{c_0}} \right)\left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^2 + {k_1}\int_\Omega {\left( {G + {h_1}\left| {{u_k}(t)} \right|} \right)dx} - k_1^2\left( {{F^2} + H_1^{2m}} \right) with k₁≥ max{3,2 + 5c₀}.

The application of Gronwall lemma to (3.10) holds ${‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} + \int_{Ω} G (x, u_{k} (t)) + h_{1} (x) | u_{k} (t) |) dx \leq C_{1} e^{- λ_{1} t} + C_{2}, t \geq 0$ \left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2 + \int_\Omega {G(x,{u_k}(t)) + {h_1}(x)\left| {{u_k}(t)} \right|)dx \le } {C_1}{e^{ - {\lambda _1}t}} + {C_2},t \ge 0 which shows (3.13) ${‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} \leq C_{1} e^{- λ_{1} t} + C_{2}, t \geq 0$ \left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2 \le {C_1}{e^{ - {\lambda _1}t}} + {C_2},t \ge 0 Where $C_{1} = C_{1} ({‖ P^{\frac{1}{2}} u_{0} ‖}_{2}, {‖ P^{\frac{1}{2}} u_{1} ‖}_{2})$ {C_1} = {C_1}\left( {{{\left\| {{P^{\frac{1}{2}}}{u_0}} \right\|}_2},{{\left\| {{P^{\frac{1}{2}}}{u_1}} \right\|}_2}} \right) , C₂ = C₂ (F,H₁).

Also, we differentiate (3.1) with respect to t and get (3.14) $(u_{k}^{″'}, ω_{j}) - (P u_{k}^{'}, ω_{j}) - (P u_{k}^{″}, ω_{j}) - (P u_{k}^{″'}, ω_{j}) + (g_{u} u_{k}^{'}, ω_{j}) = 0, j = 1, 2, \dots, k .$ \left( {u_k^{\prime\prime\prime},{\omega _j}} \right) - \left( {Pu_k^\prime,{\omega _j}} \right) - \left( {Pu_k^{\prime\prime},{\omega _j}} \right) - \left( {Pu_k^{\prime\prime\prime},{\omega _j}} \right) + \left( {{g_u}u_k^\prime,{\omega _j}} \right) = 0,j = 1,2, \ldots ,k. Multiplying (3.14) by $d_{jk}^{″} (t)$ d_{jk}^{\prime\prime}(t) and summing the resulting equations over j, we obtain (3.15) $E_{3}^{'} (t) + {‖ P^{\frac{1}{2}} u_{k}^{″} (t) ‖}_{2}^{2} + \int_{Ω} g_{u} u_{k}^{'} u_{k}^{″} d x = 0$ E_3^\prime(t) + \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(t)} \right\|_2^2 + \int_\Omega {{g_u}u_k^\primeu_k^{\prime\prime}dx = 0} with (3.16) $E_{3} (t) = \frac{1}{2} ({‖ u_{k}^{″} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{″} (t) ‖}_{2}^{2}) \leq C_{0} {({‖ P^{\frac{1}{2}} u_{k}^{″} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2})}_{2}, t \geq 0$ {E_3}(t) = \frac{1}{2}\left( {\left\| {u_k^{\prime\prime}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(t)} \right\|_2^2} \right) \le {C_0}{\left( {\left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2} \right)_2},t \ge 0 in which the Sobolev embedding theorem has been used.

