In recent years, there have been extensive studies on well-posedness of the following nonlinear variational wave equation with general data:

∂ t 2 u - c ( u ) ∂ x c ( u ) ∂ x u = 0 in ( 0 , ∞ ) × R , u | t = 0 = u 0 on R , ∂ t u | t = 0 = u 1 on R ,

where c(·) is given smooth, bounded, and positive function with c'(·) ≥ 0 and c'(u ₀) > 0,u ₀ ∈ H ¹(R),u ₁(x) ∈ L ²(R). Equation (1.1) appears naturally in the study for liquid crystals 1 2 3 4 . In addition, Chang et al. 5 , Su 6 and Kian 7 discussed globally Lipschitz continuous solutions to a class one dimension quasilinear wave equations

u t t - p ρ ( x ) , u x x = ρ ( x ) h ρ ( x ) , u , u x , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = ω 0 ( x ) ,

where (x,t) ∈ R × R ⁺, u ₀(x),ω ₀(x) ∈ R. Furthermore, Nishihara 8 and Hayashi 9 obtained the global solution to one dimension semilinear damped wave equation

u t t + u t - u x x = f ( u ) , ( t , x ) ∈ R + × R + ( u , u t ) ( 0 , x ) = ( u 0 , u 1 ) ( x ) .

Ikehata 10 and Vitillaro 11 proved global existence of solutions for semilinear damped wave equations in R ^N with noncompactly supported initial data or in the energy space, in where the nonlinear term f(u) = |u| ^p or f(u) = 0 is too special; some authors 12 13 14 discussed the regularity of invariant sets in semilinear wave equation, but they didn't refer to any the initial value condition of it. Unfortunately, it is difficulty to classify a class wave operator equations, since the differential operator structure is too complex to identify whether have variational property. Our aim is to classify a class of nonlinear damped wave operator equations in order to research them more extensively and go beyond the results of 12 .

In this article, we are interested in the existence of global solutions of the following nonlinear damped wave operator equations:

d 2 u d t 2 + k d u d t = G ( u ) , k > 0 u ( x , 0 ) = φ ( x ) , u t ( x , 0 ) = ψ ( x ) ,

where G : X 2 × R + → X 1 * is a mapping, X ₂ ⊂ X ₁, X ₁, X ₂ are Banach spaces and X 1 * is the dual spaces of X ₁, R ⁺ = [0, ∞), u = u(x,t). If k > 0, (1.4) is called damped wave equation. We obtain the existence, uniformly bounded and regularity of solutions by dividing the differential operator G(u) into two parts, variational and non-variational structure.

First we introduce a sequence of function spaces:

X ⊂ H 2 ⊂ X 2 ⊂ X 1 ⊂ H , X 2 ⊂ H 1 ⊂ H ,

where H, H ₁, H ₂ are Hilbert spaces, X is a linear space, X ₁, X ₂ are Banach spaces and all inclusions are dense embeddings. Suppose that

L : X → X 1 is one to one dense linear operator , L u , v H = u , v H , ∀ u , v ∈ X .

In addition, the operator L has an eigenvalue sequence

L e k = λ k e k , ( k = 1 , 2 , . . . )

such that {e _k } ⊂ X is the common orthogonal basis of H and H ₂. We investigate the existence of global solutions of the Equation (1.4), so we need define its solution. Firstly, in Banach space X, introduce

L p ( ( 0 , T ) , X ) = u : ( 0 , T ) → X | ∫ 0 T u p d t < ∞ ,

where p = (p ₁, p ₂,..., p _m ),p _i ≥ 1(1 ≤ i ≤ m),

u p = ∑ k = 1 m u k p k ,

where | · | _k is semi-norm in X, and ⋅ X = ∑ i = 1 m ⋅ i . Similarily, we can define

W 1 , p ( ( 0 , T ) , X ) = u : ( 0 , T ) → X | u , u ′ ∈ L p ( ( 0 , T ) , X ) .

Let L loc p ( ( 0 , ∞ ) , X ) = u ( t ) ∈ X | u ∈ L p ( ( 0 , T ) , X ) , ∀ T > 0 . .

Definition 2.1. Set (φ, ψ) ∈ X ₂ × H ₁, u ∈ W loc 1 , ∞ 0 , ∞ , H 1 ⋂ L loc ∞ ( ( 0 , ∞ ) , X 2 ) is called a globally weak solution of (1.4), if for ∀v ∈ X ₁, it has

u t , v H + k u , v H = ∫ 0 t G u , v d t + k φ , v H + ψ , v H .

