Research Article | Open Access
Nasser-Eddine Tatar, "Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type", Journal of Applied Mathematics, vol. 2012, Article ID 936140, 17 pages, 2012. https://doi.org/10.1155/2012/936140
Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type
A wave equation of the Kirchhoff type with several nonlinearities is stabilized by a viscoelastic damping. We consider the case of nonconstant (and unbounded) coefficients. This is a nondissipative case, and as a consequence the nonlinear terms cannot be estimated in the usual manner by the initial energy. We suggest a way to get around this difficulty. It is proved that if the solution enters a certain region, which we determine, then it will be attracted exponentially by the equilibrium.
We will consider the following wave equation with a viscoelastic damping term: where is a bounded domain in with smooth boundary and . The functions and are given initial data, and the (nonnegative) functions , and are at least absolutely continuous and will be specified later on. This problem arises in viscoelasticity where it has been shown by experiments that when subject to sudden changes, the viscoelastic response not only does depend on the current state of stress but also on all past states of stress. This gives rise to the integral term called the memory term. One may find a rich literature in this regard (with or without the Kirchhoff terms) treating mainly the stabilization of such systems for different classes of functions . We refer the reader to [1–25] and the references therein. For problems of the Kirchhoff type, one can consult [26–35] and in particular [36–46] where the equations are supplemented by a nonlinear source. Several questions, such as well-posedness and asymptotic behavior, have been discussed in these references, to cite but a few.
As is clear from the equation in (1.1), we consider here several nonlinearities and the relaxation function is not necessarily decreasing or even nonincreasing. These issues are important but do not constitute the main contribution in the present paper. In case that and are not nonincreasing, then we are in a nondissipative situation. This is the case also when the relaxation function oscillates (in case are nonincreasing). Our argument here is simple and flexible. It relies on a Gronwall-type inequality involving several nonlinearities. We prove that there exists a sufficiently large and a constant after which (the modified energy of) global solutions are bounded below by or decay to zero exponentially. We were not able to find conditions directly on the initial data because the Gronwall inequality is applicable only after some large values of time.
For simplicity we shall consider the simpler case and .
Theorem 1.1. Assume that and is a nonnegative summable kernel. If when and when , then there exists a unique solution to problem (1.1) such that for small enough.
The plan of the paper is as follows. In the next section we prepare some materials needed to prove our result. Section 3 is devoted to the statement and proof of our theorem.
In this section we define the different functionals we will work with. We prove an equivalence result between two functionals. Further, some useful lemmas are presented. We define the (classical) energy by where denotes the norm in . Then by (1.1) it is easy to see that for The first term in the right-hand side of (2.2) may be written as the derivative of some expression; namely, where Therefore, if we modify to we obtain for Assuming that makes a nonnegative functional. The following functionals are standard and will be used here. The next ones have been introduced by the present author in  where and and are two nonnegative functions which will be precised later (see (H2), (H3)). The functional for some , to be determined is equivalent to .
Proposition 2.1. There exist such that for all and small .
Proof. By the inequalities
where is the Poincaré constant, we have
On the other hand,
Therefore, for some constant and small such that and .
The identity to follow is easy to justify and is helpful to prove our result.
Lemma 2.2. One has for and
The next lemma is crucial in estimating (partially) our nonlinear terms. It can be found in .
Let , and let . We write if is nondecreasing in .
Lemma 2.3. Let be a positive continuous function in are nonnegative continuous functions, are nondecreasing continuous functions in , with for , and is a nonnegative continuous functions in . If in , then the inequality implies that where , and is chosen so that the functions are defined for .
Lemma 2.4. Assume that if or if . Then there exists a positive constant such that for .
3. Asymptotic Behavior
In this section we state and prove our result. To this end we need some notation. For every measurable set , we define the probability measure by The nondecreasingness set and the non-decreasingness rate of are defined by respectively.
The following assumptions on the kernel will be adopted. for all and . is absolutely continuous and of bounded variation on and for some nonnegative summable function (= where exists) and almost all . There exists a nondecreasing function such that is a nonincreasing function: and .