Furthermore, the growth condition (2.2) and the Hölder inequality mean that $\int_{Ω} | g_{u} u_{k}^{'} u_{k}^{″} | dx \leq k_{2} \int_{Ω} (| h_{2} | | u_{k}^{'} | + {| u_{k} |}^{α - 1} | u_{k}^{'} | | u_{k}^{″} |) dx \leq C_{0} ({‖ h_{2} ‖}_{n / 2} + {‖ P^{\frac{1}{2}} u_{m} ‖}_{2}^{(α - 1)}) {‖ P^{\frac{1}{2}} u_{k}^{'} ‖}_{2} {‖ P^{\frac{1}{2}} u_{k}^{″} ‖}_{2} .$ \int_\Omega {\left| {{g_u}u_k^\primeu_k^{\prime\prime}} \right|dx \le } {k_2}\int_\Omega {\left( {\left| {{h_2}} \right|\left| {u_k^\prime} \right| + {{\left| {{u_k}} \right|}^{\alpha - 1}}\left| {u_k^\prime} \right|\left| {u_k^{\prime\prime}} \right|} \right)dx \le {C_0}} \left( {{{\left\| {{h_2}} \right\|}_{n/2}} + \left\| {{P^{\frac{1}{2}}}{u_m}} \right\|_2^{({\rm{\alpha }} - 1)}} \right){\left\| {{P^{\frac{1}{2}}}u_k^\prime} \right\|_2}{\left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}} \right\|_2}. Therefore, we get (3.17) $\int_{Ω} | g_{u} u_{k}^{'} u_{k}^{″} | dx \leq \frac{1}{2} {‖ P^{\frac{1}{2}} u_{k}^{″} (t) ‖}_{2}^{2} + C_{0} {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} ({‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2 (α - 1)} + H_{2}^{2 m})$ \int_\Omega {\left| {{g_u}u_k^\primeu_k^{\prime\prime}} \right|dx \le } \frac{1}{2}\left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(t)} \right\|_2^2 + {C_0}\left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2\left( {\left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^{2({\rm{\alpha }} - 1)} + H_2^{2m}} \right) and (3.18) $E_{3}^{'} (t) + \frac{1}{2} {‖ P^{\frac{1}{2}} u_{k}^{″} (t) ‖}_{2}^{2} \leq C_{0} {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} ({‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2 (α - 1)} + H_{2}^{2 m}) .$ E_3^\prime(t) + \frac{1}{2}\left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(t)} \right\|_2^2 \le {C_0}\left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2\left( {\left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^{2({\rm{\alpha }} - 1)} + H_2^{2m}} \right). Then, the applications of the estimates (3.13) and (3.15)–(3.18) give that ∃λ 1 ≥ λ 2 > 0, depending on C₀, such that (3.19) $E_{3}^{'} (t) + λ_{2} E_{3} (t) \leq C_{0} {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} (1 + {‖ P^{\frac{1}{2}} u_{k} (t) ‖}_{2}^{2 (α - 1)} + H_{2}^{2 m}) \leq C_{3} e^{- λ_{1} t} + C_{4} .$ E_3^\prime(t) + {\lambda _2}{E_3}(t) \le {C_0}\left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2\left( {1 + \left\| {{P^{\frac{1}{2}}}{u_k}(t)} \right\|_2^{2({\rm{\alpha }} - 1)} + H_2^{2m}} \right) \le {C_3}{e^{ - {\lambda _1}t}} + {C_4}. Here, assume $C_{3} = C_{3} ({‖ P^{\frac{1}{2}} u_{0} ‖}_{2}, {‖ P^{\frac{1}{2}} u_{1} ‖}_{2}, F, H_{1}, H_{2})$ {C_3} = {C_3}\left( {{{\left\| {{P^{\frac{1}{2}}}{u_0}} \right\|}_2},{{\left\| {{P^{\frac{1}{2}}}{u_1}} \right\|}_2},F,{H_1},{H_2}} \right) , C₄ = C₄ (,F,H₁,H₂). Then (3.19) means that (3.20) $E_{2} (t) \leq E_{3} (0) e^{- λ_{2} t} + C_{3} e^{- λ_{2} t} + λ_{2}^{- 1} c_{4}, t \geq 0 .$ {E_2}(t) \le {E_3}(0){e^{ - {\lambda _2}t}} + {C_3}{e^{ - {\lambda _2}t}} + \lambda _2^{ - 1}{c_4},t \ge 0. We show that E₃ (0) is uniformly bounded for k under the conditions in Theorem 3 now. It follows by (3.1) that (3.21) $(u_{k}^{″} (t) - {Pu}_{k} (t) - P u_{k}^{'} (t) - P u_{k}^{″} (t), u_{k}^{″} (t)) = (f, u_{k}^{'} (t)) - (g, u_{k}^{″} (t)) .$ \left( {u_k^{\prime\prime}(t) - P{u_k}(t) - Pu_k^\prime(t) - Pu_k^{\prime\prime}(t),u_k^{\prime\prime}(t)} \right) = \left( {f,u_k^\prime(t)} \right) - \left( {g,u_k^{\prime\prime}(t)} \right). Especially, suppose t = 0, we get (3.22) $\begin{array}{l} {‖ u_{k}^{″} (0) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{″} (0) ‖}_{2}^{2} + \int_{Ω} P^{\frac{1}{2}} u_{k}^{″} (0) . (P^{\frac{1}{2}} u_{k} (0) + P^{\frac{1}{2}} u_{k}^{'} (0)) dx \\ = \int_{Ω} f (x) u_{k}^{″} (0) dx - \int_{Ω} (g, u_{k} (0)) u_{k}^{″} (0) dx . \end{array}$ \begin{array}{*{20}{l}} {\left\| {u_k^{\prime\prime}(0)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0)} \right\|_2^2 + \int_\Omega {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0).\left( {{P^{\frac{1}{2}}}{u_k}(0) + {P^{\frac{1}{2}}}u_k^\prime(0)} \right)} dx}\\ { = \int_\Omega {f(x)u_k^{\prime\prime}(0)dx - \int_\Omega {\left( {g,{u_k}(0)} \right)} u_k^{\prime\prime}(0)dx.} } \end{array} By Young inequality with ɛ, (3.23) $\begin{matrix} \int_{Ω} | P^{\frac{1}{2}} u_{k}^{″} (0) . P^{\frac{1}{2}} u_{k} (0) | dx \leq ε {‖ P^{\frac{1}{2}} u_{k}^{″} (0) ‖}_{2}^{2} + C_{ε} {‖ P^{\frac{1}{2}} u_{k} (0) ‖}_{2}^{2}, \\ \int_{Ω} | P^{\frac{1}{2}} u_{k}^{″} (0) . P^{\frac{1}{2}} u_{k}^{'} (0) | dx \leq ε {‖ P^{\frac{1}{2}} u_{k}^{″} (0) ‖}_{2}^{2} + C_{ε} {‖ P^{\frac{1}{2}} u_{k}^{'} (0) ‖}_{2}^{2}, \\ \int_{Ω} | g (x, u_{k} (0)) u_{k}^{″} (0) dx | \leq {‖ u_{k}^{″} (0) ‖}_{\frac{2 n}{n - 2}} {‖ g ‖}_{μ_{1}} \leq ε {‖ P^{\frac{1}{2}} u_{k}^{″} (0) ‖}_{2}^{2} + C_{ε} {‖ g ‖}_{μ_{1}}^{2}, \end{matrix}$ \begin{array}{*{20}{c}} {\int_\Omega {\left| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0).{P^{\frac{1}{2}}}{u_k}(0)} \right|dx \le \varepsilon \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0)} \right\|_2^2 + {C_\varepsilon }\left\| {{P^{\frac{1}{2}}}{u_k}(0)} \right\|_2^2,} }\\ {\int_\Omega {\left| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0).{P^{\frac{1}{2}}}u_k^\prime(0)} \right|dx \le \varepsilon \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0)} \right\|_2^2 + {C_\varepsilon }\left\| {{P^{\frac{1}{2}}}u_k^\prime(0)} \right\|_2^2,} }\\ {\int_\Omega {\left| {g\left( {x,{u_k}(0)} \right)u_k^{\prime\prime}(0)dx} \right| \le {{\left\| {u_k^{\prime\prime}(0)} \right\|}_{\frac{{2n}}{{n - 2}}}}{{\left\| g \right\|}_{{\mu _1}}} \le \varepsilon \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0)} \right\|_2^2 + {C_\varepsilon }\left\| g \right\|_{{\mu _1}}^2,} } \end{array} and $\int_{Ω} | f (x) u_{k}^{″} (0) | dx \leq ε {‖ P^{\frac{1}{2}} u_{k}^{″} (0) ‖}_{2}^{2} + C_{ε} {‖ f ‖}_{2}^{2}$ \int_\Omega {\left| {f(x)u_k^{\prime\prime}(0)} \right|dx \le } \varepsilon \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0)} \right\|_2^2 + {C_\varepsilon }\left\| f \right\|_2^2 with μ₁ = 2n/(n + 2). Since μ₁α = 2nα/(n + 2) ≤ 2n/(n − 2), we obtain by (2.2) that (3.24) $\int_{Ω} {| g |}^{μ_{1}} dx \leq C_{0} \int_{Ω} ({| u_{k} (0) |}^{μ_{1} α} + {| h_{2} |}^{μ_{1}}) dx \leq C_{0} ({‖ P^{\frac{1}{2}} u_{0} ‖}_{2}^{μ_{1} α} + {‖ h_{2} ‖}_{2}^{μ_{1}}) .$ \int_\Omega {{{\left| g \right|}^{{\mu _1}}}dx \le {C_0}} \int_\Omega {\left( {{{\left| {{u_k}(0)} \right|}^{{\mu _1}{\rm{\alpha }}}} + {{\left| {{h_2}} \right|}^{{\mu _1}}}} \right)dx \le {C_0}\left( {\left\| {{P^{\frac{1}{2}}}{u_0}} \right\|_2^{{\mu _1}{\rm{\alpha }}} + \left\| {{h_2}} \right\|_2^{{\mu _1}}} \right).} Suppose 0 < ɛ ≤ 1/6. Then, from (3.22) to (3.24) that (3.25) $\begin{array}{l} E_{3} (0) & \leq {‖ P^{\frac{1}{2}} u_{k}^{″} (0) ‖}_{2}^{2} + {‖ u_{k}^{″} (0) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (0) ‖}_{2}^{2} \\ \leq C_{0} ({‖ P^{\frac{1}{2}} u_{k}^{'} (0) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k} (0) ‖}_{2}^{2} + F^{2} + {‖ g ‖}_{μ_{1}}^{2}) \\ \leq C_{0} ({‖ P^{\frac{1}{2}} u_{1} ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{0} ‖}_{2}^{2} + F^{2} + {‖ P^{\frac{1}{2}} u_{0} ‖}_{2}^{2 α} + {‖ h_{2} ‖}_{2}^{2}) \equiv C_{3} . \end{array}$ \begin{array}{*{20}{l}} {{E_3}(0)}&{ \le \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(0)} \right\|_2^2 + \left\| {u_k^{\prime\prime}(0)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(0)} \right\|_2^2}\\ {}&{ \le {C_0}\left( {\left\| {{P^{\frac{1}{2}}}u_k^\prime(0)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_k}(0)} \right\|_2^2 + {F^2} + \left\| g \right\|_{{\mu _1}}^2} \right)}\\ {}&{ \le {C_0}\left( {\left\| {{P^{\frac{1}{2}}}{u_1}} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_0}} \right\|_2^2 + {F^2} + \left\| {{P^{\frac{1}{2}}}{u_0}} \right\|_2^{2{\rm{\alpha }}} + \left\| {{h_2}} \right\|_2^2} \right) \equiv {C_3}.} \end{array} Therefore, the inequality (3.20) shows (3.26) ${‖ P^{\frac{1}{2}} u_{k}^{″} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{'} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{k}^{″} (t) ‖}_{2}^{2} \leq C_{3} e^{- λ_{2} t} + λ_{2}^{- 1} C_{4}, t \geq 0$ \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^\prime(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u_k^{\prime\prime}(t)} \right\|_2^2 \le {C_3}{e^{ - {\lambda _2}t}} + \lambda _2^{ - 1}{C_4},t \ge 0 and the estimates (3.13) and (3.26) give that (3.27) ${\begin{array}{l} {u_{k} (t)} is bounded in L^{\infty} ([0, \infty); H_{0}^{m} (Ω)), \\ {u_{k}^{'} (t)} is bounded in L^{\infty} ([0, \infty); H_{0}^{m} (Ω)), \\ {u_{k}^{″} (t)} is bounded in L^{\infty} ([0, \infty); H_{0}^{m} (Ω)) . \end{array}$ \left\{ {\begin{array}{*{20}{l}} {\left\{ {{u_k}(t)} \right\}{\rm{is}}\,{\rm{bounded}}\,{\rm{in}}\,{L^\infty }([0,\infty );H_0^m(\Omega )),}\\ {\left\{ {u_k^\prime(t)} \right\}{\rm{is}}\,{\rm{bounded}}\,{\rm{in}}\,{L^\infty }([0,\infty );H_0^m(\Omega )),}\\ {\left\{ {u_k^{\prime\prime}(t)} \right\}{\rm{is}}\,{\rm{bounded}}\,{\rm{in}}\,{L^\infty }([0,\infty );H_0^m(\Omega )).} \end{array}} \right. So, there exists a subsequences in {u_k} (still showed by {u_k}) such that (3.28) ${\begin{array}{l} u_{k} \to u weakly star in L^{\infty} ([0, \infty); H_{0}^{m} (Ω)), \\ u_{k}^{'} \to u^{'} weakly star in L^{\infty} ([0, \infty); L^{2} (Ω)), \\ u_{k}^{″} \to u^{″} weakly star in L^{2} ([0, \infty); H_{0}^{m} (Ω)), . \end{array}$ \left\{ {\begin{array}{*{20}{l}} {{u_k} \to u\,{\rm{weakly}}\,{\rm{star}}\,{\rm{in}}\,{L^\infty }([0,\infty );H_0^m(\Omega )),}\\ {u_k^\prime \to {u^\prime}\,{\rm{weakly}}\,{\rm{star}}\,{\rm{in}}\,{L^\infty }([0,\infty );{L^2}(\Omega )),}\\ {u_k^{\prime\prime} \to {u^{\prime\prime}}\,{\rm{weakly}}\,{\rm{star}}\,{\rm{in}}\,{L^2}([0,\infty );H_0^m(\Omega )),.} \end{array}} \right. From applying the fact that L^∞ ([0,∞); $H_{0}^{m} (Ω)) ↪ L^{2} (| 0, \infty)$ \left. {H_0^m\left( \Omega \right)} \right) \hookrightarrow {L^2}\left( {|0,\infty } \right)\ ; $H_{0}^{m} (Ω))$ \left. {H_0^m \left( \Omega \right)} \right) ) and the Lions-Aubin compactness Lemma in [20], we obtain from (3.27) and (3.28) that (3.29) $u_{k} \to u, u_{k}^{'} \to strongly in L^{2} ([0, \infty); L^{2} (Ω))$ {u_k} \to u,u_k^\prime \to \user1{strongly}\,\user1{in}\,{L^2}\left( {[0,\infty );{L^2}\left( \Omega \right)} \right) and then u_k→ u a.e in Ω× [0,∞).

Using the growth condition (2.2), for any T > 0, the integral $\int_{0}^{T} \int_{Ω} {| g (x, u_{k}) (x, t) |}^{\frac{α + 1}{α}} dxdt$ \int_0^T {\int_\Omega {{{\left| {g\left( {x,{u_k}} \right)\left( {x,t} \right)} \right|}^{\frac{{\alpha + 1}}{\alpha }}}dxdt} } is bounded. Accordingly, by Lemma 2 in Chap. 1 [17], we conclude (3.30) $g (x, u_{k}) \to g (x, u) weakly in L^{\frac{α + 1}{α}} ([0, T]; L^{\frac{α + 1}{α}} (Ω))$ g\left( {x,{u_k}} \right) \to g\left( {x,u} \right){\rm{weakly}}\,{\rm{in}}\,{L^{\frac{{\alpha + 1}}{\alpha }}}\left( {\left[ {0,T} \right];{L^{\frac{{\alpha + 1}}{\alpha }}}\left( \Omega \right)} \right) with these convergences, by using the limit in the approximate equation (3.5), we get (3.31) $(u^{″} (t), v) - (Pu, v) - ({Pu}^{'}, v) - ({Pu}^{″}, v) + (g (x, u), v) = f (f, v), \forall v \in H_{0}^{m} (Ω),$ \left( {u\prime\prime\left( t \right),v} \right) - \left( {Pu,v} \right) - \left( {Pu\prime,v} \right) - \left( {Pu\prime\prime,v} \right) + \left( {g\left( {x,u} \right),v} \right) = f\left( {f,v} \right),\forall v \in H_0^m\left( \Omega \right), So, u(t) is a weak solution of (1.1) and supplies (2.5) and (2.6), and the proof of existence for the solution u(t) of (1.1) is completed.