Definition 2.2. Let Y ₁,Y ₂ be Banach spaces, the solution u(t, φ, ψ) of (1.4) is called uniformly bounded in Y ₁ × Y ₂, if for any bounded domain Ω₁ × Ω₂⊂Y ₁ × Y ₂, there exists a constant C which only depends the domain Ω₁ × Ω₂, such that

u Y 1 + u t Y 2 ≤ C , ∀ ( φ , ψ ) ∈ Ω 1 × Ω 2 and t ≥ 0 .

Definition 2.3. A mapping G : X 2 → X 1 * is called weakly continuous, if for any sequence {u _n } ⊂ X ₂, u _n ⇀ u ₀ in X ₂,

lim n → ∞ G ( u n ) , v = G ( u 0 ) , v , ∀ v ∈ X 1 .

Lemma 2.1. 15 Let H ₂, H be Hilbert spaces, and H ₂ ⊂ H be a continuous embedding. Then there exists a orthonormal basis {e _k } of H, and also is one orthogonal basis of H ₂.

Proof. Let I : H ₂ → H be imbedded. According to assume I is a linear compact operator, we define the mapping A : H ₂ → H as follows

A u , v H 2 = I u , v H = u , v H , ∀ v ∈ H 2 .

obviously, A : H ₂ → H ₂ is linear symmetrical compact operator and positive definite. Therefore, A has a complete eigenvalue sequence {λ _k } and eigenvector sequence ẽ k ⊂ H 2 such that

A ẽ k = λ k ẽ k , k = 1 , 2 , . . . ,

and ẽ k is orthogonal basis of H ₂. Hence

ẽ i , ẽ j H = A ẽ i , ẽ j H 2 = λ i ẽ i , ẽ j H 2 = 0 , if i ≠ j .

it implies ẽ i is also orthogonal sequence of H. Since H ₂ ⊂ H is dense, ẽ i is also orthogonal sequence of H, so { e i } = ẽ i / ẽ i H is norm orthogonal basis of H. The proof is completed.

Now, we introduce an important inequality

Lemma 2.2. 16 (Gronwall inequality) Let x(t), y(t), z(t) be real function on [a, b], where x(t) ≥ 0,∀a ≤ t ≤ b, z(t) ∈ C[a, b], y(t) is differentiable on [a, b]. If the inequality as follows is hold

z ( t ) ≤ y ( t ) + ∫ a t x ( τ ) z ( τ ) d τ , a ≤ t ≤ b ,

then

z ( t ) ≤ y ( a ) e ∫ a t x ( s ) d s + ∫ a t e ∫ a t x ( τ ) d y d s d s .

Suppose that G = A + B : X 2 × R + → X 1 * . Throughout of this article, we assume that

(i) There exists a function F ∈ C ¹ : X ₂ → R ¹ such that

A u , L v = - D F ( u ) , v , ∀ u , v ∈ X

(ii) Function F is coercive, if

F ( u ) → ∞ ⇔ u X 2 → ∞

(iii) B as follows

B u , L v ≤ C 1 F ( u ) + C 2 v H 1 2 , ∀ u , v ∈ X

for some g ∈ L loc 1 ( 0 , ∞ ) .

Theorem 3.1. Set G : X 2 × R + → X 1 * is weakly continuous, (φ, ψ) ∈ X ₂ × H ₁, then we obtain the results as follows:

(1) If G = A satisfies the assumption (i) and (ii), then there exists a globally weak solution of (1.4)

u ∈ W loc 1 , ∞ ( 0 , ∞ ) , H 1 ⋂ L loc ∞ ( ( 0 , ∞ ) , X 2 )

and u is uniformly bounded in X ₂ × H ₁;

(2) If G = A + B satisfies the assumption (i), (ii) and (iii), then there exists a globally weak solution of (1.4)

u ∈ W loc 1 , ∞ ( ( 0 , ∞ ) , H 1 ) ⋂ L loc ∞ ( ( 0 , ∞ ) , X 2 ) ;

(3) Furthermore, if G = A + B satisfies

G u , v ≤ 1 2 v H 2 + C F ( u ) + g ( t )

for some g ∈ L loc 1 ( 0 , ∞ ) , then u ∈ W loc 2 , 2 ( 0 , ∞ ) , H .