Note that a wide class of functions satisfies the assumption (H3). In particular, exponentially and polynomially (or power type) decaying functions are in this class.
Let be a number such that . We denote by the set .
Lemma 3.1. One has for and where is the total variation of .
Proof. This lemma is proved by a direct differentiation of along solutions of (1.1) and estimation of the different terms in the obtained expression of the derivative. Indeed, we have
For all measurable sets and such that , it is clear that
For , the first term in the right-hand side of (3.8) satisfies
and the third one fulfills
Back to (3.8) we may write The last term in the right-hand side of (3.7) will be estimated as follows: For the fourth term in (3.7), it holds that Moreover, from Lemma 2.4, for if and if , we find The definition of in (2.5) allows us to write
Gathering all the relations (3.11)–(3.15) together with (3.7), we obtain for
In the following theorem we will assume that just to fix ideas. The result is also valid for . It suffices to interchange and in the proof following it. The case is easier.
We will make use of the following hypotheses for some positive constants , and to be determined. is a continuously differentiable function such that . is a continuously differentiable function such that . if and if .. .
Theorem 3.2. Assume that the hypotheses (H1)–(H3), (A)–(C) hold and . If , then, for global solutions and small , there exist and such that or for some positive constants and as long as (D) holds. If , then there exist and such that or for some positive constants and as long as (E) holds.
Proof. A differentiation of with respect to along trajectories of (1.1) gives
and Lemma 2.2 implies
Next, a differentiation of and yields
Taking into account Lemma 3.1 and the relations (2.6), (3.20)-(3.21), we see that Next, as in , we introduce the sets and observe that where is the null set where is not defined and is as in (3.2). Furthermore, if we denote , then because for all and . Moreover, we designate by the sets In (3.22), we take and . Choosing , it is clear that for small and , large and , if . We deduce that Furthermore, if , then with and a small . Pick and such that Note that this is possible if is so large that even though
Taking the relations (3.22)–(3.30) into account and selecting so that and small enough so that we find for , large , small , and for some positive constant . Take small, and (i.e., for some ) and (i.e., for some ) to derive that for some positive constant .
If , then there exist a and such that for . Thus, in virtue of Proposition 2.1, for , we have where If there exists a such that , where , then from (3.39) and it follows that for with . Now we apply Lemma 2.3 to get where and . If, in addition, , then is uniformly bounded by a positive constant . Thus and by continuity (3.44) holds for all .
If , then for any there exists a such that for . Therefore, for some . The previous argument carries out with replaced by .
In case that , we reverse the roles of and in the argument above. The case is clear.
Remark 3.3. The case where the derivative of the kernel does not approach zero on (as is the case, for instance, when on is interesting. Indeed, the right-hand side in condition (3.34) will be replaced by with a possibly large constant .
Remark 3.4. The argument clearly works for all kinds of kernels previously treated where derivatives cannot be positive or even take the value zero. In these cases there will be no need for the smallness conditions on the kernels. This work shows that derivatives may be positive (i.e., kernels may be increasing) on some “small” subintervals and open the door for (optimal) estimations and improvements of these sets.
Remark 3.5. The assumptions and may be relaxed to and , respectively. In this case and would depend on .
Remark 3.6. The assertion in Theorem 3.2 is an “alternative” statement. As a next step it would be nice to discuss the (sufficient conditions of) occurrence of each case in addition to the global existence.
The author is grateful for the financial support and the facilities provided by the King Fahd University of Petroleum and Minerals.
- M. Aassila, M. M. Cavalcanti, and J. A. Soriano, “Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1581–1602, 2000.
- F. Alabau-Boussouira and P. Cannarsa, “A general method for proving sharp energy decay rates for memory-dissipative evolution equations,” Comptes Rendus Mathématique, vol. 347, no. 15-16, pp. 867–872, 2009.
- J. A. D. Appleby, M. Fabrizio, B. Lazzari, and D. W. Reynolds, “On exponential asymptotic stability in linear viscoelasticity,” Mathematical Models & Methods in Applied Sciences, vol. 16, no. 10, pp. 1677–1694, 2006.
