Abstract
In this work, the exponential growth of solutions for a coupled nonlinear Klein–Gordon system with distributed delay, strong damping, and source terms is proved. Take into consideration some suitable assumptions.
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1 Introduction
In modeling in the biological, physical, and social sciences, it is sometimes necessary to take account of optimal control or time delays inherent in the phenomena (see for example [4, 16]). The inclusion of delays explicitly in the equations is often a simplification or idealization that is introduced because a detailed description of the underlying processes is too complicated to be modeled mathematically, or because some of the details are unknown. More generally, how does the qualitative behavior depend on the form and magnitude of the delays? In this paper we examine how we can apply the distributed delay term for knowing the behavior of growth of solutions for a coupled nonlinear Klein–Gordon system with strong damping, source terms.
We consider the following system:
where Ω is a bounded domain in \(\mathbb{R}^{n}\) with smooth boundary ∂Ω and the source terms are defined as follows:
and \(m_{1},m_{2},\omega _{1},\omega _{2},\mu _{1},\mu _{3},a_{1},b_{1}>0\), and \(\tau _{1},\tau _{2}\) are the time delay with \(0\leq \tau _{1}<\tau _{2}\), and \(\mu _{2},\mu _{4}\) are \(L^{\infty }\) functions, and \(g,h\) are differentiable functions.
Viscous materials are the opposite of elastic materials that possess the ability to store and dissipate mechanical energy. As the mechanical properties of these viscous substances are of great importance when they appear in many applications of natural sciences, many authors have given attention to this problem since the beginning of the new millennium.
In the case of only one equation and if \(\omega _{1}=0\) (i.e., \(\Delta u_{t}=0\)), and \(\mu _{1}=\mu _{2}=0\). Our problem (1.1) has been studied in [7]. By using the Galerkin method they established the local existence result. Also, they showed the local solution is global in time under suitable conditions and with the same rate of decaying (polynomial or exponential) of the kernel g. They proved that the dissipation given by the viscoelastic integral term is strong enough to stabilize the oscillations of the solution. Moreover, their result has been obtained under weaker conditions than those used in [11]. In [12], the authors proved the exponential decay of the following problem:
This later result has been improved in [7], in which they showed that the viscoelastic dissipation alone is strong enough to stabilize the problem even with an exponential rate.
In many works on this field under assumptions of the kernel g. For problem (1.1) and with \(\mu _{1}\neq 0\), for example, in [18], the authors proved a blow-up result for the following problem:
where g satisfies \(\int _{0}^{\infty }g(s)\,ds<(2p-4)/(2p-3)\), initial data were supported with negative energy like that \(\int u_{0}u_{1}\,dx>0\).
If \((w>0)\). In [29], the authors considered the following problem:
Under suitable assumptions on g that there were solutions of (1.5) with initial energy, they showed the blow-up in a finite time. For the same problem (1.5), in [30], Song et al. proved that there were solutions of (1.5) with positive initial energy that blows up in finite time. In addition, in [19] the authors showed a blow-up result if \(p>m\) and established the global existence of the following problem:
In the case of coupled of equations, in [2], the authors studied the following system of equations:
with nonlinear functions \(f_{1}\) and \(f_{2}\) satisfying appropriate conditions. Under certain restrictions imposed on the parameters and the initial data, they obtained numerous results on the existence of weak solutions. They also showed that any weak solution with negative initial energy blows up for a finite period of time by using the same techniques as in [17].
In [6], the authors considered the system:
where they stated and proved the blow-up in finite time of solution under some restrictions on the initial data and (with positive initial energy) for some conditions on the functions \(f_{1}\) and \(f_{2}\).
Later, in [23], the authors extended the result of [6], where they considered the following nonlinear viscoelastic system:
and proved that the solutions of the system of wave equations with viscoelastic term, degenerate damping, and strong nonlinear sources acting in both equations at the same time are globally nonexisting provided that the initial data are sufficiently large in a bounded domain of Ω.
To complement the above works, we are working to prove under appropriate assumptions that the solution of problem (1.1) grows exponentially:
The paper is organized as follows. In Sect. 2, some necessary assumptions related to the problem are given. Then, in Sect. 3, the main result is proved.
2 Assumptions
We consider the following assumptions:
-
(A1)
\(g,h:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) are differentiable and decreasing functions such that
$$ \textstyle\begin{cases} g(t)\geq 0, \qquad 1-\int _{0}^{\infty }g ( s ) \,ds=l_{1}>0, \\ h(t)\geq 0,\qquad 1-\int _{0}^{\infty }h ( s ) \,ds=l_{2}>0.\end{cases} $$(2.1) -
(A2)
There exist constants \(\xi _{1}, \xi _{2}>0\) such that
$$ \textstyle\begin{cases} g^{\prime } ( t ) \leq -\xi _{1}g ( t ),\quad t \geq 0, \\ h^{\prime } ( t ) \leq -\xi _{2}h ( t ),\quad t \geq 0.\end{cases} $$(2.2) -
(A3)
\(\mu _{2},\mu _{4}:[\tau _{1},\tau _{2}]\rightarrow \mathbb{R}\) are \(L^{\infty }\) functions so that, for all \(\delta >\frac{1}{2}\),
$$\begin{aligned} \begin{aligned} &\biggl(\frac{2\delta -1}{2}\biggr) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2}( \varrho ) \bigr\vert \,d\varrho < \mu _{1}, \\ &\biggl(\frac{2\delta -1}{2}\biggr) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{4}( \varrho ) \bigr\vert \,d\varrho < \mu _{3}. \end{aligned} \end{aligned}$$(2.3)
3 Main results
In this section, the blow-up result of solution of problem (1.1) is proved.