We derive the estimates for ‖Pu(t)‖₂ and ‖Pu_t (t)‖₂ now. Also, we write u instead of u_k for convenience and view the estimates for u as a limit of u_k. Supposing v = −Pu in (3.31), we obtain (3.32) $E_{4}^{'} (t) + {‖ Pu (t) ‖}_{2}^{2} \leq {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2} + {‖ {Pu}_{t} (t) ‖}_{2}^{2} + C_{0} (F^{2} + {‖ g ‖}_{2}^{2})$ E_4^\prime\left( t \right) + \left\| {Pu\left( t \right)} \right\|_2^2 \le \left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2^2 + \left\| {P{u_t}\left( t \right)} \right\|_2^2 + {C_0}\left( {{F^2} + \left\| g \right\|_2^2} \right) with some C₀> 0 and (3.33) $E_{4} (t) = \frac{1}{2} {‖ Pu (t) ‖}_{2}^{2} + \int_{Ω} P^{\frac{1}{2}} u_{t} (t) P^{\frac{1}{2}} u (t) dx + \int_{Ω} {Pu}_{t} (t) Pu (t) dx .$ {E_4}\left( t \right) = \frac{1}{2}\left\| {Pu\left( t \right)} \right\|_2^2 + \int_\Omega {{P^{\frac{1}{2}}}{u_t}} \left( t \right){P^{\frac{1}{2}}}u\left( t \right)dx + \int_\Omega {P{u_t}} \left( t \right)Pu\left( t \right)dx. Also, assuming v = −Pu_t in (3.31), we get (3.34) $\begin{array}{l} \int_{Ω} {Pu}_{t} (- u_{tt} + Pu + {Pu}_{tt}) dx + {‖ {Pu}_{t} ‖}_{2}^{2} & = \int_{Ω} {gP}^{\frac{1}{2}} u_{t} dx - \int_{Ω} f {Pu}_{t} dx \\ \leq \frac{1}{2} {‖ {Pu}_{t} ‖}_{2}^{2} + C_{0} (F^{2} + {‖ g ‖}_{2}^{2}) . \end{array}$ \begin{array}{*{20}{l}} {\int_\Omega {P{u_t}\left( { - {u_{tt}} + Pu + P{u_{tt}}} \right)dx} + \left\| {P{u_t}} \right\|_2^2}&{ = \int_\Omega {g{P^{\frac{1}{2}}}{u_t}dx - } \int_\Omega {fP{u_t}dx} }\\ {}&{ \le \frac{1}{2}\left\| {P{u_t}} \right\|_2^2 + {C_0}\left( {{F^2} + \left\| g \right\|_2^2} \right).} \end{array} This means that (3.35) $E_{5}^{'} + \frac{1}{2} {‖ {Pu}_{t} (t) ‖}_{2}^{2} \leq C_{0} (F^{2} + {‖ g ‖}_{2}^{2})$ E_5^\prime + \frac{1}{2}\left\| {P{u_t}\left( t \right)} \right\|_2^2 \le {C_0}\left( {{F^2} + \left\| g \right\|_2^2} \right) with (3.36) $E_{5} (t) = \frac{1}{2} ({‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2} + {‖ {Pu}_{t} (t) ‖}_{2}^{2} + {‖ Pu (t) ‖}_{2}^{2}) .$ {E_5}\left( t \right) = \frac{1}{2}\left( {\left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2^2 + \left\| {P{u_t}\left( t \right)} \right\|_2^2 + \left\| {Pu\left( t \right)} \right\|_2^2} \right). We note that (3.37) ${‖ u ‖}_{2 α}^{2 α} \leq C_{0} {‖ P^{\frac{1}{2}} u ‖}_{2}^{2 α θ} + {‖ Pu ‖}_{2}^{2 α (1 - θ)} \leq η {‖ Pu ‖}_{2}^{2} + C_{η} {‖ P^{\frac{1}{2}} u ‖}_{2}^{2 β}$ \left\| u \right\|_{2\alpha }^{2\alpha } \le {C_0}\left\| {{P^{\frac{1}{2}}}u} \right\|_2^{2\alpha \theta } + \left\| {Pu} \right\|_2^{2\alpha \left( {1 - \theta } \right)} \le \eta \left\| {Pu} \right\|_2^2 + {C_\eta }\left\| {{P^{\frac{1}{2}}}u} \right\|_2^{2\beta } with small η > 0 and 2αθ = (n − 2)α − n < 2, β = α (1 − θ )/(1 − αθ ) > 0. Then, (3.37) shows (3.38) ${‖ g ‖}_{2}^{2} \leq C_{0} ({‖ u ‖}_{2 α}^{2 α} + H_{2}^{2 m}) \leq η {‖ Pu ‖}_{2}^{2} C_{η} {‖ P^{\frac{1}{2}} u ‖}_{2}^{2 β} + C_{0} H_{2}^{2 m} .$ \left\| g \right\|_2^2 \le {C_0}\left( {\left\| u \right\|_{2\alpha }^{2\alpha } + H_2^{2m}} \right) \le \eta \left\| {Pu} \right\|_2^2{C_\eta }\left\| {{P^{\frac{1}{2}}}u} \right\|_2^{2\beta } + {C_0}H_2^{2m}. Then, by (2.5), (3.35) and (3.38) that (3.39) $\begin{array}{l} E_{5}^{'} (t) + \frac{1}{2} {‖ Pu (t) ‖}_{2}^{2} & \leq {‖ Pu (t) ‖}_{2}^{2} + C_{η} {‖ P^{\frac{1}{2}} u (t) ‖}_{2}^{2 β} + C_{0} (F^{2} + H_{2}^{2 m}) \\ \leq C_{1} e^{- λ_{1} β t} + η {‖ Pu (t) ‖}_{2}^{2} + C_{2} . \end{array}$ \begin{array}{*{20}{l}} {E_5^\prime\left( t \right) + \frac{1}{2}\left\| {Pu\left( t \right)} \right\|_2^2}&{ \le \left\| {Pu\left( t \right)} \right\|_2^2 + {C_\eta }\left\| {{P^{\frac{1}{2}}}u\left( t \right)} \right\|_2^{2\beta } + {C_0}\left( {{F^2} + H_2^{2m}} \right)}\\ {}&{ \le {C_1}{e^{ - {\lambda _1}\beta t}} + \eta \left\| {Pu\left( t \right)} \right\|_2^2 + {C_2}.} \end{array} Assume φ (t) = k₁E₅ (t) + E₄ (t). We get from (3.32) and (3.39) that (3.40) $ϕ^{'} (t) + \frac{k_{1} - 1}{2} {‖ {Pu}_{t} (t) ‖}_{2}^{2} + (1 - (1 + k_{1} / 2) η) {‖ Pu (t) ‖}_{2}^{2} \leq C_{1} e^{- λ_{1} β t} + C_{2} .$ {\phi ^\prime}\left( t \right) + \frac{{{k_1} - 1}}{2}\left\| {P{u_t}\left( t \right)} \right\|_2^2 + \left( {1 - \left( {1 + {k_1}/2} \right)\eta } \right)\left\| {Pu\left( t \right)} \right\|_2^2 \le {C_1}{e^{ - {\lambda _1}\beta t}} + {C_2}. Suppose k₁≥ 3 and η is small that 1 − η (1 + k₁/2) ≥4/5. Then, (3.40) shows (3.41) $ϕ^{'} (t) + {‖ {Pu}_{t} ‖}_{2}^{2} + \frac{1}{2} {‖ Pu (t) ‖}_{2}^{2} \leq C_{1} e^{- λ_{1} β t} + C_{2} .$ {\phi ^\prime}\left( t \right) + \left\| {P{u_t}} \right\|_2^2 + \frac{1}{2}\left\| {Pu\left( t \right)} \right\|_2^2 \le {C_1}{e^{ - {\lambda _1}\beta t}} + {C_2}. We note that (3.42) $E_{4} (t) \leq \frac{3}{5} {‖ Pu ‖}_{2}^{2} + 3 {‖ {Pu}_{t} (t) ‖}_{2}^{2} + \frac{1}{2} ({‖ P^{\frac{1}{2}} u ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} ‖}_{2}^{2})$ {E_4}\left( t \right) \le \frac{3}{5}\left\| {Pu} \right\|_2^2 + 3\left\| {P{u_t}\left( t \right)} \right\|_2^2 + \frac{1}{2}\left( {\left\| {{P^{\frac{1}{2}}}u} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2^2} \right) and (3.43) $\begin{array}{l} ϕ (t) & \leq (\frac{3}{5} + \frac{k_{1}}{2}) {‖ Pu ‖}_{2}^{2} + (3 + \frac{k_{1}}{2}) {‖ {Pu}_{t} ‖}_{2}^{2} + \frac{1}{2} ({‖ P^{\frac{1}{2}} u ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} ‖}_{2}^{2}) \\ \leq C_{0} ({‖ Pu (t) ‖}_{2}^{2} + {‖ {Pu}_{t} (t) ‖}_{2}^{2} + C_{1} e^{- λ_{1} β t} + C_{2}) . \end{array}$ \begin{array}{*{20}{l}} {\varphi \left( t \right)}&{ \le \left( {\frac{3}{5} + \frac{{{k_1}}}{2}} \right)\left\| {Pu} \right\|_2^2 + \left( {3 + \frac{{{k_1}}}{2}} \right)\left\| {P{u_t}} \right\|_2^2 + \frac{1}{2}\left( {\left\| {{P^{\frac{1}{2}}}u} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2^2} \right)}\\ {}&{ \le {C_0}\left( {\left\| {Pu\left( t \right)} \right\|_2^2 + \left\| {P{u_t}\left( t \right)} \right\|_2^2 + {C_1}{e^{ - {\lambda _1}\beta t}} + {C_2}} \right).} \end{array} Also (3.41) and (3.43) give that ∃λ 1β ≥ λ 3 > 0, depending on C₀, such that (3.44) $ϕ^{'} (t) + λ_{3} ϕ (t) \leq C_{1} e^{- λ_{1} β t} + C_{2}, t \geq 0 .$ \varphi \prime\left( t \right) + {\lambda _3}\varphi \left( t \right) \le {C_1}{e^{ - {\lambda _1}\beta t}} + {C_2},t \ge 0. So, (3.45) $ϕ (t) \leq ϕ (0) e^{- λ_{3} t} + C_{1} e^{- λ_{3} t} + C_{2} λ_{3}^{- 1}, t \geq 0 .$ \varphi \left( t \right) \le \varphi \left( 0 \right){e^{ - {\lambda _3}t}} + {C_1}{e^{ - {\lambda _3}t}} + {C_2}{\lambda _3}^{ - 1},t \ge 0. Otherwise, we get (3.46) $\begin{array}{l} ϕ (t) & = k_{1} E_{4} (t) + E_{3} (t) \geq \frac{k_{1}}{2} ({‖ P^{\frac{1}{2}} u_{t} ‖}_{2}^{2} + {‖ {Pu}_{t} ‖}_{2}^{2} + {‖ Pu ‖}_{2}^{2}) \\ - \frac{1}{2} ({‖ P^{\frac{1}{2}} u_{t} ‖}_{2}^{2} + {‖ {Pu}_{t} ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u ‖}_{2}^{2}) \\ \geq \frac{k_{1} - 1}{2} {‖ P^{\frac{1}{2}} u_{t} ‖}_{2}^{2} + (\frac{k_{1}}{2} - 1) {‖ {Pu}_{t} ‖}_{2}^{2} + \frac{k_{1} - c_{0}}{2} {‖ Pu ‖}_{2}^{2} \\ \geq {‖ P^{\frac{1}{2}} u_{t} ‖}_{2}^{2} + {‖ {Pu}_{t} ‖}_{2}^{2} + {‖ Pu ‖}_{2}^{2}, \end{array}$ \begin{array}{*{20}{l}} {\varphi \left( t \right)}&{ = {k_1}{E_4}\left( t \right) + {E_3}\left( t \right) \ge \frac{{{k_1}}}{2}\left( {\left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2^2 + \left\| {P{u_t}} \right\|_2^2 + \left\| {Pu} \right\|_2^2} \right)}\\ {}&{\,\,\,\,\, - \frac{1}{2}\left( {\left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2^2 + \left\| {P{u_t}} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u} \right\|_2^2} \right)}\\ {}&{ \ge \frac{{{k_1} - 1}}{2}\left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2^2 + \left( {\frac{{{k_1}}}{2} - 1} \right)\left\| {P{u_t}} \right\|_2^2 + \frac{{{k_1} - {c_0}}}{2}\left\| {Pu} \right\|_2^2}\\ {}&{ \ge \left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2^2 + \left\| {P{u_t}} \right\|_2^2 + \left\| {Pu} \right\|_2^2,} \end{array} where the facts k₁≥ {4,2 + c₀} and Sobolev imbedding theorem (see [17]) ${‖ P^{\frac{1}{2}} u ‖}_{2}^{2} \leq c_{0} {‖ Pu ‖}_{2}^{2} \forall u \in H^{2 m} (Ω) \cap H_{0}^{m} (Ω)$ \left\| {{P^{\frac{1}{2}}}u} \right\|_2^2 \le {c_0}\left\| {Pu} \right\|_2^2\,\,\,\,\forall u \in {H^{2m}}\left( \Omega \right) \cap H_0^m\left( \Omega \right) have been used. So, by the estimates (3.45) and (3.46) that (3.47) ${‖ P^{\frac{1}{2}} u_{t} ‖}_{2}^{2} + {‖ Pu (t) ‖}_{2}^{2} + {‖ {Pu}_{t} (t) ‖}_{2}^{2} \leq C_{5} e^{- λ_{3} t} + C_{4} λ_{3}^{- 1}, t \geq 0$ \left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2^2 + \left\| {Pu\left( t \right)} \right\|_2^2 + \left\| {P{u_t}\left( t \right)} \right\|_2^2 \le {C_5}{e^{ - {\lambda _3}t}} + {C_4}\lambda _3^{ - 1},t \ge 0 with C₄ = C₄ (F,H₁,H₂), C₅ = C₅ (‖Pu₀‖₂, ‖Pu₁‖₂ ,F,H₁,H₂).