Proof. Let {e _k } ⊂ X be the public orthogonal basis of H and H ₂, satisfies (2.3).

Note

X n = ∑ i = 1 n α i e i | α i ∈ R 1 , X ̃ n = ∑ j = 1 n β j ( t ) e j | β j ∈ C 2 0 , ∞ .

From the assumption, we know L X n = X n , L X n ̃ = X n ̃ , apply the Galerkin method to make truncate in X n ̃ :

d 2 u i d t 2 + k d u i d t = G ( u n ) , e i , 1 ≤ i ≤ n u i ( x , 0 ) = φ , e i H , u ′ i ( x , 0 ) = ψ , e i H

there exists u n = ∑ i = 1 n u i ( t ) e i ∈ C 2 ( 0 , ∞ ) , X n for any v ∈ X n ̃ satisfies

∫ 0 t d 2 u n d t 2 + k d u n d t , v H d t = ∫ 0 t G u n , v d t

for any v ∈ X _n , it yields that

d u n d t , v H + k u n , v H = ∫ 0 t G u n , v d t + k φ , v H + ψ , v H

(1) If G = A , u n ∈ X n ̃ substitute v = d d t L u n into (3.7), we get

∫ 0 t d 2 u n d t 2 + k d u n d t , d d t L u n H 1 d t = ∫ 0 t G u n , d d t L u n d t

combine condition (2.2) with (3.1), we get

∫ 0 t ∫ Ω d 2 u n d t 2 d u n d t d x d t + ∫ 0 t ∫ Ω k d u n d t d u n d t d x d t + ∫ 0 t D F ( u n ) d u n d t d x d t = 0 ∫ 0 t 1 2 d d t d u n d t H 1 2 d t + k ∫ 0 t d u n d t H 1 2 d t + ∫ 0 t d d t F ( u n ) d t = 0 1 2 d u n d t H 1 2 - 1 2 ψ n H 1 2 + k ∫ 0 t d u n d t H 1 2 d t + F ( u n ) - F ( φ n ) = 0

consequently, we get

F ( u n ) + 1 2 u ′ n H 1 2 + k ∫ 0 t u ′ n H 1 2 d t = F ( u n ) + 1 2 ψ n H 1 2 .

Assume φ ∈ H ₂, combine(2.2)with(2.3), we know {e _n } is also the orthogonal basis of H ₁, then φ _n → φ in H ₂, ψ _n → ψ in H ₁, owing to H ₂ ⊂ X ₂ is embedded, so

φ n → φ i n X 2 ψ n → ψ i n X 1

due to the condition (3.6), from (3.9)and (3.10) we easily know

{ u n } ⊂ W loc 1 , ∞ ( ( 0 , ∞ ) , H 1 ) ⋂ L loc ∞ ( ( 0 , ∞ ) , X 2 ) i s b o u n d e d .

consequently, assume that

u n ⇀ u 0 i n W loc 1 , ∞ ( ( 0 , ∞ ) , H 1 ) ⋂ L loc ∞ ( ( 0 , ∞ ) , X 2 ) a . e . t > 0

i.e. u _n ⇀ u ₀ in X ₂ a.e. t > 0, and G is weakly continuous, so

lim n → ∞ G u n , v = G u 0 , v .

By (3.8), we have

lim n → ∞ d u n d t H + k u n , v H = lim n → ∞ ∫ 0 t G u n , v d t + k φ , v H + ψ , v H d u 0 d t , v H + k u 0 , v H = ∫ 0 t G u 0 , v d t + k φ , v H + ψ , v H

it indicates for any v ∈ ⋃ n = 1 ∞ X n ⊂ X 2 , it holds. Hence, for any v ∈ X ₂, we have

d u 0 d t , v H + k u 0 , v H = ∫ 0 t G u 0 , v d t + k φ , v H + ψ , v H .

Consequently, u ₀ is a globally weak solution of (1.4).

Furthermore, by (3.9) and (3.10), for any R > 0, there exists a constant C such that if

φ X 2 + ψ H 1 ≤ R

then the weak solution u(t, φ, ψ) of (1.4) satisfies

u ( t , φ , ψ ) X 2 + u t ( t , φ , ψ ) H 1 ≤ C . ∀ t ≥ 0

Assume (φ,ψ) ∈ X ₂ × H ₁ satisfies (3.12), by H ₂ ⊂ X ₂ is dense. May fix φ _n ∈ H ₂ such that

φ n X 2 + ψ H 1 ≤ R , lim n → ∞ φ n = φ in X 2

by (3.13), the solution {u(t, φ _n , ψ)} of (1.4) is bounded in W loc 1 , ∞ ( 0 , ∞ ) , H 1 ⋂ L loc ∞ ( ( 0 , ∞ ) , X 2 ) a.e. t > 0.

