This paper deals with global stability dynamics for the Klein–Gordon–Zakharov system in . We first establish that this system admits a family of linear mode unstable explicit quasi-periodic wave solutions. Next, we prove that the Kelvin–Voigt damping can help to stabilize those linear mode unstable explicit quasi-periodic wave solutions for the Klein–Gordon–Zakharov system in the Sobolev space for any . Moreover, the Kelvin–Voigt damped Klein–Gordon–Zakharov system admits a unique Sobolev regular solution exponentially convergent to some special solutions (including quasi-periodic wave solutions) of it. Our result can be extended to the n-dimension dissipative Klein–Gordon–Zakharov system for any .
In this paper, we devote to the study of stabilizability for the two dimension Klein–Gordon–Zakharov system:
where , represent the Klein–Gordon component and the wave component respectively. The Klein–Gordon–Zakharov system describes the interaction between Langmuir waves and ion sound waves in plasma [12,48]. For this model, the results of global well-posedness, see for instance [13,15,32,36,37]. Masmoudi and Nakanishi [32] yielded the existence and uniqueness of solution to the Klein–Gordon–Zakharov system in the energy space . Ozawa, Tsutaya and Tsutsumi [36] proved that there exist the unique global solutions of the Klein–Gordon–Zakharov equations for the small initial data. The authors also showed the time local well-posedness of the Klein–Gordon–Zakharov equations and the unique global existence of solutions for the small initial data in the energy space [37]. Dong [13] established the small global solution to the Klein–Gordon–Zakharov equations and investigated the pointwise asymptotic behavior of the solution. Later, the author applied Alinhac’s ghost weight energy estimates to obtain the global solution of the Klein–Gordon–Zakharov equations with uniform energy bounds [15]. Moreover, small energy scattering for the three-dimensional Klein–Gordon–Zakharov system with radial symmetry was studied in [20]. In [21], the authors obtained the global dynamical behavior below ground state and the dichotomy between scattering and finite time blow up.
The study of both nonlinear wave equations and nonlinear Klein–Gordon equations have attracted lots of researchers’ interest and many interesting results have been established. The relationship between two models was shown in [9]. The global well-posedness of coupled wave and Klein–Gordon equations has been established in [25,31]. Katayama [25] obtained the global solution for coupled systems of nonlinear wave and Klein–Gordon equations in three space dimensions. Ma [31] developed a method by distinguishing the roles of different type of decay factors, then established the global existence result of Wave–Klein–Gordon system in two dimensions. Ionescu and Pausader [24] considered a coupled Wave–Klein–Gordon system in three dimensions and proved global regularity and modified scattering for the small and smooth initial data with suitable decay at infinity. Dong [14] investigated the small data global existence result and the sharp pointwise asymptotic behaviour of the solution to the coupled wave and Klein–Gordon system under null condition in dimension two. In addition to, the results of other kinds of wave and Klein–Gordon equations have been studied. Cavalcanti and Gonzalez demonstrated that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space [8]. Fang, Wang and Yang derived the global dynamic properties of the Maxwell coupled with a massive Klein–Gordon scalar field with a general class of data [17].
As a general abstract model, the Zakharov system also attracted much attention. It describes the interaction between laser and plasma, which has a wide range of physics and multitude of applications, such as laser fusion, electron beam fusion, solar radio bursts etc.(we can refer to [41]). The well-posedness and properties of solutions for Zakharov system, we refer the readers to [4,33,38]. Bourgain [4] proved local and global existence and regularity theorems for the one-dimensional Zakharov model in the space periodic case by Fourier analysis. The global existence of solutions to the initial value problem for the Zakharov system in one spatial dimension has been obtained by Pecher in [38]. A blow-up solution of the Zakharov system in was established in [33]. It has been shown that below the ground state radial solutions to Zakharov system are global and scatter [7]. However, in the non-radial case, all solutions with data below the ground state are global in time [19].