- M. M. Cavalcanti, V. N. Domingos Cavalcanti, and P. Martinez, “General decay rate estimates for viscoelastic dissipative systems,” Nonlinear Analysis, vol. 68, no. 1, pp. 177–193, 2008.
- M. Fabrizio and S. Polidoro, “Asymptotic decay for some differential systems with fading memory,” Applicable Analysis, vol. 81, no. 6, pp. 1245–1264, 2002.
- K. M. Furati and N.-E. Tatar, “Uniform boundedness and stability for a viscoelastic problem,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1211–1220, 2005.
- X. S. Han and M. X. Wang, “Global existence and uniform decay for a nonlinear viscoelastic equation with damping,” Nonlinear Analysis, vol. 70, no. 9, pp. 3090–3098, 2009.
- W. Liu, “Uniform decay of solutions for a quasilinear system of viscoelastic equations,” Nonlinear Analysis, vol. 71, no. 5-6, pp. 2257–2267, 2009.
- W. Liu, “General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms,” Journal of Mathematical Physics, vol. 50, no. 11, Article ID 113506, 17 pages, 2009.
- M. Medjden and N.-E. Tatar, “On the wave equation with a temporal non-local term,” Dynamic Systems and Applications, vol. 16, no. 4, pp. 665–671, 2007.
- M. Medjden and N.-E. Tatar, “Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1221–1235, 2005.
- S. A. Messaoudi, “General decay of solutions of a viscoelastic equation,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1457–1467, 2008.
- S. A. Messaoudi and N.-E. Tatar, “Global existence and uniform stability of solutions for a quasilinear viscoelastic problem,” Mathematical Methods in the Applied Sciences, vol. 30, no. 6, pp. 665–680, 2007.
- S. A. Messaoudi and N.-E. Tatar, “Exponential and polynomial decay for a quasilinear viscoelastic equation,” Nonlinear Analysis, vol. 68, no. 4, pp. 785–793, 2008.
- S. A. Messaoudi and N.-E. Tatar, “Exponential decay for a quasilinear viscoelastic equation,” Mathematische Nachrichten, vol. 282, no. 10, pp. 1443–1450, 2009.
- J. E. Muñoz Rivera and M. G. Naso, “On the decay of the energy for systems with memory and indefinite dissipation,” Asymptotic Analysis, vol. 49, no. 3-4, pp. 189–204, 2006.
- V. Pata, “Exponential stability in linear viscoelasticity,” Quarterly of Applied Mathematics, vol. 64, no. 3, pp. 499–513, 2006.
- N.-E. Tatar, “On a problem arising in isothermal viscoelasticity,” International Journal of Pure and Applied Mathematics, vol. 8, no. 1, pp. 1–12, 2003.
- N.-E. Tatar, “Long time behavior for a viscoelastic problem with a positive definite kernel,” The Australian Journal of Mathematical Analysis and Applications, vol. 1, no. 1, article 5, pp. 1–11, 2004.
- N.-E. Tatar, “Polynomial stability without polynomial decay of the relaxation function,” Mathematical Methods in the Applied Sciences, vol. 31, no. 15, pp. 1874–1886, 2008.
- N.-E. Tatar, “How far can relaxation functions be increasing in viscoelastic problems?” Applied Mathematics Letters, vol. 22, no. 3, pp. 336–340, 2009.
- N.-E. Tatar, “Exponential decay for a viscoelastic problem with a singular kernel,” Zeitschrift für Angewandte Mathematik und Physik, vol. 60, no. 4, pp. 640–650, 2009.
- N.-E. Tatar, “On a large class of kernels yielding exponential stability in viscoelasticity,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2298–2306, 2009.
- N.-E. Tatar, “Oscillating kernels and arbitrary decays in viscoelasticity,” Mathematische Nachrichten, vol. 285, no. 8-9, pp. 1130–1143, 2012.
- S. Q. Yu, “Polynomial stability of solutions for a system of non-linear viscoelastic equations,” Applicable Analysis, vol. 88, no. 7, pp. 1039–1051, 2009.
