Journal of Applied Mathematics

Volume 2013 (2013), Article ID 353757, 8 pages

http://dx.doi.org/10.1155/2013/353757

Research Article

Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

Yu-Zhu Wang

School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

Received 17 December 2012; Revised 15 May 2013; Accepted 30 May 2013

Academic Editor: Roberto Natalini

Copyright © 2013 Yu-Zhu Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider the Cauchy problem for the damped nonlinear hyperbolic equation in n-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

1. Introduction

We investigate the Cauchy problem for the following damped nonlinear hyperbolic equation:

with the initial value

Here is the unknown function of and , and are constants. The nonlinear term 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 , , , 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 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 , . Local existence and uniqueness of classical solutions to (1), (2) were also established, provided that , . Their method is to first establish local-in-time 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 boundary value problems for (1) are investigated (see [4, 5]). Equation (1) is a fifth-order wave equation. For more higher order wave equations, we refer to [6–8] and references therein.

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), (2)

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 , , and . Here , , and is assumed to be suitably small. When , our result allows for the initial data , . But in [2], the authors proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that , , so our result improves the regularity of the initial condition for the time derivative. This improvement is due to the strong damping term since the strong damping term 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].

The global existence and asymptotic behavior of solutions to hyperbolic-type equations have been investigated by many authors. We refer to [10–15] for hyperbolic equations, [16–21] for damped wave equation, and [22, 23] for various aspects of dissipation of the plate equation.

We give some notations which are used in this paper. Let denote the Fourier transform of defined by

and we denote its inverse transform by .

For , denotes the usual Lebesgue space with the norm . The usual Sobolev space of is defined by with the norm ; the homogeneous Sobolev space of is defined by with the norm ; especially , . Moreover, we know that for .

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.

2. Solution Formula

The aim of this section is to derive the solution formula for the problem (1), (2). We first investigate (4). Taking the Fourier transform, we have

The corresponding initial value is given as

The characteristic equation of (8) is

Let be the corresponding eigenvalues of (10), and we obtain

The solution to the problem (8)-(9) is given in the form

where

We define and by and , respectively, where denotes the inverse Fourier transform. Then, applying to (12), we obtain

By the Duhamel principle, we obtain the solution formula to (1), (2)

3. Decay Property

The aim of this section is to establish decay estimates of the solution operators and appearing in the solution formula (15).

Lemma 1. The solution of the problem (8), (9) satisfies

for and , where

Proof. Multiplying (8) by and taking the real part yield

Multiplying (8) by and taking the real part, we obtain

Multiplying both sides of (19) and (20) by and 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.

Lemma 2. Let and be the fundamental solution of (4) in the Fourier space, which are given in (13) and (14), respectively. Then one has the estimates

for and , where

Proof. If , from (12), we obtain

Substituting the equalities into (17) with , we get (29).

In what follows, we consider , and it follows from (12) that

Substituting the equalities into (17) with , we get the desired estimate (30). The lemma is proved.

Lemma 3. Let and be nonnegative integer. Then one has

Here in (34) and in (38).

Proof. By the Plancherel theorem and (29), Hausdorff-Young inequality, we obtain

Here and is a small positive constant in Lemma 1. Thus (34) follows.

Similarly, using (29) and (30), respectively, we can prove (35)–(37).

In what follows, we prove (38). By the Plancherel theorem, (29), and Hausdorff-Young inequality, we have

where 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.

4. Decay Estimate of Solutions to (4), (2)

Theorem 4. Assume that , . Then the classical solution to (4), (2), which is given by the formula (15), satisfies the decay estimate

Proof. Firstly, we prove (42). Using (34) and (35), for , we obtain

For , it follows from (36) and (37) that

Equation (44) follows from (42) and Gagliardo-Nirenberg inequality. The lemma is proved.

5. 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]).

Lemma 5. Let and be positive integers, , ,  , satisfy , and let . Assume that is class of and satisfies

If and , then for one has

Lemma 6. Let and be positive integers, let , , , satisfy , and let . Let be a function that satisfies the assumptions of Lemma 5. Moreover, assume that

If , and , , then for , one has

Based on the estimates (42)–(44) of solutions to the linear problem (4), (2), one defines the following solution space:

where

For , one defines

where depends on the initial value, which is chosen in the proof of main result.

For , using Gagliardo-Nirenberg inequality, one obtains

Theorem 7. Assume that , , and integer . satisfies the assumptions of Lemmas 5 and 6. Put

If is suitably small, the Cauchy problem (1)-(2) has a unique global classical solution satisfying

Moreover, the solution satisfies the decay estimate

for and .