The stabilization in nonlinear partial differential equations was studied widely. Buffe [5] considered the damped wave equation with Ventcel boundary condition and obtained the stabilization result of it. Ammari, Hassine and Robbiano [3] showed the stabilization for the wave equation with Kelvin–Voigt damping in a bounded domain. In [47], the authors established the stabilization of a multidimensional wave equation on a cuboidal domain. For related contributions, one can refer to [28,29], as well as to the basic monograph by Eden, Foiaş, Nicolaenko, and Temam [16]. Besides, lots of other kinds of partial differential equations were studied, and the stability results were obtained. The global stabilization in small time of the viscous Burgers equation with three scalar controls was proved in [11]. Laurent, Rosier and Zhang arrived at the global exponential stabilizability of the Korteweg–de Vries equation on a periodic domain [27]. In [10], the authors considered the one dimensional Schödinger equation with a bilinear control and proved the rapid stabilization of the linearized equation around the ground state. Guo and Huang [18] established the exponential stability of undamped Euler–Bernoulli beam with both ends free by the operator semigroup technique.
Inspired by the existing results, we are interested in studying the stability of the Klein–Gordon–Zakharov equations in . We will state a fact that the Klein–Gordon–Zakharov system admits a family of linear mode unstable time-periodic solutions. Then we show the nonlinear stability of the Kelvin–Voigt damped Klein–Gordon–Zakharov system.
Assume that the solution of (1.1) only depends on the time variable, then the Klein–Gordon–Zakharov system (1.1) can be reduced into an ordinary differential equation:
from which, we get a general expression of exact solutions:
where , represent constants, and
In particularly, if , the Klein–Gordon–Zakharov system (1.1) admits a family of time quasi-periodic solutions:
with , where the frequency is denoted by .
We now state the first result in this paper.
The Klein–Gordon–Zakharov system (
1.1
) admits a family of linear mode unstable time-periodic solutions (
1.3
).
Scattering below the ground state in the non-radial case is an open question (see [6]). Our second result concerns with the nonlinear stability of smooth solution for dissipative Klein–Gordon–Zakharov system. It can be seen as a scattering result for the dissipative case.
The linear system of (1.1) is
the solution of (1.4) is denoted by , we suppose
Our purpose is to show the stability of the Kelvin–Voigt damped Klein–Gordon–Zakharov system:
with the initial data
Let
substituting them into (1.6)–(1.8), we obtain the perturbation system
with the initial data
where
By (1.5), we can obtain
The second result is to establish the scattering result for the Kelvin–Voigt damped Klein–Gordon–Zakharov system (1.6)–(1.8). This result implies that the internal damping affects stability of equations.
Assume thatis a small smooth solution of linear system (
1.4
) with the integerand satisfies (
1.5
). If there exists a small positive constant ε such thatThen the Kelvin–Voigt damped Klein–Gordon–Zakharov system (
1.6
)–(
1.8
) is exponentially stable in the sense that for a given initial data (
1.9
) supported inwith a fixed positive constant C, and the system (
1.6
)–(
1.8
) admits a solutionMoreover, there exists a positive constant c such that
In general, the solution of quasilinear wave equation takes the blowup phenomenon if the initial data is large. The Klein–Gordon–Zakharov system is a quasilinear coupled wave system. The global well-posedness of it with small initial data in two dimension has been proved by Dong [13] in recently. It is still unknown for the case of large initial data. In Theorem 1.2, we construct a large solution near the special periodic solution of the Klein–Gordon–Zakharov system with Kelvin–Voigt damping. It is a scattering result of this kind of damping system.
In Section 2, we prove Theorem 1.1 relying on the method of spectral analysis. Then in Section 3, we obtain the global well-posedness of linear Kelvin–Voigt damped Klein–Gordon–Zakharov system with variable coefficients. We confirm Theorem 1.2 by Nash–Moser iteration scheme. In Section 4, we construct an approximate solution of nonlinear approximation perturbation systems. Finally, the proof of Theorem 1.2 is shown in Section 5. The readers can refer to [22,23,34,35,39,40] for more details of Nash–Moser iteration scheme, and it has been used in [42–46].