Therefore, assume u(t, φ _n , ψ) ⇀ u in W loc 1 , ∞ ( 0 , ∞ ) , H 1 ⋂ L loc ∞ ( ( 0 , ∞ ) , X 2 ) then u(t) is a weak solution of (1.4), it satisfies uniformly bounded of (3.13). So the conclusion (1) is proved.

(2) If G = A + B , u n ∈ X n ̃ , substitute v = d d t L u n into (3.7), we get

∫ 0 t d 2 u n d t 2 , d d t L u n H 1 + k d u n d t , d d t L u n 1 H 1 d t = ∫ 0 t A u n , d u n d t + B u n , d u n d t d t

combine the condition (2.2) and (3.1), we have

∫ 0 t ∫ Ω d 2 u n d t 2 d u n d t d x d t + k ∫ 0 t ∫ Ω d u n d t 2 d u n d t d x d t + ∫ 0 t D F ( u n ) d u n d t d t = ∫ 0 t B u n , d u n d t d t ∫ 0 t 1 2 d d t d u n d t H 1 2 d t + k ∫ 0 t d u n d t H 1 2 d t + ∫ 0 t d d t F ( u n ) d t = ∫ 0 t B u n , d u n d t d t 1 2 u ′ n H 1 2 - 1 2 ψ n H 1 2 + k ∫ 0 t u ′ n H 1 2 d t + F ( u n ) + F ( φ n ) = ∫ 0 t B u n , d u n d t d t

consequently, we have

F ( u n ) + 1 2 u ′ n H 1 2 + k ∫ 0 t u ′ n H 1 2 d t = ∫ 0 t B u n , d u n d t d t + F ( φ n ) + 1 2 ψ n H 1 2

by the condition (3.3),(3.14)implies

F ( u n ) + 1 2 u ′ n H 1 2 ≤ C ∫ 0 t F ( u n ) + 1 2 u ′ n H 1 2 d t + f ( t )

where f ( t ) = ∫ 0 t g ( τ ) d t + 1 2 ψ H 1 2 + sup n F ( φ n ) . .

by Gronwall inequality [Lemma(2.2)], from (3.15) we easily know:

F ( u n ) + 1 2 u ′ n H 1 2 ≤ f ( 0 ) e C t + ∫ 0 t f ′ ( τ ) e C ( t - τ ) d τ

it implies that, for any 0 < T < ∞

{ u n } ⊂ W 1 , ∞ ( 0 , T ) , X 2 ⋂ L ∞ 0 , T , X 2 is bounded .

now, use the same way as (1), we can obtain the result (2).

(3) If the condition (3.4) is hold, u n ∈ X n ̃ , substitute v = d 2 u d t 2 into (3.7), we can get

∫ 0 t d 2 u n d t 2 , d 2 u n d t 2 H + k d u n d t , d 2 u n d t 2 H d t = ∫ 0 t G u n , d 2 u n d t 2 d t

then

∫ 0 t d 2 u n d t 2 , d 2 u n d t 2 H d t + k 2 ∫ 0 t d d t u ′ n ( t ) H 2 d t ≤ ∫ 0 t 1 2 u ″ n ( t ) H 2 + C F ( u n ) + g ( t ) d t ∫ 0 t d 2 u n d t 2 , d 2 u n d t 2 H d t + k 2 u ′ n H 2 ≤ k 2 ψ n H 2 + ∫ 0 t 1 2 d 2 u n d t 2 H 2 + C F ( u n ) + g ( τ ) d τ

by (3.16), it implies that

∫ 0 t d 2 u n d t 2 H 2 d τ ≤ C , ( C > 0 )

consequently, for any 0 < T < ∞

{ u n } ⊂ W 2 , 2 ( ( 0 , T ) , H ) is bounded .

it implies that u ∈ W ^2,2((0,T), H), the main theorem (3.1) has been proved.

The authors declare that they have no competing interests.

All authors typed, read and approved the final manuscript.