Proof. Define the mapping

Using (34)-(35), (38), Lemma 5, and (54), for , we obtain

Thus

It follows from (58) that

Using (36)-(37), (39), Lemma 5, and (54), for , we have

Thus

Combining (60), (63) and taking and suitably small yield

For , , by using (58), we have

Exploiting (65), (38) Lemma 6, and (54), for , we obtain

which implies

Similarly for , from (61), (39), and (54), we have

which implies

Noting , by using (67), (69) and taking suitably small, yields

From (64) and (70), we know that is strictly contracting mapping. Consequently, we conclude that there exists a fixed point of the mapping , which is a classical solution to (1), (2). We have completed the proof of Theorem 7.

Acknowledgments

This work was supported in part by the NNSF of China (Grant no. 11101144) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province.

References

H. T. Banks, D. S. Gilliam, and V. I. Shubov, “Global solvability for damped abstract nonlinear hyperbolic systems,” Differential and Integral Equations, vol. 10, no. 2, pp. 309–332, 1997.

G. Chen and F. Da, “Blow-up of solution of Cauchy problem for three-dimensional damped nonlinear hyperbolic equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 358–372, 2009.

C. Song and Z. Yang, “Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation,” Mathematical Methods in the Applied Sciences, vol. 33, no. 5, pp. 563–575, 2010.

G. Chen, “Initial boundary value problem for a damped nonlinear hyperbolic equation,” Journal of Partial Differential Equations, vol. 16, no. 1, pp. 49–61, 2003.

G. Chen, R. Song, and S. Wang, “Local existence and global nonexistence theorems for a damped nonlinear hyperbolic equation,” Journal of Mathematical Analysis and Applications, vol. 368, no. 1, pp. 19–31, 2010.

Y.-Z. Wang, “Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 465–482, 2009.

Y.-Z. Wang, F. G. Liu, and Y. Z. Zhang, “Global existence and asymptotic behavior of solutions for a semi-linear wave equation,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 836–853, 2012.

S. Wang and H. Xu, “On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term,” Journal of Differential Equations, vol. 252, no. 7, pp. 4243–4258, 2012.

V. Pata and S. Zelik, “Smooth attractors for strongly damped wave equations,” Nonlinearity, vol. 19, no. 7, pp. 1495–1506, 2006.

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques and Applications, Springer, Berlin, Germany, 1997.

C. D. Sogge, Lecures on Nonlinear Wave Equations, vol. 2 of Monographs in Analysis, International Press, 1995.

W.-R. Dai and D.-X. Kong, “Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields,” Journal of Differential Equations, vol. 235, no. 1, pp. 127–165, 2007.

D.-X. Kong and T. Yang, “Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,” Communications in Partial Differential Equations, vol. 28, no. 5-6, pp. 1203–1220, 2003.

Y.-Z. Wang, “Global existence of classical solutions to the minimal surface equation in two space dimensions with slow decay initial value,” Journal of Mathematical Physics, vol. 50, no. 10, Article ID 103506, 14 pages, 2009.

Y.-Z. Wang and Y.-X. Wang, “Global existence of classical solutions to the minimal surface equation with slow decay initial value,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 576–583, 2010.

Y. Liu and W. Wang, “The pointwise estimates of solutions for dissipative wave equation in multi-dimensions,” Discrete and Continuous Dynamical Systems A, vol. 20, no. 4, pp. 1013–1028, 2008.

M. Nakao and K. Ono, “Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,” Mathematische Zeitschrift, vol. 214, no. 2, pp. 325–342, 1993.

K. Nishihara, “- estimates of solutions to the damped wave equation in 3-dimensional space and their application,” Mathematische Zeitschrift, vol. 244, no. 3, pp. 631–649, 2003.

K. Ono, “Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations,” Discrete and Continuous Dynamical Systems A, vol. 9, no. 3, pp. 651–662, 2003.

W. Wang and W. Wang, “The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions,” Journal of Mathematical Analysis and Applications, vol. 366, no. 1, pp. 226–241, 2010.

Z. Yang, “Longtime behavior of the Kirchhoff type equation with strong damping on ,” Journal of Differential Equations, vol. 242, no. 2, pp. 269–286, 2007.

Y. Liu and S. Kawashima, “Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation,” Discrete and Continuous Dynamical Systems A, vol. 29, no. 3, pp. 1113–1139, 2011.

Y. Sugitani and S. Kawashima, “Decay estimates of solutions to a semilinear dissipative plate equation,” Journal of Hyperbolic Differential Equations, vol. 7, no. 3, pp. 471–501, 2010.

T. T. Li and Y. M. Chen, Nonlinear Evolution Equations, Scientific Press, 1989 (Chinese).

S. M. Zheng, Nonlinear Evolution Equations, vol. 133 of Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, 2004.