Linear mode unstable of explicit quasi-periodic wave solutions
The aim of this section is to illuminate the linear mode unstable of explicit quasi-periodic wave solutions for Klein–Gordon–Zakharov system. Let
then substitute them into (1.1), the perturbation system is shown as:
with the initial data
We now introduce the definition of mode stable or unstable for explicit quasi-periodic wave solutions , and V gave in (1.3). Before done it, we consider the linear system of (2.1):
which admits the characteristic system
Explicit quasi-periodic wave solution gave in (1.3) is called mode stable if holds. The eigenvalue λ is called a stable eigenvalue. Otherwise, if , it is called mode unstable, then the eigenvalue λ is called an unstable eigenvalue.
The eigenvalues () are determined by
which is equivalent to
On one hand, if , we derive
on the other hand, if
a routine computation gives
where .
We should further compute the eigenvalues in two cases, if , that is , then
If , that is , then
We find positive eigenvalue from the above formulas, which means that the explicit quasi-periodic wave solutions of system (1.1) is linear mode unstable.
It remains to analyze the complex eigenvalues in low frequency and high frequency respectively, notice that
For one thing, we consider eigenvalues in the low frequency , it follows from (2.11) that
To analyze the eigenvalues and , using (2.11) and simplifying the result gives
In the same way,
For the other, we analyze the eigenvalues in high frequency , using (2.12), we have
Applying (2.12) to take two successive operations gives
Based on the observation to the eigenvalues of (2.3), we find that the explicit quasi-periodic wave solutions of Klein–Gordon–Zakharov system are linear mode unstable.
Global well-posedness for the linearized system
In this section, we focus on the global well-posedness for the linear Kelvin–Voigt damped Klein–Gordon–Zakharov system with variable coefficients. More precisely, for all , let us consider the following system:
with the initial data
where
Let the coefficients of system (3.1)–(3.3) be positive and satisfy the following conditions:
and there exists a positive constant C such that
The coefficients of the strong damping terms satisfy
moreover, we suppose
for a positive small constant ε. Here we use ∂ to denote the derivative of time or spacial variable.
Introducing two weighted functions which are positive smooth bounded functions, they satisfy
with a positive constant C.
Moreover, there exists a positive constant C such that
We first derive a weighted priori estimate of solutions for the linear system (3.1)–(3.3).
Assume that (
3.5
)–(
3.8
) hold. Then the solution of linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (
3.1
)–(
3.3
) satisfies
On one hand, we multiply both sides of (3.1) by , and then integrate it over , we can get
Direct computation gives that
Thus, according to (3.14)–(3.17), we reduce (3.13) into
On the other hand, multiplying both sides of (3.1) by , and then integrating it over , we get
Direct computation gives that
Thus, it follows from the above equations (3.20)–(3.22) that,
Furthermore, it can be obtained from the Cauchy inequality that
and
Thus by means of above estimates, it follows that
An argument similar to the one used deriving (3.24), we multiply equation (3.2) with , and integrate it over , then we apply the Cauchy inequality to get
Performing the same procedure for (3.3), we have
where
Combining (3.24), (3.25) and (3.26), we see that
where
Next step in the proof is to analyse all of coefficients in (3.33). By means of the assumptions given in (3.5)–(3.7), (3.10) and (3.11), there exists a positive constant C such that
Similarly, it follows that
By the assumptions given in (3.5)–(3.11), we can show
and
and
The analysis of the remainder terms is almost identical, we have
According to the above estimation, integrating (3.33) over , we have
Finally, we can apply Grönwall’s inequality to (3.38) to obtain
□
A direct application of Lemma 3.1 is to derive the following -estimate of solution for the linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (3.1)–(3.3).
Assume that (
3.5
)–(
3.7
) hold. Then the solution of linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (
3.1
)–(
3.3
) satisfies
For simplicity, we take weighted functions
As (3.9)–(3.12) hold, proceeding as in the proof of Lemma 3.1, we can also prove Lemma 3.2. □
Next, we derive -estimate for any and . Applying () to both sides of (3.1)–(3.3), we have
where
with the symbol
We have the following -estimate result.