To establish the uniqueness, we suppose that u(t) and v(t) are two solutions of (1.1), which supply the estimates (2.5)–(2.7) and u(0) = v(0), u^′ (0) = v^′ (0). Taking U (t) = u_t (t), V (t) = v_t (t) and W (t) = U (t) − V (t), then we see from (1.1) that (3.48) $W_{t} - PW - P W_{t} - P (u - v) = g (x, v) - g (x, u), x \in Ω, t \geq 0 .$ {W_t} - PW - P{W_t} - P\left( {u - v} \right) = g\left( {x,v} \right) - g\left( {x,u} \right),\,\,x \in \Omega ,\,\,t \ge 0. Multiplying (3.48) by W, we obtain (3.49) $\begin{array}{l} \frac{1}{2} \frac{d}{dt} ({‖ W (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} W (t) ‖}_{2}^{2}) + {‖ P^{\frac{1}{2}} W (t) ‖}_{2}^{2} + \int_{Ω} P^{\frac{1}{2}} (u - v) P^{\frac{1}{2}} Wdx \\ = \int_{Ω} (g (x, v) - g (x, u)) Wdx \end{array}$ \begin{array}{l} \frac{1}{2}\frac{d}{{dt}}\left( {\left\| {W\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}W\left( t \right)} \right\|_2^2} \right) + \left\| {{P^{\frac{1}{2}}}W\left( t \right)} \right\|_2^2 + \int_\Omega {{P^{\frac{1}{2}}}\left( {u - v} \right){P^{\frac{1}{2}}}Wdx} \\ = \int_\Omega {\left( {g\left( {x,v} \right) - g\left( {x,u} \right)} \right)Wdx} \end{array} and (3.50) $\begin{array}{l} {‖ W (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} W (t) ‖}_{2}^{2} + 2 \int_{0}^{t} {‖ P^{\frac{1}{2}} W (s) ‖}_{2}^{2} ds + 2 \int_{0}^{t} \int_{Ω} P^{\frac{1}{2}} (u (s) - v (s)) P^{\frac{1}{2}} W (s) dxds \\ = 2 \int_{0}^{t} \int_{Ω} (g (x, v (s)) - g (x, u (s))) W (s) dxds . \end{array}$ \begin{array}{l} \left\| {W\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}W\left( t \right)} \right\|_2^2 + 2\int_0^t {\left\| {{P^{\frac{1}{2}}}W\left( s \right)} \right\|_2^2ds} + 2\int_0^t {\int_\Omega {{P^{\frac{1}{2}}}\left( {u\left( s \right) - v\left( s \right)} \right){P^{\frac{1}{2}}}W\left( s \right)dxds} } \\ = 2\int_0^t {\int_\Omega {\left( {g\left( {x,v\left( s \right)} \right) - g\left( {x,u\left( s \right)} \right)} \right)W\left( s \right)dxds.} } \end{array} Since $| P^{\frac{1}{2}} (u (s) - v (s)) | \leq \int_{0}^{s} | P^{\frac{1}{2}} (u_{τ} (τ) - v_{τ} (τ)) | d τ = \int_{0}^{s} | P^{\frac{1}{2}} W (τ) | d τ$ \left| {{P^{\frac{1}{2}}}\left( {u\left( s \right) - v\left( s \right)} \right)} \right| \le \int_0^s {\left| {{P^{\frac{1}{2}}}\left( {{u_\tau }\left( \tau \right) - {v_\tau }\left( \tau \right)} \right)} \right|d\tau } = \int_0^s {\left| {{P^{\frac{1}{2}}}W\left( \tau \right)} \right|d\tau } then ${‖ P^{\frac{1}{2}} (u (s) - v (s)) ‖}_{2} \leq s^{1 / 2} {(\int_{0}^{s} {‖ P^{\frac{1}{2}} W (τ) ‖}_{2}^{2} d τ)}^{1 / 2}$ {\left\| {{P^{\frac{1}{2}}}\left( {u\left( s \right) - v\left( s \right)} \right)} \right\|_2} \le {s^{1/2}}{\left( {\int_0^s {\left\| {{P^{\frac{1}{2}}}W\left( \tau \right)} \right\|_2^2d\tau } } \right)^{1/2}} and (3.51) $\begin{array}{l} \int_{0}^{t} \int_{Ω} | P^{\frac{1}{2}} (u (s) - v (s)) P^{\frac{1}{2}} W (s) | dxds & \leq \int_{0}^{t} \int_{Ω} \int_{0}^{s} | P^{\frac{1}{2}} W (s) | | P^{\frac{1}{2}} W (τ) | dxd τ ds \\ \leq \int_{0}^{t} \int_{0}^{s} {‖ P^{\frac{1}{2}} W (s) ‖}_{2} {‖ P^{\frac{1}{2}} W (τ) ‖}_{2} d τ ds \\ \leq t \int_{0}^{t} {‖ P^{\frac{1}{2}} W (s) ‖}_{2}^{2} ds . \end{array}$ \begin{array}{*{20}{l}} {\int_0^t {\int_\Omega {\left| {{P^{\frac{1}{2}}}\left( {u\left( s \right) - v\left( s \right)} \right){P^{\frac{1}{2}}}W\left( s \right)} \right|dxds} } }&{ \le \int_0^t {\int_\Omega {\int_0^s {\left| {{P^{\frac{1}{2}}}W\left( s \right)} \right|} \left| {{P^{\frac{1}{2}}}W\left( \tau \right)} \right|dxd\tau ds} } }\\ {}&{ \le \int_0^t {\int_0^s {{{\left\| {{P^{\frac{1}{2}}}W\left( s \right)} \right\|}_2}} {{\left\| {{P^{\frac{1}{2}}}W\left( \tau \right)} \right\|}_2}d\tau } ds}\\ {}&{ \le t\int_0^t {\left\| {{P^{\frac{1}{2}}}W\left( s \right)} \right\|_2^2ds.} } \end{array} Now, taking U_ɛ (s) = ɛu(s) + (1 − ɛ)v(s), 0 ≤ ɛ ≤ 1, we get $\begin{array}{l} G & = \int_{0}^{t} \int_{Ω} | g (x, u (s)) - g (x, v (s)) | | W (s) | dxds = \int_{0}^{t} \int_{Ω} | \int_{0}^{1} \frac{d}{d ε} g (x, U_{ε}) d ε | | W (s) | dxds \\ \leq \int_{0}^{t} \int_{Ω} \int_{0}^{1} | g_{u} (x, U_{ε}) | | u (s) - v (s) | | W (s) | d ε dxds \\ \leq k_{2} \int_{0}^{t} \int_{Ω} ({| u |}^{α - 1} + {| v |}^{α - 1} + h_{2} (x)) | u (s) - v (s) | | W (s) | dxds \\ \leq c_{0} \int_{0}^{t} ({‖ u (s) ‖}_{σ_{1}}^{σ_{1}} + {‖ v (s) ‖}_{σ_{1}}^{σ_{1}} + {‖ h_{2} (s) ‖}_{σ_{2}}^{σ_{2}}) {‖ P^{\frac{1}{2}} (u (s) - v (s)) ‖}_{2} {‖ P^{\frac{1}{2}} W (s) ‖}_{2} ds \end{array}$ \begin{array}{*{20}{l}} G&{ = \int_0^t {\int_\Omega {\left| {g\left( {x,u\left( s \right)} \right) - g\left( {x,v\left( s \right)} \right)} \right|\left| {W\left( s \right)} \right|dxds = \int_0^t {\int_\Omega {\left| {\int_0^1 {\frac{d}{{d\varepsilon }}g\left( {x,{U_\varepsilon }} \right)d\varepsilon } } \right|\left| {W\left( s \right)} \right|dxds} } } } }\\ {}&{ \le \int_0^t {\int_\Omega {\int_0^1 {\left| {{g_u}\left( {x,{U_\varepsilon }} \right)} \right|\left| {u\left( s \right) - v\left( s \right)} \right|\left| {W\left( s \right)} \right|d\varepsilon dxds} } } }\\ {}&{ \le {k_2}\int_0^t {\int_\Omega {\left( {{{\left| u \right|}^{\alpha - 1}} + {{\left| v \right|}^{\alpha - 1}} + {h_2}\left( x \right)} \right)\left| {u\left( s \right) - v\left( s \right)} \right|\left| {W\left( s \right)} \right|dxds} } }\\ {}&{ \le {c_0}\int_0^t {\left( {\left\| {u\left( s \right)} \right\|_{{\sigma _1}}^{{\sigma _1}} + \left\| {v\left( s \right)} \right\|_{{\sigma _1}}^{{\sigma _1}} + \left\| {{h_2}\left( s \right)} \right\|_{{\sigma _2}}^{{\sigma _2}}} \right){{\left\| {{P^{\frac{1}{2}}}\left( {u\left( s \right) - v\left( s \right)} \right)} \right\|}_2}{{\left\| {{P^{\frac{1}{2}}}W\left( s \right)} \right\|}_2}ds} } \end{array} where σ1 = n(α − 1)/2 ≤2n/(n − 2), σ2 = n/2.

