Definition 1. Let $$ and $$ satisfy the hypotheses (I)-(IV). For every τ ∈ R, a function u ∈ L²(τ, T; V_g)∩L^∞(τ, T; H_g), ∀T > τ is called a weak solution of problem (1) if it fulfils

Theorem 2. Let $$ satisfies the assumptions (I)-(IV), there exists a unique solution

such that (6) and (7) holds.

Proof. We apply the Faedo-Galerkin methods. Since V_g is separable and ℵ is dense in V_g, there exists a sequence $$, which forms a complete orthonormal system in H_g and a basic for V_g. Let m be a positive integer, for each m, we define an approximate solution u_m of (6) as $$, which satisfies

for t ∈ [0, T], j = 1, ⋯, m and u_m(0) = u_0m, where u_0m is the orthegonal projection in H_g of u₀ onto the space spanned by w₁, ⋯, w_m. Then, we can obtain

We can write the differential equations in the usual form

where α_ij, α_ijk, β_ij ∈ R.

Let ϕ_im(0) be the ith component of u_0m. The nonlinear ordinary differential system (18) has a maximal solution defined on some interval [0, t_m]. If t_m < T, then |u_m(t)|⟶∞ as t⟶t_m. The following we will prove t_m = T. We need several estimates to do.

We multiply (16) by ϕ_jm(t) and add these equations for j = 1, ⋯, m to obtain

Then, we have

so that

Let $$, then

By the Gronwall inequality, we have

Hence,

which implies that the sequence u_m remains in bounded set of L^∞(0, T; H_g). From (22), we have

Then,

We intergrate (26) from 0 to T; we have

So, the u_m remains in a bounded set of L²(0, T; V_g).

Let $$ denotes the function from R into V_g, which is equal to u_m on [0, T] and to 0 on the complement of this interval. The Fourier transform of $$ is denoted by $$. Then, we will show that there exist a positive constant C and γ such that

Since the u_m remains in a bounded set of L²(0, T; V_g), the $$ remains in a bounded set of H^γ(R; V_g, H_g). Since $$ has two discontinuities at 0 and T, the distribute derivative of $$ is given by

where δ₀ and δ_T are the dirac distributions at 0 and T, and $$ is the derivative of u_m on [0, T]. We obtain that for j = 1, ⋯, m, where δ₀ and δ_T are distributions at 0 and T, f_m = f + h − νAu_m − bu_m − νRu_m − αu_m and $$ on [0, T]. By the Fourier transform, we have where $$ and $$ denoting the Fourier transforms of $$ and $$, respectively. We multiply (31) by $$ and add the resulting equations for j = 1, ⋯, m; we get

We obtain

So, f_m(t) belongs to a bounded set in the space $$. For ∀m, we have $$ Since u_m(0)| and |u_m(T)| are bounded, from (31), we obtain

Let γ < (1/4), we have

then

Since u_m ∈ L²(0, T; V_g), by the Parseval equality $$ and by the Schwarz inequality and the Parseval equality, we obtain

So, u_m ∈ H^γ(R; V_g, H_g), and u_m remains in a bounded set of L^∞(0, T; H_g), L²(0, T; V_g) and H^γ(R; V_g, H_g). There exists an element u ∈ L²(0, T; V_g)∩L^∞(0, T; H_g) and a subsequence $$ such that $$ in L²(0, T; V_g) weakly and $$ in L^∞(0, T; V_g) weak-star as m^′⟶∞. For any $$, we have $$ strongly in $$.

For any support $$ of w_j, we have $$ strongly in $$. Let ψ be a continuously differentiable function on [0, T] with ψ(T) = 0, we multiply (16) by ψ(t), then integrate by parts,

We have

where ∀v ∈ V_g.

Finally, we prove that u satisfies (7). We multiply (6) by ψ and integrate

We compare (39) with (40) to obtain (u(0) − u₀, v)ψ(0) = 0. Let ψ(0) = 1, then we have (u(0) − u₀, v) = 0, ∀v ∈ V_g. So, u(0) = u₀.

Now, we will prove the solution of (6) and (7) is unique. We let u₁ and u₂ be the solutions of (9) and u = u₁ − u₂. We have

We take the scalar product of (41) with u(t), then

Therefore,

Then,

We have

Hence, |u(t)|² = 0, ∀t ∈ [0, T]. So, u₁ = u₂.

From [15], we can define a family of two parametric maps {U_f(t, τ)} = {U_f(t, τ) | t ≥ τ, τ ∈ R} in H_g,

Here, $$ is called the time symbol of the system. We have the following concepts and conclusions from [15].

Definition 3. For the given time symbol $$, a family of two-parametric maps {U(t, τ)} with t ≥ τ ≥ 0 is called a process in H_g, if

Now, we define translation operator in $$. $$.

We have

Denote Σ = {T(h)f(x, s) = f(x, s + h), ∀h ∈ R}, where T(·) is the positive invariant semigroups acting on Σ and satifying T(h)Σ ⊂ Σ, ∀h ≥ 0 and

Let $$ be a constant, obviously

Let E be the Banach space; we use $$ to denote the set of all bounded sets on E and consider a family of processes {U_f(t, τ)} with f ∈ Σ, the parameter f is called the symbols of the process family {U_f(t, τ)}, Σ is called the symbol space, and we assume that Σ is a complete metric space.