Assume that (
3.5
)–(
3.7
) hold. Then the solution of linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (
3.1
)–(
3.3
) satisfies
To prove above Lemma 3.3, we need to use the induction. Let , then the linear system (3.40)–(3.42) can be written as
where
Notice that the structure of equations (3.46)–(3.48) are similar to linear system (3.1)–(3.3). We shall adopt the same procedure as in getting (3.33), multiplying equation (3.46) by , (3.47) by , and (3.48) by , then we can obtain
We now deal with the right hand side terms of (3.52). Based on (3.49) and Cauchy inequality, satisfies
and
The estimations of and are quite similar to and so are omitted. Based on the above estimations, summing up (3.52) from to , we find
where
and
and
Here , , , , , , , , and are given in (3.27)–(3.32) and (3.34)–(3.37), respectively.
Similar to the coefficient estimation of inequality (3.33), we can further analyze the coefficient of inequality (3.53). By the assumptions given in (3.5)–(3.11), there exists a positive constant C such that
and
and
and
With the help of the above results of coefficients estimation, we can find a positive constant such that
In light of Lemma 3.1 and applying Grönwall’s inequality to (3.54), we have
where we use the result of Lemma 3.1 to estimate the terms , and .
When , we suppose that the following inequality holds
Then, it is sufficient to show that the case also holding.
We multiply equation (3.40)–(3.42) by , and , respectively, then an argument similar to the one used in getting (3.52) shows that
We now consider the right hand side of (3.57). According to (3.43) and Cauchy inequality, it holds
and
and
and
We omit the details estimation of the remaining two terms since the corresponding estimates on (3.44) and (3.45) can be obtained in the same way. In fact, from the above identities, we have
where
and
and
where , , , , , , , , and are given in (3.27)–(3.32) and (3.34)–(3.37), respectively.
We now analyze all of the coefficients for inequality (3.58). By the assumptions (3.5)–(3.11), there exists a positive constant C such that
No estimation will be given for the remaining terms here, due to them have been estimated previously.
Based on the above estimations, (3.58) can be deduced into
It is not difficult to verify that can be controlled by (3.56), and then apply Grönwall’s inequality to (3.59) to obtain
□
Directly derive from Lemma 3.3, we have the following result.
Assume that (
3.5
)–(
3.7
) hold. Then the solution of linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (
3.1
)–(
3.3
) satisfies
Based on the above results, the well-posedness for the linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (3.1)–(3.3) can be given.
Assume that (
3.5
)–(
3.8
) hold. The linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (
3.1
)–(
3.3
), with the initial data (
3.4
), exists a unique global solutionMoreover, it satisfies
Our first goal is to show the local existence of smooth solution for the linear Kelvin–Voigt damped Klein–Gordon–Zakharov system (3.1)–(3.3) with the initial data (3.4). The argument is analogous to that in Theorem 3.2 of [40] and only include the outline here.
Let , treating ϕ as a new variable, we have
with the initial data
the form of operator is as follows
with
and
We now solve the approximation problem by means of the technique of the fixed point iteration scheme
Indeed, there exists a Cauchy sequence in , whose tend to and it is a sequence of solutions of the linearized system (3.61) in , where T denotes a positive constant.
Furthermore, according to the results of Lemma 3.1–3.4, it follows that
Thus, the constructed local solution can be extended to the global solution in time.
To see ϕ is unique assume otherwise, so there exists which is also the solution of (3.61) with , set , and
Due to (3.60), we find which is a contradiction. The proof is completed. □
Construction of approximation solutions
In order to overcome the possible loss of derivatives due to the quasilinear linear term, we introduce a family of smooth operators in smooth bounded regions. More details can be found in [1,2]. We define smooth operators , in time and space variables which are based on Proposition 1.6 of [2, p. 83] or [26, p. 72]. Let such that in , otherwise, . For fixed positive constants and ,
where Ψ satisfies
thus
We define the operator
it is not difficult to verify that
where C represent a positive constant and . The proof depends on the proof given in [1, p. 192] or [2].
In our iteration scheme, we set
where is a fixed positive constant. It is more convenient for our purposes to set it by . Then, by (4.1), it holds
The first iteration step
Let and . We denote the initial approximation function by , and it satisfies the following assumptions:
A nonhomogeneous linear Kelvin–Voigt damped Klein–Gordon–Zakharov system with variable coefficients can be obtained by linearization of the perturbation system (1.10)–(1.12) at the initial approximation function , we denote them by
For any , let us consider the Kelvin–Voigt damped Klein–Gordon–Zakharov system with variable coefficients derive external forces as follows
with the initial data
where the external force is related to the error term at the initial approximation function.
