Research Article | Open Access
Zhigang Pan, Hong Luo, Tian Ma, "Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms", Journal of Applied Mathematics, vol. 2012, Article ID 805158, 15 pages, 2012. https://doi.org/10.1155/2012/805158
Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms
We consider the global existence of strong solution , corresponding to a class of fully nonlinear wave equations with strongly damped terms in a bounded and smooth domain in , where is a given monotone in nonlinearity satisfying some dissipativity and growth restrictions and is in a sense subordinated to . By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution .
We are concerned with the following mixed problem for a class of fully nonlinear wave equations with strongly damped terms in a bounded and domain : where Equations of type (1.1) are a class essential nonlinear wave equations describing the speed of strain waves in a viscoelastic configuration (e.g., a bar if the space dimension and a plate if ) made up of the material of the rate type [1, 2]. They can also be seen as field equations governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model . Concerning damped cases, there is much to the global existence of solutions for the problem: they discussed the global existence of weak solutions and regularity in and [4–8]. On the other hand, Ikehata and Inoue  considered the global existence of weak solutions for two-dimensional problem in an exterior domain with a compact smooth boundary for a semilinear strongly damped wave equation with a power-type nonlinearity and : Cholewa and Dlotko  discussed the global solvability and asymptotic behavior of solutions to semilinear Cauchy problem for strongly damped wave equation in the whole of . They assume the nonlinear term grows like and if . Similar problems attracted attention of the researchers for many years [11–13]. Especially, Yang  studied the global existence of weak solutions to the more general equation including (1.4), but he did not discuss the regularity of weak solution for the quasilinear wave equation. We are interested in discussing the global existence and regularity of weak solutions for strongly damped wave equation with the dissipative terms containing and the nonlinear terms containing . Here is a given monotone in nonlinearity satisfying some dissipativity and growth restrictions and is in a sense subordinated to .
In , we have investigated the existence of global solutions to a class of nonlinear damped wave operator equations. In this paper, our first aim is to study the global existence of strong solutions to the more general equation including (1.4), which is the motivation that we establish our abstract strongly damped wave equation model with. The second aim is to deal with the global existence of strong solutions to a class of fully nonlinear wave equations with strongly damped terms under some weakly growing conditions.
This paper is organized as follows:(i)in Section 2, we recall some preliminary tools and definitions;(ii)in Section 3, we put forward our abstract strongly damped wave equation model and proof the global existence of strong solution of it;(iii)in Section 4, we provide the proof of the main results about the mixed problem (1.1).
We introduce two spatial sequences: where , , , and are Hilbert spaces, is a linear space, and are Banach spaces.
All embeddings of (2.1) are dense. Let
Furthermore, has eigenvectors satisfying and constitutes common orthogonal basis of and .
We consider the following abstract wave equation model: where is a map, , and is a bounded linear operator, satisfying
Definition 2.1. We say that is a global weak solution of the (2.4) provided for for each and .
Definition 2.2. Let , . We say that in is uniformly weakly convergent if is bounded, and
Lemma 2.3 (see ). Let be bounded sequences and uniformly weakly convergent to . Then, for each , it follows that
Lemma 2.4 (see ). Let be an open set and satisfy Caratheodory condition and If is bounded and convergent to in for all bounded , then for each , the following equality holds
3. Model Results
Let . Assume(A1) there is a functional such that (A2) functional is coercive, that is, (A3) satisfies
Theorem 3.1. Set , for each , then the following assertions hold.(1)If satisfies (A1) and (A2), then (2.4) has a globally weak solution (2)If satisfies (A1)–(A3), then (2.4) has a global weak solution (3)Furthermore, if is symmetric sectorial operator, that is, , and satisfies then .
Proof. Let be a common orthogonal basis of and , satisfying (2.3). Set
Clearly, , .
By using Galerkin method, there exits satisfying for , and for .
Firstly, we consider . Let in (3.9). Taking into account (2.2) and (3.1), it follows that
Let . From (2.1) and (2.2), it is known that are orthogonal basis of . We find that in , and in . Since is imbedding, it follows that
From (3.2), (3.11), and (3.12), we obtain that,
Let which implies that in is uniformly weakly convergent from that is compact imbedding.
If we have the following equality: then is a weak solution of (2.4) in view of (3.8), (3.14).