From (2.5) and Sobolev imbedding theorem, there is C₃> 0 such that ${‖ u (s) ‖}_{σ_{1}}^{σ_{1}} + {‖ v (s) ‖}_{σ_{1}}^{σ_{1}} + {‖ h_{2} ‖}_{σ_{2}}^{σ_{2}} \leq C_{0} ({‖ P^{\frac{1}{2}} u (s) ‖}_{2}^{σ_{1}} + {‖ P^{\frac{1}{2}} v (s) ‖}_{2}^{σ_{1}} + {‖ h_{2} ‖}_{σ_{2}}^{σ_{2}}) \leq C_{3} \forall s \geq 0 .$ \left\| {u\left( s \right)} \right\|_{{\sigma _1}}^{{\sigma _1}} + \left\| {v\left( s \right)} \right\|_{{\sigma _1}}^{{\sigma _1}} + \left\| {{h_2}} \right\|_{{\sigma _2}}^{{\sigma _2}} \le {C_0}\left( {\left\| {{P^{\frac{1}{2}}}u\left( s \right)} \right\|_2^{{\sigma _1}} + \left\| {{P^{\frac{1}{2}}}v\left( s \right)} \right\|_2^{{\sigma _1}} + \left\| {{h_2}} \right\|_{{\sigma _2}}^{{\sigma _2}}} \right) \le {C_3}\forall s \ge 0. Then, (3.52) $G \leq C_{3} \int_{0}^{t} s^{1 / 2} {(\int_{0}^{s} {‖ P^{\frac{1}{2}} W (τ) ‖}_{2}^{2} d τ)}^{1 / 2} {‖ P^{\frac{1}{2}} W (s) ‖}_{2} ds \leq C_{3} t \int_{0}^{t} {‖ P^{\frac{1}{2}} W (τ) ‖}_{2}^{2} d τ .$ G \le {C_3}\int_0^t {{s^{1/2}}} {\left( {\int_0^s {\left\| {{P^{\frac{1}{2}}}W\left( \tau \right)} \right\|_2^2d\tau } } \right)^{1/2}}{\left\| {{P^{\frac{1}{2}}}W\left( s \right)} \right\|_2}ds \le {C_3}t\int_0^t {\left\| {{P^{\frac{1}{2}}}W\left( \tau \right)} \right\|_2^2d\tau } . Then, the estimates (3.50)–(3.52) indicate that (3.53) ${‖ W (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} W (t) ‖}_{2}^{2} + 2 \int_{0}^{t} {‖ P^{\frac{1}{2}} W (s) ‖}_{2}^{2} \leq (C_{3} + 1) t \int_{0}^{t} {‖ P^{\frac{1}{2}} W (s) ‖}_{2}^{2} ds .$ \left\| {W\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}W\left( t \right)} \right\|_2^2 + 2\int_0^t {\left\| {{P^{\frac{1}{2}}}W\left( s \right)} \right\|_2^2 \le \left( {{C_3} + 1} \right)t\int_0^t {\left\| {{P^{\frac{1}{2}}}W\left( s \right)} \right\|_2^2ds} .} The integral inequality (3.53) represents that there exists T₁> 0, such that W (t) = 0 in [0,T₁]. As a result, u(t) − v(t) = u(0) − v(0) = 0 in [0,T₁].

Then, we conduce that u(t) = v(t) on [T₁,2T₁], [2T₁,3T₁],..., and u(t) = v(t) on [0,∞). This shows the proof of uniqueness.

Now, we establish u ∈ C ([0,∞); $H_{0}^{m} (Ω))$ \left. {H_0^m \left( \Omega \right)} \right) ). Assume t > s ≥ 0. Then, (3.54) $\begin{array}{l} {‖ P^{\frac{1}{2}} (u (t)) - u (s) ‖}_{2}^{2} & = {\int_{Ω} | \int_{s}^{t} P^{\frac{1}{2}} u_{τ} (τ) d τ |}^{2} dx \\ \leq (t - s) \int_{s}^{t} {‖ P^{\frac{1}{2}} u_{τ} (τ) d τ ‖}_{2}^{2} d τ \to 0 as t \to s . \end{array}$ \begin{array}{*{20}{l}} {\left\| {{P^{\frac{1}{2}}}\left( {u\left( t \right)} \right) - u\left( s \right)} \right\|_2^2}&{ = {{\int_\Omega {\left| {\int_s^t {{P^{\frac{1}{2}}}{u_\tau }\left( \tau \right)d\tau } } \right|} }^2}dx}\\ {}&{ \le \left( {t - s} \right)\int_s^t {\left\| {{P^{\frac{1}{2}}}{u_\tau }\left( \tau \right)d\tau } \right\|_2^2d\tau } \to 0\,{\rm{as}}\,t \to s.} \end{array} This indicates u(t) ∈ C ([0,∞); $H_{0}^{m} (Ω))$ \left. {H_0^m \left( \Omega \right)} \right) . Also, we get (3.55) $\begin{array}{l} {‖ P (u (t) - u (s)) ‖}_{2}^{2} & = \int_{Ω} {| \int_{s}^{t} {Pu}_{τ} (τ) d τ |}^{2} dx \\ \leq (t - s) \int_{s}^{t} {‖ {Pu}_{τ} (τ) ‖}_{2}^{2} d τ \to 0 as t \to s . \end{array}$ \begin{array}{*{20}{l}} {\left\| {P\left( {u\left( t \right) - u\left( s \right)} \right)} \right\|_2^2}&{ = \int_\Omega {{{\left| {\int_s^t {P{u_\tau }\left( \tau \right)d\tau } } \right|}^2}dx} }\\ {}&{ \le \left( {t - s} \right)\int_s^t {\left\| {P{u_\tau }\left( \tau \right)} \right\|_2^2d\tau } \to 0\,{\rm{as}}\,t \to s.} \end{array} and u(t) ∈ C [0,∞); $H^{2 m} (Ω) \cap H_{0}^{m} (Ω))$ H^{2m} \left( \Omega \right) \cap \left. {H_0^m \left( \Omega \right)} \right) .

Moreover, we get (3.56) ${‖ P^{\frac{1}{2}} (u_{t} (t) - u_{t}) (s) ‖}_{2}^{2} \leq (t - s) \int_{s}^{t} {‖ P^{\frac{1}{2}} u_{tt} (τ) ‖}_{2}^{2} d τ \to 0 as t \to s .$ \left\| {{P^{\frac{1}{2}}}\left( {{u_t}\left( t \right) - {u_t}} \right)\left( s \right)} \right\|_2^2 \le \left( {t - s} \right)\int_s^t {\left\| {{P^{\frac{1}{2}}}{u_{tt}}\left( \tau \right)} \right\|_2^2d\tau } \to 0\,{\rm{as}}\,t \to s. This shows that u(t) ∈ C¹ ([0,∞); $H_{0}^{m}$ {H_0^m } ) and the proof of Theorem 3 is completed.

Global attractor for the problem (1)

By Theorem 3, we see that the solution operatör S (t)(u₀,u₁) = (u(t),u_t (t)), t ≥ 0 of the problem (1.1) creates a semigroup on $X = (H^{2 m} (Ω) \cap H_{0}^{m} (Ω)) \times (H^{2 m} (Ω) \cap H_{0}^{m} (Ω))$ X = \left( {H^{2m} \left( \Omega \right) \cap H_0^m \left( \Omega \right)} \right) \times \left( {H^{2m} \left( \Omega \right) \cap H_0^m \left( \Omega \right)} \right) , which supplies these properties: (1)

S (t) : X → X for all t ≥ 0;

(2)

S (t + s) = S (t)S (s) for t,s ≥ 0;

(3)

S (t)(u₀,u₁) → S (s)(u₀,u₁) in X as t → s for any (u₀,u₁) ∈ X.

For establishing the existence of the (X,X)-global attractor for the problem (1.1), firstly, we show the continuity of S (t) relating to the initial data (u₀,u₁).

The proof of Theorem 4

Suppose u_k (t), u(t) is corresponding solution of the problem (1.1) with the initial data (u_0k,u₁k) and (u₀,u₁) respectively, k = 1,2,....