- M. Abdelli and A. Benaissa, “Energy decay of solutions of a degenerate Kirchhoff equation with a weak nonlinear dissipation,” Nonlinear Analysis, vol. 69, no. 7, pp. 1999–2008, 2008.
- M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, “Existence and exponential decay for a Kirchhoff-Carrier model with viscosity,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 40–60, 1998.
- M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano, and J. S. Prates Filho, “Existence and asymptotic behaviour for a degenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions,” Revista Matemática Complutense, vol. 14, no. 1, pp. 177–203, 2001.
- M. Hosoya and Y. Yamada, “On some nonlinear wave equations II: global existence and energy decay of solutions,” Journal of the Faculty of Science, The University of Tokyo IA, vol. 38, no. 1, pp. 239–250, 1991.
- R. Ikehata, “A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms,” Differential and Integral Equations, vol. 8, no. 3, pp. 607–616, 1995.
- L. A. Medeiros and M. A. Milla Miranda, “On a nonlinear wave equation with damping,” Revista Matemática de la Universidad Complutense de Madrid, vol. 3, no. 2-3, pp. 213–231, 1990.
- J. E. Muñoz Rivera and F. P. Quispe Gómez, “Existence and decay in non linear viscoelasticity,” Bollettino della Unione Matematica Italiana B, vol. 6, no. 1, pp. 1–37, 2003.
- K. Ono, “Global solvability for degenerate Kirchhoff equations with weak dissipation,” Mathematica Japonica, vol. 50, no. 3, pp. 409–413, 1999.
- K. Ono, “On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,” Mathematical Methods in the Applied Sciences, vol. 20, no. 2, pp. 151–177, 1997.
- G. A. P. Menzala, “On global classical solutions of a nonlinear wave equation,” Applicable Analysis, vol. 10, no. 3, pp. 179–195, 1980.
- A. Arosio and S. Garavaldi, “On the mildly degenerate Kirchhoff string,” Mathematical Methods in the Applied Sciences, vol. 14, no. 3, pp. 177–195, 1991.
- T. Kobayashi, H. Pecher, and Y. Shibata, “On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity,” Mathematische Annalen, vol. 296, no. 2, pp. 215–234, 1993.
- L. A. Medeiros, J. Limaco, and C. L. Frota, “On wave equations without global a priori estimates,” Boletim da Sociedade Paranaense de Matemática 3, vol. 30, no. 2, pp. 19–32, 2012.
- S. Mimouni, A. Benaissa, and N.-E. Amroun, “Global existence and optimal decay rate of solutions for the degenerate quasilinear wave equation with a strong dissipation,” Applicable Analysis, vol. 89, no. 6, pp. 815–831, 2010.
- J. E. Muñoz Rivera, “Global solution on a quasilinear wave equation with memory,” Bollettino della Unione Matematica Italiana B, vol. 8, no. 2, pp. 289–303, 1994.
- K. Narashima, “Nonlinear vibration of an elastic string,” Journal of Sound and Vibration, vol. 8, no. 1, pp. 134–146, 1968.
- K. Ono, “Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings,” Journal of Differential Equations, vol. 137, no. 2, pp. 273–301, 1997.
- K. Ono, “On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 321–342, 1997.
- J.-Y. Park and J.-J. Bae, “On the existence of solutions of strongly damped nonlinear wave equations,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 6, pp. 369–382, 2000.
- R. M. Torrejón and J. M. Yong, “On a quasilinear wave equation with memory,” Nonlinear Analysis, vol. 16, no. 1, pp. 61–78, 1991.
- S.-T. Wu and L.-Y. Tsai, “On global existence and blow-up of solutions for an integro-differential equation with strong damping,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 979–1014, 2006.
- M. Pinto, “Integral inequalities of Bihari-type and applications,” Funkcialaj Ekvacioj, vol. 33, no. 3, pp. 387–403, 1990.
Copyright © 2012 Nasser-Eddine Tatar. 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.