According to the assumptions of (4.3)–(4.7), we can check that the assumptions of (3.5)–(3.8) hold. Note that for any and
Thus, we can arrive at the global existence of the solution for the linear system (4.11) by using Proposition 3.1.
The linear wave system (
4.11
) with the initial data (
4.12
) admits a unique global solutionMoreover, it satisfies
The general iteration step
For , we define
with the integer .
Suppose that the m-th approximation step of quasilinear system (1.10)–(1.12) is denoted by with , where
it gives
Linearizing (1.10)–(1.12) around , we can get the following initial value problem:
where the error terms are
and
which are also the nonlinear terms of quasilinear system (1.10)–(1.12) at . Meanwhile, due to , it follows that
and
An argument similar to the one used in getting Proposition 3.1, we can establish the global existence of m-th approximation solution.
The linearized problem (
4.14
) admits a unique global solutionMoreover, it satisfies
On one hand, the m-th approximation solution will be found if we can find such that
By substituting (4.18) into quasilinear system (1.10)–(1.12), we can acquire
then let
notice that the above Kelvin–Voigt damped Klein–Gordon–Zakharov system with variable coefficients has the same form as (4.8)–(4.10), if is replaced by , and the error term
On the other hand, we obtain
where we denote the s-th derivatives of time or spacial variables by the symbol , and for a sufficient small positive parameter ε, it holds
thus, we can see that
it suggests that the leading term of the m-th approximation solution is the initial approximation function . Therefore, the linear system of m-th approximation solutions has the same structure as the linear system (4.11), and a similar assumption given in (3.5)–(3.8) can be satisfied. According to the same arguments as in the proof of Proposition 4.1, we can show that the linear problem (4.14) admits a global solution
Meanwhile, (4.17) can be acquired. This completes the proof. □
We now illustrate that is a global solution of the nonlinear system (1.10)–(1.12). It suffices to prove the convergence of series , and . We now give the same estimate of error term in each iteration scheme.
Let. The error term verifies
We notice that error terms are
and the highest order of derivatives on x of it is 2. Due to Cauchy’s inequality and (5.2), we can derive (5.1). □
We now demonstrate the convergence of iteration scheme. For any , let and
which gives that
We prove the Theorem 1.2 by the induction argument. Our first goal is to deal with the case of zero initial data
Later, we consider the case
Note that with . For all , we assert that there exists a sufficient small positive constant ε such that
When , by (4.17), letting , it holds
Moreover, according to (5.1) and above estimate,
and
which implies that .
Let us assume that the case of holds, that is,
Then it is sufficient to show that the case of m holds. Upon (4.17) and the second inequality of (5.5), we derive
combining above inequality with (5.1)–(5.3) receives
We choose a sufficiently small positive constant ε such that
Thus, by (5.7) we have
so, the error term goes to 0 as , that is,
On the other hand, note that , by (5.5)–(5.6), it follows that
This implies that . Hence, we conclude that (5.4) holds.
Therefore, in the case of zero initial data, the nonlinear system (1.10)–(1.12) admits a unique global Sobolev solution
and using (4.4)–(4.7) to get
Finally, we need to discuss the case of small non-zero initial data. For any , we introduce the auxiliary function
Thus, the nonlinear system (1.10)–(1.12) is transformed into equations of and the initial data reduce to
Then, we can construct a unique global Sobolev solution by following the above iteration scheme. Furthermore, the global Sobolev solution of the nonlinear system (1.10)–(1.12) with a small non-zero initial data takes the form
Due to the uniqueness of each iteration step , here the global solution is unique.
We shall return to the initial approximation function , it should satisfy (4.3), i.e.
and by (4.17) and (5.8), it holds
thus it follows from (5.9) that
Note that
hence
The proof is completed. □
Footnotes
Acknowledgements
The third author is supported by Guangxi Natural Science Foundation No. 2021JJG110002 and National Natural Science Foundation of China No. 12161006.
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