Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

(x 1 , . . . , x n ) ∈ Rn and t > 0, k 1 > 0 and k 2 > 0 are constants. The nonlinear termf(u) = O(u1+θ) and θ is a positive integer. Equation (1) is a model in variational form for the neoHookean elastomer rod and describes the motion of a neoHookean elastomer rod with internal damping; for more detailed physical background, we refer to [1]. In [1], the authors have studied a general class of abstract evolution equations


Introduction
We investigate the Cauchy problem for the following damped nonlinear hyperbolic equation: with the initial value  = 0 :  =  0 () ,   =  1 () . ( Here  = (,) is the unknown function of  = ( 1 , . . .,   ) ∈ R  and  > 0,  1 > 0 and  2 > 0 are constants.The nonlinear term () = ( 1+ ) and  is a positive integer.Equation ( 1) is a model in variational form for the neo-Hookean elastomer rod and describes the motion of a neo-Hookean elastomer rod with internal damping; for more detailed physical background, we refer to [1].In [1], the authors have studied a general class of abstract evolution equations where  1 ,  2 , , and  satisfy certain assumptions.For quite general conditions on the nonlinear term, global existence, uniqueness, regularity, and continuous dependence on the initial value of a generalized solution to (3) in a bounded domain of R  were obtained.Equation (1) fits the abstract framework of [1].The local well-posedness for the Cauchy problem for (1), (2) in three-dimensional space was obtained by Chen and Da [2].More precisely, they proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that  0 ∈  6 (R 3 ),  1 ∈  4 (R 3 ).
Local existence and uniqueness of classical solutions to (1), (2) were also established, provided that  0 ∈  12 (R 3 ),  1 ∈  10 (R 3 ).Their method is to first establish local-intime well-posedness of a periodic version of (1), (2) and then construc a solution to (1), (2) as a limit of periodic solutions with divergent periods.This paper also arrived at some sufficient conditions for blow-up of the solution in finite time, and an example was given.Song and Yang [3] studied the existence and nonexistence of global solutions to the Cauchy problem for (1) in one-dimensional space.
The main purpose of this paper is to establish global existence and asymptotic behavior of solutions to (1), (2) by using the contraction mapping principle.Firstly, we consider the decay property of the following linear equation: We obtain the following decay estimate of solutions to (4), ( Based on the above estimates, we define a solution space with time weighted norms, and then global existence and asymptotic behavior of solutions to (1), ( 2) are obtained by using the contraction mapping principle.More precisely, we prove global existence and the following decay estimate of solution to (1), ( 2): for  ≤  + 4,  ≤ , and  ≥ [/2] + 1.
is assumed to be suitably small.When  = 3, our result allows for the initial data  0 ∈  6 (R 3 ),  1 ∈  2 (R 3 ).But in [2], the authors proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that  0 ∈  6 (R 3 ),  1 ∈  4 (R 3 ), so our result improves the regularity of the initial condition for the time derivative.This improvement is due to the strong damping term Δ 2   since the strong damping term Δ 2   has stronger dissipative effect than the damping   .The stronger dissipative effect has been exhibited in the study of the strongly damped wave equation and related problems; see, for instance, [9].
We give some notations which are used in this paper.Let F[] denote the Fourier transform of  defined by and we denote its inverse transform by F −1 .For 1 ≤  ≤ ∞,   =   (R  ) denotes the usual Lebesgue space with the norm ‖ ⋅ ‖   .The usual Sobolev space of  is defined by    = ( − Δ) Finally, in this paper, we denote every positive constant by the same symbol  or  without confusion.[⋅] is the Gauss symbol.
The paper is organized as follows.In Section 2 we derive the solution formula of our semilinear problem.We study the decay property of the solution operators appearing in the solution formula in Section 3.Then, in Section 4, we discuss the linear problem and show the decay estimates.Finally, we prove global existence and asymptotic behavior of solutions for the Cauchy problem (1), (2) in Section 5.

Decay Property
The aim of this section is to establish decay estimates of the solution operators () and () appearing in the solution formula (15).for  ∈ R  and  ≥ 0, where Proof.Multiplying ( 8) by û and taking the real part yield Multiplying ( 8) by û and taking the real part, we obtain Multiplying both sides of ( 19) and ( 20) by 2 and  1 || 4 and summing up the resulting equation yield where A simple computation implies that where Note that It follows from (23) that Using ( 21) and (26), we get Thus which together with (23) proves the desired estimates (17).
Then we have completed the proof of the lemma.
where  0 is a small positive constant in Lemma 1. Thus (38) follows.Similarly, we can prove (39).Thus we have completed the proof of the lemma.

Global Existence and Asymptotic Behavior
The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem ( 1), (2).We need the following lemma, which comes from [24] (see also [25]).
Based on the estimates (42)-( 44) of solutions to the linear problem (4), (2), one defines the following solution space: where For  > 0, one defines where  depends on the initial value, which is chosen in the proof of main result.For ℎ ≤ −[/2]+3, using Gagliardo-Nirenberg inequality, one obtains ] + 1), and integer  ≥ 1. () satisfies the assumptions of Lemmas 5 and 6.Put If  0 is suitably small, the Cauchy problem (1)-( 2) has a unique global classical solution (, ) satisfying Moreover, the solution satisfies the decay estimate for  ≤  + 4 and  ≤ .