Since (u_0k,u_1k) → (u₀,u₁) in X, {(u_0k,u_1k)} is bounded in X. Set w_k (t) = u_k (t) − u(t). Then, w_k holds (4.1) ${\begin{matrix} w_{k}^{″} - P w_{k} - P w_{k}^{'} - P w_{k}^{″} = g (x, u) - g (x, u_{k}) = G_{k}, (x, t) \in Ω \times (0, \infty), \\ w_{k} (x, 0) = u_{0 k} (x) - u_{0} (x), w_{k}^{'} (x, 0) = u_{1 k} (x) - u_{1} (x), x \in Ω, \\ w_{k} (x, t) = 0, (x, t) \in \partial Ω \times [0, \infty) . \end{matrix}$ \left\{ {\begin{array}{*{20}{c}} {w_k^{\prime\prime} - P{w_k} - Pw_k^\prime - Pw_k^{\prime\prime} = g\left( {x,u} \right) - g\left( {x,{u_k}} \right) = {G_k},\left( {x,t} \right) \in \Omega \times \left( {0,\infty } \right),}\\ {{w_k}\left( {x,0} \right) = {u_{0k}}\left( x \right) - {u_0}\left( x \right),w_k^\prime\left( {x,0} \right) = {u_{1k}}\left( x \right) - {u_1}\left( x \right),x \in \Omega ,}\\ {{w_k}\left( {x,t} \right) = 0,\left( {x,t} \right) \in \partial \Omega \times [0,\infty ).} \end{array}} \right. Multiplying the equation in (4.1) by w^′_k,−Pw_k and −Pw^′_k, we get (4.2) $\frac{1}{2} \frac{d}{dt} ({‖ w_{k}^{'} ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k} ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2}) + (1 - η) {‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2} \leq C_{η} {‖ G_{k} ‖}_{2}^{2}$ \frac{1}{2}\frac{d}{{dt}}\left( {\left\| {w_k^\prime} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{w_k}} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2} \right) + \left( {1 - \eta } \right)\left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2 \le {C_\eta }\left\| {{G_k}} \right\|_2^2 and (4.3) $\begin{array}{l} \frac{d}{dt} (\frac{1}{2} {‖ P w_{k} ‖}_{2}^{2} + \int_{Ω} (P w_{k}^{'} P w_{k} + P^{\frac{1}{2}} w_{k} P^{\frac{1}{2}} w_{k}^{'}) dx) + (1 - η) {‖ P w_{k} ‖}_{2}^{2} \\ \leq {‖ P w_{k}^{'} ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2} + C_{n} {‖ G_{k} ‖}_{2}^{2} \leq c_{0} {‖ P w_{k}^{'} ‖}_{2}^{2} + C_{η} {‖ G_{k} ‖}_{2}^{2} \end{array}$ \begin{array}{*{20}{l}} {\,\,\frac{d}{{dt}}\left( {\frac{1}{2}\left\| {P{w_k}} \right\|_2^2 + \int_\Omega {\left( {Pw_k^\primeP{w_k} + {P^{\frac{1}{2}}}{w_k}{P^{\frac{1}{2}}}w_k^\prime} \right)dx} } \right) + \left( {1 - \eta } \right)\left\| {P{w_k}} \right\|_2^2}\\ { \le \left\| {Pw_k^\prime} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2 + {C_n}\left\| {{G_k}} \right\|_2^2 \le {c_0}\left\| {Pw_k^\prime} \right\|_2^2 + {C_\eta }\left\| {{G_k}} \right\|_2^2} \end{array} and (4.4) $\frac{1}{2} \frac{d}{dt} ({‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2} + {‖ P w_{k} ‖}_{2}^{2} + {‖ P w_{k}^{'} ‖}_{2}^{2}) + (1 - η) {‖ P w_{k}^{'} ‖}_{2}^{2} \leq C_{η} {‖ G_{k} ‖}_{2}^{2}$ \frac{1}{2}\frac{d}{{dt}}\left( {\left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2 + \left\| {P{w_k}} \right\|_2^2 + \left\| {Pw_k^\prime} \right\|_2^2} \right) + \left( {1 - \eta } \right)\left\| {Pw_k^\prime} \right\|_2^2 \le {C_\eta }\left\| {{G_k}} \right\|_2^2 with small η > 0. Then, by (4.2) and (4.4) we obtain (4.5) $y_{k}^{'} (t) + (k (1 - η) - c_{0}) {‖ P w_{k}^{'} ‖}_{2}^{2} + (1 - η) {‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2} + (1 - η) {‖ P w_{k} (t) ‖}_{2}^{2} \leq k C_{η} {‖ G_{k} ‖}_{2}^{2}$ y_k^\prime\left( t \right) + \left( {k\left( {1 - \eta } \right) - {c_0}} \right)\left\| {Pw_k^\prime} \right\|_2^2 + \left( {1 - \eta } \right)\left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2 + \left( {1 - \eta } \right)\left\| {P{w_k}\left( t \right)} \right\|_2^2 \le k{C_\eta }\left\| {{G_k}} \right\|_2^2 where (4.6) $\begin{array}{l} y_{k} (t) & = \frac{k_{1} + 1}{2} ({‖ P w_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2}) \\ + \frac{1}{2} ({‖ P^{\frac{1}{2}} w_{k} (t) ‖}_{2}^{2} + {‖ w_{k}^{'} (t) ‖}_{2}^{2}) + \frac{k_{1}}{2} {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2} \\ + \int_{Ω} (P w_{k}^{'} (t) P w_{k} (t) + P^{\frac{1}{2}} w_{k} (t) P^{\frac{1}{2}} w_{k}^{'} (t)) dx \\ \leq \frac{k_{1} + 2}{2} ({‖ P w_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2}) \\ + \frac{k_{1} + 1}{2} {‖ P w_{k}^{'} (t) ‖}_{2}^{2} + {‖ w_{k}^{'} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2} \\ \leq C_{0} ({‖ P w_{k}^{'} (t) ‖}_{2}^{2} + {‖ P w_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2}) \end{array}$ \begin{array}{*{20}{l}} {{y_k}(t)}&{ = \frac{{{k_1} + 1}}{2}\left( {\left\| {P{w_k}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2} \right)}\\ {}&{\,\,\,\,\, + \frac{1}{2}\left( {\left\| {{P^{\frac{1}{2}}}{w_k}(t)} \right\|_2^2 + \left\| {w_k^\prime(t)} \right\|_2^2} \right) + \frac{{{k_1}}}{2}\left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2}\\ {}&{\,\,\,\,\, + \int_\Omega {\left( {Pw_k^\prime(t)P{w_k}(t) + {P^{\frac{1}{2}}}{w_k}(t){P^{\frac{1}{2}}}w_k^\prime(t)} \right)dx} }\\ {}&{\, \le \frac{{{k_1} + 2}}{2}\left( {\left\| {P{w_k}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2} \right)}\\ {}&{\,\,\,\,\, + \frac{{{k_1} + 1}}{2}\left\| {Pw_k^\prime(t)} \right\|_2^2 + \left\| {w_k^\prime(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2}\\ {}&{ \le {C_0}\left( {\left\| {Pw_k^\prime(t)} \right\|_2^2 + \left\| {P{w_k}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2} \right)} \end{array} By taking k₁≥ 3 (4.7) $\begin{array}{l} y_{k} (t) & = \frac{k_{1} + 1}{2} ({‖ P w_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2}) \\ + \frac{1}{2} ({‖ P^{\frac{1}{2}} w_{k} (t) ‖}_{2}^{2} + {‖ w_{k}^{'} (t) ‖}_{2}^{2}) + \frac{k_{1}}{2} {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2} \\ - \frac{1}{2} ({‖ P w_{k}^{'} ‖}_{2}^{2} + {‖ P w_{k}^{'} ‖}_{2}^{2}) - \frac{1}{2} ({‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2}) \\ \geq {‖ P w (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} (t) ‖}_{2}^{2} + {‖ P w_{k}^{'} (t) ‖}_{2}^{2}, t \geq 0 . \end{array}$ \begin{array}{*{20}{l}} {{y_k}(t)}&{\, = \frac{{{k_1} + 1}}{2}\left( {\left\| {P{w_k}(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2} \right)}\\ {}&{ + \frac{1}{2}\left( {\left\| {{P^{\frac{1}{2}}}{w_k}(t)} \right\|_2^2 + \left\| {w_k^\prime(t)} \right\|_2^2} \right) + \frac{{{k_1}}}{2}\left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2}\\ {}&{ - \frac{1}{2}\left( {\left\| {Pw_k^\prime} \right\|_2^2 + \left\| {Pw_k^\prime} \right\|_2^2} \right) - \frac{1}{2}\left( {\left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2} \right)}\\ {}&{ \ge \left\| {Pw(t)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime(t)} \right\|_2^2 + \left\| {Pw_k^\prime(t)} \right\|_2^2,\,\,t \ge 0.} \end{array} Otherwise, we obtain from assumption (A₂), (4.8) $\begin{array}{l} {‖ G_{k} ‖}_{2}^{2} & = \int_{Ω} {| g (x, u_{k}) - g (x, u) |}^{2} d x = \int_{Ω} g_{u}^{2} w_{k}^{2} d x \\ \leq c_{0} \int_{Ω} ({| u_{k} |}^{2 (α - 1)} + {| u |}^{2 (α - 1)} + h_{2}^{2}) w_{k}^{2} d x . \end{array}$ \begin{array}{*{20}{l}} {\left\| {{G_k}} \right\|_2^2}&{ = \int_\Omega {{{\left| {g\left( {x,{u_k}} \right) - g\left( {x,u} \right)} \right|}^2}dx} = \int_\Omega {g_u^2w_k^2dx} }\\ {}&{ \le {c_0}\int_\Omega {\left( {{{\left| {{u_k}} \right|}^{2\left( {\alpha - 1} \right)}} + {{\left| u \right|}^{2\left( {\alpha - 1} \right)}} + h_2^2} \right)w_k^2dx.} } \end{array} The application of Sobolev imbedding theorem and the estimate (2.7) gives (4.9) $\int_{Ω} {| u_{k} |}^{2 (α - 1)} w_{k}^{2} d x \leq {‖ w_{k} ‖}_{2 μ_{2}}^{2} {‖ u_{k} ‖}_{2 (α - 1) μ_{3}}^{2 (α - 1)} \leq C_{3} {‖ w_{k} ‖}_{2 μ_{2}}^{2} \leq C_{3} {‖ w_{k} ‖}_{2}^{2}$ \int_\Omega {{{\left| {{u_k}} \right|}^{2\left( {\alpha - 1} \right)}}} w_k^2dx \le \left\| {{w_k}} \right\|_{2{\mu _2}}^2\left\| {{u_k}} \right\|_{2\left( {\alpha - 1} \right){\mu _3}}^{2\left( {\alpha - 1} \right)} \le {C_3}\left\| {{w_k}} \right\|_{2{\mu _2}}^2 \le {C_3}\left\| {{w_k}} \right\|_2^2 with μ₂ = n/(n − 4)⁺ and μ₃ = μ₂/(μ₂ − 1). Similarly, (4.10) $\int_{Ω} {| u_{k} |}^{2 (α - 1)} w_{k}^{2} d x \leq {‖ w_{k} ‖}_{2 μ_{2}}^{2} {‖ u ‖}_{2 (α - 1) μ_{3}}^{2 (α - 1)} \leq C_{3} {‖ w_{k} ‖}_{2 μ_{2}}^{2} \leq C_{3} {‖ P w_{k} ‖}_{2}^{2}$ \int_\Omega {{{\left| {{u_k}} \right|}^{2\left( {\alpha - 1} \right)}}} w_k^2dx \le \left\| {{w_k}} \right\|_{2{\mu _2}}^2\left\| u \right\|_{2\left( {\alpha - 1} \right){\mu _3}}^{2\left( {\alpha - 1} \right)} \le {C_3}\left\| {{w_k}} \right\|_{2{\mu _2}}^2 \le {C_3}\left\| {P{w_k}} \right\|_2^2 and (4.11) $\int_{Ω} h_{2}^{2} w_{k}^{2} d x \leq {‖ w_{k} ‖}_{2 μ_{2}}^{2} {‖ h_{2} ‖}_{N / 2}^{2} \leq C_{3} {‖ w_{k} ‖}_{2 μ_{2}}^{2} \leq C_{3} {‖ P w_{k} ‖}_{2}^{2} .$ \int_\Omega {h_2^2} w_k^2dx \le \left\| {{w_k}} \right\|_{2{\mu _2}}^2\left\| {{h_2}} \right\|_{N/2}^2 \le {C_3}\left\| {{w_k}} \right\|_{2{\mu _2}}^2 \le {C_3}\left\| {P{w_k}} \right\|_2^2. Then, we get from (4.5) to (4.11) that λ 4 > 0, such that (4.12) $y_{k}^{'} (t) + λ_{4} y_{k} (t) \leq C_{3} {‖ G_{k} ‖}_{2}^{2} \leq C_{3} {‖ w_{k} ‖}_{2 μ_{2}}^{2} \leq C_{3} {‖ P w_{k} ‖}_{2 μ_{2}}^{2} \leq C_{3} y_{k} (t)$ y_k^\prime\left( t \right) + {\lambda _4}{y_k}\left( t \right) \le {C_3}\left\| {{G_k}} \right\|_2^2 \le {C_3}\left\| {{w_k}} \right\|_{2{\mu _2}}^2 \le {C_3}\left\| {P{w_k}} \right\|_{2{\mu _2}}^2 \le {C_3}{y_k}\left( t \right) where C₃ is as in (2.6), independent of k. The differential inequality (4.12) means (4.13) $y_{k} (t) \leq y_{k} (0) e^{(C_{3} - λ_{4}) t}, t \geq 0 .$ {y_k}\left( t \right) \le {y_k}\left( 0 \right){e^{\left( {{C_3} - {\lambda _4}} \right)t}},t \ge 0. Then, from (4.6) and (4.7), we obtain (4.14) $y_{k} (0) \leq C_{0} ({‖ P^{\frac{1}{2}} (u_{1 k} - u_{1}) ‖}_{2}^{2} + {‖ P (u_{0 k} - u_{0}) ‖}_{2}^{2} + {‖ P (u_{1 k} - u_{1}) ‖}_{2}^{2}) \to 0 as k \to \infty$ {y_k}\left( 0 \right) \le {C_0}\left( {\left\| {{P^{\frac{1}{2}}}\left( {{u_{1k}} - {u_1}} \right)} \right\|_2^2 + \left\| {P\left( {{u_{0k}} - {u_0}} \right)} \right\|_2^2 + \left\| {P\left( {{u_{1k}} - {u_1}} \right)} \right\|_2^2} \right) \to 0\,{\rm{as}}\,k \to \infty and (4.15) ${‖ P w_{k} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{k}^{'} ‖}_{2}^{2} + {‖ P w_{k}^{'} ‖}_{2}^{2} \leq y_{k} (t) \leq y_{k} (0) e^{(C_{3} - λ_{4}) t} \to 0 as k \to \infty .$ \left\| {P{w_k}\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_k^\prime} \right\|_2^2 + \left\| {Pw_k^\prime} \right\|_2^2 \le {y_k}\left( t \right) \le {y_k}\left( 0 \right){e^{\left( {{C_3} - {\lambda _4}} \right)t}} \to 0\,\,{\rm{as}}\,k \to \infty . This indicates that S (t) : X → X is continuous. Now we show that {S (t)}_t≥0 is asymptotically compact in X from the method in [9].