Definition 4. A family of processes {U_f(t, τ)}, f ∈ Σ is called uniformly bounded (w.r.t.f ∈ Σ), if any $$, both

Definition 5. A set B₀ ⊂ E is said to be uniformly absorbing for the family of processes {U_f(t, τ)}, f ∈ Σ}, if for any τ ∈ R and each $$, there exists t₀ = t₀(τ, B) ≥ τ, such that for all t ≥ t₀,

Definition 6. A set P ⊂ E is said uniformly atttracting set of {U_f(t, τ)}, f ∈ Σ}, if for any τ ∈ R, there is

A family of processes {U_f(t, τ)}, f ∈ Σ} is said to uniformly compact, if there exists a compacted uniformly absorbed set in {U_f(t, τ)}, f ∈ Σ}. A family of processes {U_f(t, τ)}, f ∈ Σ} is said to uniformly asymptotic compact, if there exists a compacted uniformly atttracting set in {U_f(t, τ)}, f ∈ Σ}.

Definition 7. A closed set $$ is said to be the uniform attractor of the family of processes {U_f(t, τ)}, f ∈ Σ}, if

• (1)

  $$ is uniformly attractive

• (2)

  $$is included in any uniformly attracting set of {U_f(t, τ)}, f ∈ Σ}, that is $$

Theorem 8. Let {f_γ(θ): γ ∈ Γ} ⊂ C = C([−r, 0]; X) be equicontinuous and for any ∀θ ∈ [−r, 0], {f_γ(θ): γ ∈ Γ} is quasicompact in X, then {f_γ(θ): γ ∈ Γ} is relatively compact in C([−r, 0]; X).

Lemma 9 (Uniform Gronwall lemma). Let g, h, y be local integrable function on (t₀, ∞), y^′ is also local integrable on (t₀, ∞), and y^′(t) ≤ g(t)y(t) + h(t), ∀t ≥ t₀.$$, where r, a₁, a₂, a₃ is positive constant. Then,

Lemma 10. Let for any τ ≤ t, $$, assume that (I)-(IV) hold, then there exist bounded absorbing sets $$ of process family {U_f(t, τ): t ≥ τ} in $$.

Proof. Since $$ is bounded, then there exists $$, such that

For any $$, we define u(·) = u(·; τ, (u₀, ϕ)), then taking the inner product of (9) with u(t), we have

Let $$, then

Integrating both sides from τ to t, then

Then,

for

Let s − τ = θ, then

Taking m ∈ (0, m₀), such that m + σ + C_g − 2νλ₁β < 1, then $$, so

Let t ≥ τ + h, ∀θ ∈ [−h, 0], then

Then,

Let $$, we will prove the existence of the uniformly absorbing bounded set in $$. First, we must prove the boundedness of $$.

Lemma 11. Given that $$, then there exist $$ and constant $$, such that

where $$ denotes any bounded set on the $$.

Proof. Taking the inner product of (9) with u(t),

Then,

Integrating on both sides in [t, t + 1], we have

Then,

When $$, that is $$, we have

where

Lemma 12. For any τ ≤ t, $$. Assume that (I)-(IV) hold, then there exists uniformly bounded absorbing set $$ of process family {U_f(t, τ): t ≥ τ} in $$.

Proof. Let $$, taking the inner product of (9) with A_gu, we obtain

for

Then,

Applying Lemma 9,

where $$$$$$ If taking $$, then

Let u(·) = u(·; t − s, (u₀, ϕ)), so $$. Then,

Definition 13. Let E be Banach space, if ∀ε > 0, there exists η > 0, such that

Then, $$ is called normal function.

Theorem 14. Suppose that nonlinear term h satisfies (I)-(III), f is translation compact function in $$, then process family {U_f(·, ·) | f ∈ Σ} exist uniform attractor $$, and $$.

Proof. Since B₂ is bounded set in $$ and uniform absorbed set of {U_f(·, ·) | f ∈ Σ}. For each τ ∈ R, we take a set

where j denotes any compact self-adjoint operator, then B₃ ⊂ B₂ ⊂ B₁, and B₃ is another uniform absorbing bounded set of {U_f(·, ·) | f ∈ Σ} in $$. Now, we will prove B₃ is relatively compact in $$. From Theorem 8, we only need to prove B₃ is equicontinuous and uniform bounded in $$. From the definition of B₃, we can obtain it is uniformly bounded. Now, we will prove B₃ is equicontinuous. For any θ₁, θ₂ ∈ [−r, 0], ϕ ∈ B₂, f ∈ Σ,

Let θ₂ > θ₁, and denote u(·) = u(·; τ, j(ϕ), f) as u(·), then

We estimate the items on the right end of the above formula, let θ₁⟶θ₂,

Since

we let $$$$

When $$, and θ₂ > θ₁, then

Let

Then,

So,

And $$ When θ₁⟶θ₂, ∀ϕ ∈ B₂, f ∈ Σ, we have

Then, B₃ is equicontinuous, and B₃ is relatively compact in $$, so $$ is compacted uniformly absorbing set of {U_f(·, ·) | f ∈ Σ} in $$, Let $$, since $$ and the embedding mapping is continuous, so $$ is compact in $$; $$ is also compacted uniformly absorbing set of {U_f(·, ·) | f ∈ Σ} in $$. Then, process family {U_f(·, ·) | f ∈ Σ} exists uniform attractor $$