Abstract
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 nonvariational structure, we obtain the existence, uniformly bounded and regularity of solutions.
Mathematics Subject Classification 2000: 35L05; 35A01; 35L35.
Keywords:
nonlinear damped wave operator equations; global solutions; uniformly bounded; regularity1 Introduction
In recent years, there have been extensive studies on wellposedness of the following nonlinear variational wave equation with general data:
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 [14]. 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^{+}, u_{0}(x),ω_{0}(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 [1214] 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, X_{2 }⊂ X_{1}, X_{1}, X_{2 }are Banach spaces and 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 nonvariational structure.
2 Preliminaries
First we introduce a sequence of function spaces:
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
In addition, the operator L has an eigenvalue sequence
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
where p = (p_{1}, p_{2},..., p_{m}),p_{i }≥ 1(1 ≤ i ≤ m),
where  · _{k }is seminorm in X, and . Similarily, we can define
Definition 2.1. Set (φ, ψ) ∈ X_{2 }× H_{1}, is called a globally weak solution of (1.4), if for ∀v ∈ X_{1}, it has
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 Ω_{1 }× Ω_{2}⊂Y_{1 }× Y_{2}, there exists a constant C which only depends the domain Ω_{1 }× Ω_{2}, such that
Definition 2.3. A mapping is called weakly continuous, if for any sequence {u_{n}} ⊂ X_{2}, u_{n }⇀ u_{0 }in X_{2},
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
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 such that
and is orthogonal basis of H_{2}. Hence
it implies is also orthogonal sequence of H. Since H_{2 }⊂ 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
3 Main results
Suppose that . Throughout of this article, we assume that
(i) There exists a function F ∈ C^{1 }: X_{2 }→ R^{1 }such that
(ii) Function F is coercive, if
(iii) B as follows
Theorem 3.1. Set 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)
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)
(3) Furthermore, if G = A + B satisfies
Proof. Let {e_{k}} ⊂ X be the public orthogonal basis of H and H_{2}, 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 ∈ 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
due to the condition (3.6), from (3.9)and (3.10) we easily know
consequently, assume that
i.e. u_{n }⇀ u_{0 }in X_{2 }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 ∈ X_{2}, we have
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
then the weak solution u(t, φ, ψ) of (1.4) satisfies
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
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
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.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
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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