Global solutions to a class of nonlinear damped wave operator equations

This study investigates the existence of global solutions to a class of nonlinear damped wave operator equations. Dividing the differential operator into two parts, variational and non-variational structure, we obtain the existence, uniformly bounded and regularity of solutions.

Mathematics Subject Classification 2000: 35L05; 35A01; 35L35.

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

where c(·) is given smooth, bounded, and positive function with c'(·) ≥ 0 and c'(u₀) > 0,u₀ ∈ H¹(R),u₁(x) ∈ L²(R). Equation (1.1) appears naturally in the study for liquid crystals [1-4]. In addition, Chang et al. [5], Su [6] and Kian [7] discussed globally Lipschitz continuous solutions to a class one dimension quasilinear wave equations

where (x,t) ∈ R × R⁺, u₀(x),ω₀(x) ∈ R. Furthermore, Nishihara [8] and Hayashi [9] obtained the global solution to one dimension semilinear damped wave equation

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 [12-14] 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:

where is a mapping, X₂ ⊂ X₁, X₁, X₂ are Banach spaces and is the dual spaces of X₁, 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:

where H, H₁, H₂ are Hilbert spaces, X is a linear space, X₁, X₂ are Banach spaces and all inclusions are dense embeddings. Suppose that

In addition, the operator L has an eigenvalue sequence

such that {e_k} ⊂ X is the common orthogonal basis of H and H₂. We investigate the existence of global solutions of the Equation (1.4), so we need define its solution. Firstly, in Banach space X, introduce

where p = (p₁, p₂,..., p_m),p_i ≥ 1(1 ≤ i ≤ m),

where | · |_k is semi-norm in X, and . Similarily, we can define

Let .

Definition 2.1. Set (φ, ψ) ∈ X₂ × H₁, is called a globally weak solution of (1.4), if for ∀v ∈ X₁, it has

Definition 2.2. Let Y₁,Y₂ be Banach spaces, the solution u(t, φ, ψ) of (1.4) is called uniformly bounded in Y₁ × Y₂, if for any bounded domain Ω₁ × Ω₂⊂Y₁ × Y₂, there exists a constant C which only depends the domain Ω₁ × Ω₂, such that

Definition 2.3. A mapping is called weakly continuous, if for any sequence {u_n} ⊂ X₂, u_n ⇀ u₀ in X₂,

Lemma 2.1. [15]Let H₂, H be Hilbert spaces, and H₂ ⊂ H be a continuous embedding. Then there exists a orthonormal basis {e_k} of H, and also is one orthogonal basis of H₂.

Proof. Let I : H₂ → H be imbedded. According to assume I is a linear compact operator, we define the mapping A : H₂ → H as follows

obviously, A : H₂ → H₂ is linear symmetrical compact operator and positive definite. Therefore, A has a complete eigenvalue sequence {λ_k} and eigenvector sequence such that

and is orthogonal basis of H₂. Hence

it implies is also orthogonal sequence of H. Since H₂ ⊂ H is dense, is also orthogonal sequence of H, so 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

then

Suppose that . Throughout of this article, we assume that

(i) There exists a function F ∈ C¹ : X₂ → R¹ such that

(ii) Function F is coercive, if

(iii) B as follows

for some .

Theorem 3.1. Set is weakly continuous, (φ, ψ) ∈ X₂ × H₁, 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)

and u is uniformly bounded in X₂ × H₁;

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

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

for some , then .

Proof. Let {e_k} ⊂ X be the public orthogonal basis of H and H₂, satisfies (2.3).

Note

From the assumption, we know , apply the Galerkin method to make truncate in :

there exists for any satisfies

for any v ∈ X_n, it yields that

(1) If substitute into (3.7), we get

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

consequently, we get

Assume φ ∈ H₂, combine(2.2)with(2.3), we know {e_n} is also the orthogonal basis of H₁, then φ_n → φ in H₂, ψ_n → ψ in H₁, owing to H₂ ⊂ X₂ is embedded, so

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

consequently, assume that

i.e. u_n ⇀ u₀ in X₂ a.e. t > 0, and G is weakly continuous, so

By (3.8), we have

it indicates for any , it holds. Hence, for any v ∈ X₂, we have

Consequently, u₀ 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

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

Assume (φ,ψ) ∈ X₂ × H₁ satisfies (3.12), by H₂ ⊂ X₂ is dense. May fix φ_n ∈ H₂ such that

by (3.13), the solution {u(t, φ_n, ψ)} of (1.4) is bounded in a.e. t > 0.

Therefore, assume u(t, φ_n, ψ) ⇀ u in 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 , substitute into (3.7), we get

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

consequently, we have

by the condition (3.3),(3.14)implies

where .

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

it implies that, for any 0 < T < ∞

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

(3) If the condition (3.4) is hold, , substitute into (3.7), we can get

then

by (3.16), it implies that

consequently, for any 0 < T < ∞

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.

The author is very grateful to the anonymous referees whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. Foundation item: the National Natural Science Foundation of China (No. 10971148).

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