We will show (3.15) as follows. It follows that from (2.5),
Taking into account (2.2), (2.5), and (3.9), we get that
From (2.1) and (3.14), we have
Then, we get
In view of (3.9), (3.14), we obtain for all
Since is dense in , for all , (3.20) holds for all . Thus, we have
From (3.14) and being compact imbedding, it follows that
Then, (3.14) follows from (3.19)–(3.21), which implies assertion (1).
Secondly, we consider . Let in (3.9). In view of (2.2) and (2.8), it follows that
From (3.3), we have where .
By using Gronwall inequality, it follows that which implies that for all ,
From (3.25) and (3.21), it follows that
Let which implies that in is uniformly weakly convergent from that is compact imbedding.
The left proof is same as assertion (1).
Lastly, assume (3.6) hold. Let in (3.9). It follows that
From (3.26), the above inequality implies We see that for all , is bounded. Thus .
4. Main Result
Now, we begin to consider the mixed problem (1.1). Set We assume where are constant and , is the best constant satisfying
Proof. We introduce spatial sequences
where the inner products of and are defined by
where such that is an embedding.
Linear operators and are defined by It is known that and satisfy (2.2), (2.3), and (2.5). Define by
We show that is -coercively weakly continuous. Let satisfying (2.7) and
We need to prove that
From (2.7) and Lemma 2.3, we obtain
From (4.3), we get
We have the deformation
From (4.14) and Lemma 2.4, we have
From (4.12), (4.15)–(4.17), it follows that
Since , we have
From (4.14), (4.19), (4.1), (4.4), and Lemma 2.3, we get (4.13).
Let , where is same as (4.1). We get which implies Conditions (A1), (A2) of model results in Theorem 3.1.
We will show (3.3) as follows. It follows that which imply Conditions (A3) of Theorem 3.1. From Theorem 3.1, (1.1) has a solution satisfying
The authors are grateful to the anonymous reviewers for their careful reading and useful suggestions, which greatly improved the presentation of the paper. This paper was funded by the National Natural Science Foundation of China (no. 11071177) and the NSF of Sichuan Science and Technology Department of China (no. 2010JY0057).
- G. Andrews, “On the existence of solutions to the equation ,” Journal of Differential Equations, vol. 35, no. 2, pp. 200–231, 1980.
- S. Kawashima and Y. Shibata, “Global existence and exponential stability of small solutions to nonlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, 1992.
- D. D. Ang and A. P. N. Dinh, “Strong solutions of a quasilinear wave equation with nonlinear damping,” SIAM Journal on Mathematical Analysis, vol. 19, no. 2, pp. 337–347, 1988.
- K. Nishihara and H. Zhao, “Existence and nonexistence of time-global solutions to damped wave equation on half-line,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 6, pp. 931–960, 2005.
- N. Hayashi, E. I. Kaikina, and P. I. Naumkin, “Damped wave equation in the subcritical case,” Journal of Differential Equations, vol. 207, no. 1, pp. 161–194, 2004.
- M. Prizzi, “Regularity of invariant sets in semilinear damped wave equations,” Journal of Differential Equations, vol. 247, no. 12, pp. 3315–3337, 2009.
- S. Zelik, “Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,” Communications on Pure and Applied Analysis, vol. 3, no. 4, pp. 921–934, 2004.
- T. Ogawa and H. Takeda, “Non-existence of weak solutions to nonlinear damped wave equations in exterior domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3696–3701, 2009.
- R. Ikehata and Y.-k. Inoue, “Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 1, pp. 154–169, 2008.
- J. W. Cholewa and T. Dlotko, “Strongly damped wave equation in uniform spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 1, pp. 174–187, 2006.
- G. F. Webb, “Existence and asymptotic behavior for a strongly damped nonlinear wave equation,” Canadian Journal of Mathematics, vol. 32, no. 3, pp. 631–643, 1980.
- E. Feireisl, “Bounded, locally compact global attractors for semilinear damped wave equations on ,” Differential and Integral Equations, vol. 9, no. 5, pp. 1147–1156, 1996.
- R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997.
- Z. Yang, “Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 218–243, 2004.
- Z. G. Pan, Z. L. Pu, and T. Ma, “Global solutions to a class of nonlinear damped wave operator equations,” Boundary Value Problems, vol. 2012, article 42, 2012.
- T. Ma, Theories and Methods in Partial Differential Equations, Science Press, Beijing, China, 2011.
- E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics, vol. 108 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1995.
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