Assume {(u_0k,u_1k)} is a bounded sequence and {u_k (t)} be the corresponding solutions of the problem (1.1) in C [0,∞); $H^{2 m} (Ω) \cap H_{0}^{m} (Ω))$ H^{2m} \left( \Omega \right) \cap \left. {H_0^m \left( \Omega \right)} \right) . We suppose t_k→ ∞ as k → ∞. For any T > 0, assume t_n,t_k> T. Then, the application of (4.12) to w_kn (t) = u_n (t + t_n − T ) − u_k (t + t_n − T ), we get (4.16) $Y_{kn} (t) \leq Y_{kn} (0) e^{- λ_{4} t} + C_{3} \int_{0}^{t} e^{- λ_{4} (t - s)} {‖ w_{kn} (s) ‖}_{2 μ_{2}}^{2} ds, t \geq 0$ {Y_{kn}}\left( t \right) \le {Y_{kn}}\left( 0 \right){e^{ - {\lambda _4}t}} + {C_3}\int_0^t {{e^{ - {\lambda _4}\left( {t - s} \right)}}} \left\| {{w_{kn}}\left( s \right)} \right\|_{2{\mu _2}}^2ds,t \ge 0 with (4.17) $Y_{kn} (t) = {‖ P w_{kn} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} w_{kn}^{'} (t) ‖}_{2}^{2} + {‖ P w_{kn}^{'} (t) ‖}_{2}^{2} .$ {Y_{kn}}\left( t \right) = \left\| {P{w_{kn}}\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}w_{kn}^\prime\left( t \right)} \right\|_2^2 + \left\| {Pw_{kn}^\prime\left( t \right)} \right\|_2^2. Especially, we take t = T and obtain (4.18) $\begin{array}{l} {‖ P (u_{n} (t_{n})) - u_{k} (t_{k}) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} (u_{n}^{'} (t_{n}) - u_{k}^{'} (t_{k})) ‖}_{2}^{2} + {‖ P (u_{n}^{'} (t_{n}) - u_{k}^{'} (t_{k})) ‖}_{2}^{2} \\ \leq Y_{kn} (0) e^{- λ_{4} T} + C_{3} sup_{0 \leq s \leq T} {‖ u_{k} (t_{k} - T + s) - u_{n} (t_{n} - T + s) ‖}_{2 μ_{2}}^{2} . \end{array}$ \begin{array}{l} \left\| {P\left( {{u_{ & n}}\left( {{t_n}} \right)} \right) - {u_k}\left( {{t_k}} \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}\left( {\user1{u}_n^\prime\left( {{t_n}} \right) - u_k^\prime\left( {{t_k}} \right)} \right)} \right\|_2^2 + \left\| {P\left( {u_n^\prime\left( {{t_n}} \right) - u_k^\prime\left( {{t_k}} \right)} \right)} \right\|_2^2\\ \le {Y_{kn}}\left( 0 \right){e^{ - {\lambda _4}T}} + {C_3}\mathop {\sup }\limits_{0 \le s \le T} \left\| {{u_k}\left( {{t_k} - T + s} \right) - {u_n}\left( {{t_n} - T + s} \right)} \right\|_{2{\mu _2}}^2. \end{array} Since the embedding $(H^{2 m} (Ω) \cap H_{0}^{m} (Ω)) ↪ L^{2 μ_{2}} (Ω)$ \left( {{H^{2m}}\left( \Omega \right) \cap H_0^m\left( \Omega \right)} \right) \hookrightarrow {L^{2{\mu _2}}}\left( \Omega \right)\ is compact, we can remove a subsequence {u_{k_k₁}(t_{k_k₁} − T + s)} which converges in L^2μ₂ (Ω). Therefore, for any ɛ > 0, firstly we fix T > 0, such that (4.19) $Y_{kn} (0) e^{- λ_{4} T} < \frac{ε}{2} .$ {Y_{kn}}\left( 0 \right){e^{ - {\lambda _4}T}} < \frac{\varepsilon }{2}. Supposing n₀> 0 and k₁, j > n₀, we get (4.20) $C_{3} sup_{0 \leq s \leq T} {‖ u_{k_{k_{1}}} (t_{k_{k_{1}}} - T + s) - u_{k_{j}} (t_{k_{j}} - T + s) ‖}_{2 μ_{2}}^{2} < \frac{ε}{2} .$ {C_3}\mathop {\sup }\limits_{0 \le s \le T} \left\| {{u_{{k_{{k_1}}}}}\left( {{t_{{k_{{k_1}}}}} - T + s} \right) - {u_{{k_j}}}\left( {{t_{{k_j}}} - T + s} \right)} \right\|_{2{\mu _2}}^2 < \frac{\varepsilon }{2}. Then, it follows by (4.18) to (4.20) that {u_{k_k₁}(t_{k_k₁})} is a Cauchy sequence in X and we finalize that {S (t)}_t≥0 is asyptotically compact on X and now Theorem 4 is established.

Proof of Theorem 5

From Lemma 2, it is sufficient to indicate that there exists a continuous operator semigroup {S (t)} on X such that S (t)(u₀,u₁) = (u(t),u_t (t)) for each t ≥ 0. By the estimates (2.7), we accomplish that (4.21) $β_{0} = {(u, v) \in X {‖ P^{\frac{1}{2}} v ‖}_{2}^{2} + {‖ Pu ‖}_{2}^{2} + {‖ Pv ‖}_{2}^{2} \leq C_{4}}$ {\beta _0} = \left\{ {\left( {u,v} \right) \in X\left\| {{P^{\frac{1}{2}}}v} \right\|_2^2 + \left\| {Pu} \right\|_2^2 + \left\| {Pv} \right\|_2^2 \le {C_4}} \right\} is an absorbing set of {S (t)}_t≥0 and for any (u₀,u₁) ∈ X, (4.22) $dis t_{X} (S (t) (u_{0}, u_{1}), β_{0}) \leq C_{5} e^{- λ_{3} t}, t \geq 0$ \user1{dis}{\user1{t}_X}\left( {S\left( t \right)\left( {{u_0},{u_{ & 1}}} \right),{\beta _0}} \right) \le {C_5}{e^{ - {\lambda _3}t}},t \ge 0 where the constants C₄, C₅ are in (2.7). By Theorem 2, S (t) : X → X is continuous and asymptotically compact on X. From a general theory (see [1, 11]), we conclude that S (t) admits a global attractor A on X defined by (4.23) $A = ω (β_{0}) = \underset{τ \geq 0}{\cap} {[\underset{t \geq τ}{\cup S (t)} β_{0}]}_{X}$ A = \omega \left( {{\beta _0}} \right) = \mathop \cap \limits_{\tau \ge 0} {\left[ {\mathop { \cup S\left( t \right)}\limits_{t \ge \tau } {\beta _0}} \right]_X} where [D]_X is the closure of the set D in X. Then we prove the Theorem 5.

Decay property of solution for (1)

In this section, we search the decay property of solution to (1.1) with f ≣ 0. Firstly, we present a well-known Lemma that will be needed.

Lemma 7

([18])

Assume E : [0,∞) → [0,∞) is a non-increasing function and suppose that there are constants q ≥ 0 and γ > 0 such that(5.1) $\int_{S}^{\infty} E^{q + 1} (t) dt \leq γ^{- 1} E {(0)}^{q} E (s), \forall S \geq 0 .$ \int_S^\infty {{E^{q + 1}}} \left( t \right)dt \le {\gamma ^{ - 1}}E{\left( 0 \right)^q}E\left( s \right),\forall S \ge 0.Then, we get(5.2) $E (t) \leq E (0) {(\frac{1 + q}{1 + q γ t})}^{1 / q} \forall t \geq 0 if q > 0$ E\left( t \right) \le E\left( 0 \right){\left( {\frac{{1 + q}}{{1 + q\gamma t}}} \right)^{1/q}}\forall t \ge 0\,ifq > 0and(5.3) $E (t) \leq E (0) e^{1 - γ t} \forall t \geq 0 if q = 0 .$ E\left( t \right) \le E\left( 0 \right){e^{1 - \gamma t}}\forall t \ge 0ifq = 0.

Proof of Theorem 7

Suppose u(t) is a weak solution in Theorem 3 with f = 0. Show (5.4) $E (t) = \frac{1}{2} ({‖ u (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2}) + \int_{Ω} G (u (t)) dx, t \geq 0 .$ E\left( t \right) = \frac{1}{2}\left( {\left\| {u\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2^2} \right) + \int_\Omega {G\left( {u\left( t \right)} \right)dx,t \ge 0.} Then, we obtain by (1.1) that (5.5) $E^{'} (t) + {‖ P^{\frac{1}{2}} u_{t} ‖}^{2} = 0, \forall \geq 0 .$ E\prime\left( t \right) + {\left\| {{P^{\frac{1}{2}}}{u_t}} \right\|^2} = 0,\,\,\,\forall \ge 0. This indicates that E (t) is non-increasing in [0,∞).