First, as in [22], we introduce the new variables:
then
and
Let us denote
Therefore, problem (1.1) takes the form
with the initial and boundary condition
where
Theorem 3.1
Assume (2.1), (2.2), and (2.3) hold. Let
Then, for any initial data,
where
problem (3.4) has a unique solution
for some\(T>0\).
In the next theorem we give the global existence result, its proof is based on the potential well depth method in which the concept of so-called stable set appears, where we show that if we restrict our initial data in the stable set, then our local solution obtained is global in time. We will make use of arguments in [28].
Theorem 3.2
Suppose that (2.1), (2.2), (2.3), and (3.6) hold. If\(u_{0},v_{0}\in W\), \(u_{1},v_{1}\in H_{0}^{1}(\varOmega )\), \(y,z\in L^{2}(\varOmega \times (0,1)\times (\tau _{1},\tau _{2}))\), and
where\(C_{\ast }\)is the best Poincare constant, then the local solution\((u,v,y,z)\)is global in time.
In order to achieve the main result, the following lemmas are needed.
Lemma 3.1
There exists a function \(F(u,v)\) such that
where
taking \(a_{1}=b_{1}=1\) for convenience.
Lemma 3.2
([23])
There exist two positive constants \(c_{0}\) and \(c_{1}\) such that
Define the energy functional as follows.
Lemma 3.3
Assume that (2.1), (2.2), (2.3), and (3.6) hold, let\((u,v,y,z)\)be a solution of (3.4), then\(E(t)\)is nonincreasing, that is,
satisfies
where
Proof
By multiplying the first and the second equation in (3.4) respectively by \(u_{t},v_{t}\) and integrating over Ω, we get
and, from the initial and boundary condition in (3.4)
and
then
By (3.12)–(3.14), we get (3.9). Also, by using Young’s inequality, (2.1), (2.2), and (2.3) in (3.15), we obtain (3.10). □
Now, we define the functional
Theorem 3.3
Assume that (2.1)–(2.3) and (3.6) hold. Assume further that\(E(0)<0\), then the solution of problem (3.4) grows exponentially.
Proof
From (3.9 we have
Therefore,
Hence
and
Setting
where \(\varepsilon >0\) to be assigned later.
By multiplying the first and second equation on (3.4) respectively by \(u,v\) and with a derivative of (3.21), we get
Using Young’s inequality, we get
Thus
Also
so
From (3.22)
Therefore, using (3.19) and by setting \(\delta _{1},\delta _{1}\) so that \(\frac{1}{4\delta _{1}c_{3}}=\frac{\kappa }{2}\) and \(\frac{1}{4\delta _{2}c_{3}}=\frac{\kappa }{2}\), substituting in (3.27), we get
For \(0< a<1\), from (3.16)
Substituting in (3.28), we get
Using Poincare’s inequality, we obtain
In this stage, we take \(a>0\) small enough so that
and we assume
Then we choose κ so large that
We fixed κ and a, we appoint ε small enough so that
and from (3.21) we get
Thus, for some \(\beta >0\), estimate (3.31) becomes
By (3.8), for some \(\beta _{1}>0\),
and
Next, using Young’s and Poincare’s inequalities ([12]), thus from (3.21), we have
for some \(c>0\). From inequalities (3.34) and (3.37) we obtain the differential inequality
where \(\lambda >0\), depending only on β and c.
A simple integration of (3.38) gives
Therefore, we conclude that the solution grows exponentially. This completes the proof. □
4 Conclusion
In this work, the growth of solutions for a coupled nonlinear Klein–Gordon system with strong damping, source, and distributed delay terms was studied. Next, motivated by last works in [1, 3, 5, 8–10, 13–15, 20, 21, 24–27, 31], and [16], we obtained the growth and blow-up for the studied problem (1.1) by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then, the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.
References
Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Phys. A, Stat. Mech. Appl. 500, 40–49 (2018)
Agre, K., Rammaha, M.A.: Systems of nonlinear wave equations with damping and source terms. Differ. Integral Equ. 19, 1235–1270 (2007)
Alizadeh, M., Alimohammady, M.: Regularity and entropy solutions of some elliptic equations. Miskolc Math. Notes 19(2), 715–729 (2018)
Aschepkov, L.T., Dolgy, D.V., Kim, T., Agarwal, R.P.: Optimal Control p. xv+209 Springer, Cham (2016). ISBN 978-3-319-49780-8
Ball, J.: Remarks on blow-up and nonexistence theorems for nonlinear evolutions equation. Q. J. Math. 28, 473–486 (1977)
Ben Aissa, A., Ouchenane, D., Zennir, K.: Blow up of positive initial-energy solutions to systems of nonlinear wave equations with degenerate damping and source terms. Nonlinear Stud. 4, 523–535 (2012)
Berrimi, S., Messaoudi, S.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64, 2314–2331 (2006)
Boulaaras, S.: A well-posedness and exponential decay of solutions for a coupled Lamé system with viscoelastic term and logarithmic source terms. Appl. Anal. (2019, in press). https://doi.org/10.1080/00036811.2019.1648793
Boulaaras, S., Choucha, A., Ouchenane, D., Cherif, B.: Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms. Adv. Differ. Equ. 2020, 310 (2020). https://doi.org/10.1186/s13662-020-02772-0
Boulaaras, S., Ouchenane, D.: General decay for a coupled Lamé system of nonlinear viscoelastic equations. Math. Methods Appl. Sci. 43, 1717–1735 (2020)
Cavalcanti, M.M., Cavalcanti, D., Ferreira, J.: Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math. Methods Appl. Sci. 24, 1043–1053 (2001)
Cavalcanti, M.M., Cavalcanti, D., Filho, P.J.S., Soriano, J.A.: Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. Differ. Integral Equ. 14, 85–116 (2001)
El-Sayed, A.A., Agarwal, P.: Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials. Math. Methods Appl. Sci. 42, 978–3991 (2019)
Gala, S., Liu, Q., Ragusa, M.A.: A new regularity criterion for the nematic liquid crystal flows. Appl. Anal. 91(9), 1741–1747 (2012)
Gala, S., Ragusa, M.A.: Logarithmically improved regularity criterion for the Boussinesq equations in Besov spaces with negative indices. Appl. Anal. 95(6), 1271–1279 (2016)
Gan, Z.H., Zhang, J.: Global solution for coupled nonlinear Klein–Gordon system. Appl. Math. Mech. 28, 677–687 (2007)
Georgiev, V., Todorova, G.: Existence of a solution of the wave equation with nonlinear damping and source term. J. Differ. Equ. 109, 295–308 (1999)
Kafini, M., Messaoudi, S.A.: A blow-up result in a Cauchy viscoelastic problem. Appl. Math. Lett. 21, 549–553 (2008)
Liang, G., Zhaoqin, Y., Guonguang, L.: Blow up and global existence for a nonlinear viscoelastic wave equation with strong damping and nonlinear damping and source terms. Appl. Math. 6, 806–816 (2015)
Mezouar, N., Boulaaras, S.: Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation. Bull. Malays. Math. Sci. Soc. 43, 725–755 (2020)
Mezouar, N., Boularas, S.: Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term. Bound. Value Probl. 2020, 90 (2020). https://doi.org/10.1186/s13661-020-01390-9
Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21, 935–958 (2008)
Ouchenane, D., Zennir, K., Bayoud, M.: Global nonexistence of solutions for a system of nonlinear viscoelastic wave equation with degenerate damping and source terms. Ukr. Math. J. 65, 723–739 (2013)
Piskin, E.: Uniform decay and blow-up of solutions for coupled nonlinear Klein–Gordon equations with nonlinear damping terms. Math. Methods Appl. Sci. 37, 3036–3047 (2014)
Piskin, E.: Blow-up of solutions for coupled nonlinear Klein–Gordon equations with weak damping terms. Math. Sci. Lett. 3, 189–191 (2014)
Piskin, E.: Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms. Malaya J. Mat. 3, 168–174 (2015)
Polidoro, S., Ragusa, M.A.: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam. 24(3), 1011–1046 (2008)
Shun-Tang, W., Long-Yi, T.: On global existence and blow-up of solutions or an integro-differential equation with strong damping. Taiwan. J. Math. 10, 979–1014 (2006)
Song, H.T., Xue, D.S.: Blow up in a nonlinear viscoelastic wave equation with strong damping. Nonlinear Anal. 109, 245–251 (2014)
Song, H.T., Zhong, C.K.: Blow-up of solutions of a nonlinear viscoelastic wave equation. Nonlinear Anal., Real World Appl. 11, 3877–3883 (2010)
Sweilam, N.H., Nagy, A.M., El-Sayed, A.A.: Solving time-fractional order telegraph equation via sinc-Legendre collocation method. Mediterr. J. Math. 13, 5119–5133 (2016)
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The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions. The third author would like to thank his Professors/Scientists Pr. Mohamed Haiour and Pr. Azzedine Benchettah for the important content of masters and PhD courses in pure and applied mathematics which he received during his studies. Moreover, he thanks them for the additional help they provided to him during office hours in their office about the few concepts/difficulties he had encountered, and he appreciates their talent and dedication for their postgraduate students currently and previously!
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Rahmoune, A., Ouchenane, D., Boulaaras, S. et al. Growth of solutions for a coupled nonlinear Klein–Gordon system with strong damping, source, and distributed delay terms. Adv Differ Equ 2020, 335 (2020). https://doi.org/10.1186/s13662-020-02801-y
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DOI: https://doi.org/10.1186/s13662-020-02801-y