Multiplying the equation in (1.1) by E^q (t)u(t), q > 0, we obtain (5.6) $\int_{S}^{T} E^{q} \int_{Ω} u (u_{tt} - Pu - {Pu}_{tt} + g (u)) dxdt = 0, \forall T > S \geq 0 .$ \int_S^T {{E^q}} \int_\Omega {u\left( {{u_{tt}} - Pu - P{u_{tt}} + g\left( u \right)} \right)dxdt} = 0,\,\,\forall T > S \ge 0. We note that $\begin{array}{l} {\int_{S}^{T} E^{q} (t) (u, u_{tt}) dt = E^{q} (t) (u, u_{t}) |}_{S}^{T} - \int_{S}^{T} (qE {(t)}^{q - 1} E^{'} (t) (u, u_{t}) + E^{q} (t) {‖ u_{t} (t) ‖}_{2}^{2}) \\ - \int_{S}^{T} E^{q} (t) (u, Pu) dt = \int_{S}^{T} E^{q} (t) {‖ P^{\frac{1}{2}} u ‖}_{2}^{2} \\ - \int_{S}^{T} E^{q} (t) (u, {Pu}_{t}) dt = \int_{S}^{T} E^{q} (t) (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t}) dt \end{array}$ \begin{array}{l} \left. {\int_S^T {{E^q}\left( t \right)\left( {u,{u_{tt}}} \right)} dt = {E^q}\left( t \right)\left( {u,{u_t}} \right)} \right|_S^T - \int_S^T {\left( {qE{{\left( t \right)}^{q - 1}}E\prime\left( t \right)\left( {u,{u_t}} \right) + {E^q}\left( t \right)\left\| {{u_t}\left( t \right)} \right\|_2^2} \right)} \\ - \int_S^T {{E^q}\left( t \right)\left( {u,Pu} \right)dt} = \int_S^T {{E^q}\left( t \right)\left\| {{P^{\frac{1}{2}}}u} \right\|_2^2} \\ - \int_S^T {{E^q}} \left( t \right)\left( {u,P{u_t}} \right)dt = \int_S^T {{E^q}\left( t \right)\left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right)dt} \end{array} and $\begin{array}{l} - \int_{S}^{T} E^{q} (t) (u, {Pu}_{tt}) dt & = - \int_{S}^{T} (q E {(t)}^{q - 1} E^{'} (t) (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t}) + E^{q} (t) {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2}) dt \\ {+ E^{q} (t) (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t}) |}_{S}^{T} . \end{array}$ \begin{array}{*{20}{l}} { - \int_S^T {{E^q}\left( t \right)\left( {u,P{u_{tt}}} \right)dt} }&{ = - \int_S^T {\left( {qE{{\left( t \right)}^{q - 1}}E\prime\left( t \right)\left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right) + {E^q}\left( t \right)\left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2^2} \right)dt} }\\ {}&{\,\,\,\,\,\left. { + {E^q}\left( t \right)\left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right)} \right|_S^T.} \end{array} Then, we get by (5.6) that (5.7) $\begin{array}{l} 2 \int_{S}^{T} E^{q + 1} (t) dt = & - E^{q} (t) [{(u, u_{t}) + (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t}) |}_{S}^{T}] \\ + q \int_{S}^{T} E {(t)}^{q - 1} E^{'} (t) [(u, u_{t}) + (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u t)] dt \\ + 2 \int_{S}^{T} E^{q} (t) ({‖ u_{t} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2}) dt \\ + \int_{S}^{T} E^{q} (t) (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t}) dt + \int_{S}^{T} E^{q} (t) (2 G (u) - ug (u)) dt . \end{array}$ \begin{array}{*{20}{l}} {2\int_S^T {{E^{q + 1}}\left( t \right)dt = } }&{ - {E^q}\left( t \right)\left[ {\left. {\left( {u,{u_t}} \right) + \left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right)} \right|_S^T} \right]}\\ {}&{ + q\int_S^T {E{{\left( t \right)}^{q - 1}}E\prime\left( t \right)\left[ {\left( {u,{u_t}} \right) + \left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}ut} \right)} \right]} dt}\\ {}&{ + 2\int_S^T {{E^q}\left( t \right)\left( {\left\| {{u_t}\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2^2} \right)dt} }\\ {}&{ + \int_S^T {{E^q}\left( t \right)\left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right)dt} + \int_S^T {{E^q}} \left( t \right)\left( {2G\left( u \right) - ug\left( u \right)} \right)dt.} \end{array} Since G(u) ≥ 0, E (t) ≥ 0. Moreover, we get the following estimates from (5.5): (5.8) ${‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2} \leq {(- E^{'} (t))}^{1 / 2}, {‖ P^{\frac{1}{2}} u (t) ‖}_{2}^{2} \leq 2 {(E (t))}^{1 / 2}, {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2} \leq 2 {(E (t))}^{1 / 2}, \forall \geq 0,$ {\left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2} \le {\left( { - E\prime\left( t \right)} \right)^{1/2}},\,\,\,\,\,\left\| {{P^{\frac{1}{2}}}u\left( t \right)} \right\|_2^2 \le 2{\left( {E\left( t \right)} \right)^{1/2}},{\left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2} \le 2{\left( {E\left( t \right)} \right)^{1/2}},\forall \ge 0,(5.9) $| E^{q} (t) ((u, u_{t}) + (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t})) | \leq C_{0} E^{q} (t) {‖ P^{\frac{1}{2}} u ‖}_{2} {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2} \leq C_{0} E^{q + 1} (t),$ \left| {{E^q}\left( t \right)\left( {\left( {u,{u_t}} \right) + \left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right)} \right)} \right| \le {C_0}{E^q}\left( t \right){\left\| {{P^{\frac{1}{2}}}u} \right\|_2}{\left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2} \le {C_0}{E^{q + 1}}\left( t \right),(5.10) $\begin{array}{l} \int_{S}^{T} | E {(t)}^{q - 1} E^{'} (t) | [(u, u_{t}) + (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t})] dt \\ \leq C_{0} \int_{S}^{T} E {(t)}^{q - 1} | E^{'} (t) | {‖ P^{\frac{1}{2}} u ‖}_{2} {‖ P^{\frac{1}{2}} u_{t} ‖}_{2} dt \leq C_{0} E^{q + 1} (S), \end{array}$ \begin{array}{l} \int_S^T {\left| {E{{\left( t \right)}^{q - 1}}E\prime\left( t \right)} \right|\left[ {\left( {u,{u_t}} \right) + \left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right)} \right]} dt\\ \le {C_0}\int_S^T {E{{\left( t \right)}^{q - 1}}} \left| {E\prime\left( t \right)} \right|{\left\| {{P^{\frac{1}{2}}}u} \right\|_2}{\left\| {{P^{\frac{1}{2}}}{u_t}} \right\|_2}dt \le {C_0}{E^{q + 1}}\left( S \right), \end{array}(5.11) $2 \int_{S}^{T} E^{q} (t) ({‖ u_{t} (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2}) dt \leq C_{0} \int_{S}^{T} E^{q} (t) {(- E^{'} (t))}^{1 / 2} \leq C_{0} E^{q + 1} (s),$ 2\int_S^T {{E^q}} \left( t \right)\left( {\left\| {{u_t}\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2^2} \right)dt \le {C_0}\int_S^T {{E^q}} \left( t \right){\left( { - E\prime\left( t \right)} \right)^{1/2}} \le {C_0}{E^{q + 1}}\left( s \right),(5.12) $\begin{array}{l} \int_{S}^{T} E^{q} (t) (P^{\frac{1}{2}} u, P^{\frac{1}{2}} u_{t}) dt & \leq \int_{S}^{T} E^{q} (t) {‖ P^{\frac{1}{2}} u ‖}_{2} {‖ P^{\frac{1}{2}} u_{t} ‖}_{2} \\ \leq \int_{S}^{T} E^{q + 1} (t) dt + C_{1} E^{q + 1} (S) . \end{array}$ \begin{array}{*{20}{l}} {\int_S^T {{E^q}\left( t \right)\left( {{P^{\frac{1}{2}}}u,{P^{\frac{1}{2}}}{u_t}} \right)dt} }&{ \le \int_S^T {{E^q}\left( t \right){{\left\| {{P^{\frac{1}{2}}}u} \right\|}_2}{{\left\| {{P^{\frac{1}{2}}}{u_t}} \right\|}_2}} }\\ {}&{ \le \int_S^T {{E^{q + 1}}\left( t \right)dt} + {C_1}{E^{q + 1}}\left( S \right).} \end{array} Then we obtain from (5.8) to (5.12) that (5.13) $\int_{S}^{T} E^{q + 1} (t) dt \leq C_{0} E^{q + 1} (S) \leq C_{0} E^{q} (0) E (S) \equiv γ^{- 1} E^{q} (0) E (S) .$ \int_S^T {{E^{q + 1}}\left( t \right)dt} \le {C_0}{E^{q + 1}}\left( S \right) \le {C_0}{E^q}\left( 0 \right)E\left( S \right) \equiv {\gamma ^{ - 1}}{E^q}\left( 0 \right)E\left( S \right). From Lemma 10, we get $\begin{array}{l} E (t) & = \frac{1}{2} ({‖ u (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u (t) ‖}_{2}^{2} + {‖ P^{\frac{1}{2}} u_{t} (t) ‖}_{2}^{2}) + \int_{Ω} G (u (t)) dx \\ \leq E (0) {(\frac{1 + q}{1 + q γ t})}^{1 / q} \leq C_{1} {(1 + t)}^{- 1 / q} . \end{array}$ \begin{array}{*{20}{l}} {E\left( t \right)}&{ = \frac{1}{2}\left( {\left\| {u\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}u\left( t \right)} \right\|_2^2 + \left\| {{P^{\frac{1}{2}}}{u_t}\left( t \right)} \right\|_2^2} \right) + \int_\Omega {G\left( {u\left( t \right)} \right)dx} }\\ {}&{ \le E\left( 0 \right){{\left( {\frac{{1 + q}}{{1 + q\gamma t}}} \right)}^{1/q}} \le {C_1}{{\left( {1 + t} \right)}^{ - 1/q}}.} \end{array} This is the estimates (2.8) and the proof of Theorem 7 is completed.

Conclusion 8.In this paper, we obtained the global attractor and the asymptotic behavior of global solution for the higher-order evolution equation with damping term. This improves and extends many results in the literature such as (Xie and Zhong (2007); Chen et al. (2011)).

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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Global Attractors for the Higher-Order Evolution Equation

Published Online: Mar 31, 2020

Page range: 195 - 210

Received: Apr 30, 2019

Accepted: Jun 18, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00018

KeywordsGlobal attractor, existence, asymptotic behavior

© 2020 Erhan Pişkin et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Global attractor, existence, asymptotic behavior