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Existence and blowup of solutions for the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction
Advances in Difference Equations volume 2017, Article number: 293 (2017)
Abstract
This paper studies the existence and blowup of solutions for the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction, which describe the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves in a plasma considering the quantum effects. Firstly the existence and uniqueness of the local smooth solutions are obtained by the a priori estimates and the Galerkin method. Secondly, and what is more, by introducing some auxiliary functionals and invariant manifolds, the authors study and derive a sharp threshold for the global existence and blowup of solutions by applying potential well argument and the concavity method. Furthermore, two more specific conditions of how small the initial data are for the solutions to exist globally are concluded by the dilation transformation.
1 Introduction
In this paper, we consider the existence and blowup for the following modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction:
with the periodic initial conditions
where \(x\in\Omega\), \(t\geq0\). The spatial domain Ω is a bounded domain in two dimensional real Euclidean space, and the time \(t\geq0\). The unknown complex vector-valued function \(u=u(x,t)\) is the electric field, \(V=V(x,t)\) is an unknown real vector-valued function and \(n=n(x,t)\) is an unknown real scalar-valued function, which represents the density fluctuation of ions. System (1)-(3) describes the interaction between Langmuir waves and ion-acoustic waves in a plasma considering the quantum effects [1–4]. The quantum parameter H expresses the ratio between the ion plasmon energy and the electron thermal energy, which appears in the modified Zakharov equations for plasmas with a quantum correction [4]. \(L>0\) is the period and \(e_{i}\) (\(i=1,2\)) is the standard coordinate vector.
Combining (2) and (3) to eliminate the function \(V(x,t)\), we find that system (1)-(3) is equivalent to
which combines the Klein-Gordon equation and the modified Zakharov equations for plasmas with a quantum correction. The latter equation describes the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves considering the quantum effects [4]. The importance of quantum effects in ultrasmall electronic devices, in dense astrophysical plasma systems and in laser plasmas has produced an increasing interest in the investigation of the quantum counterpart of some of the classical plasma physics phenomena [5]. That is why we call system (1)-(3) the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction. Thus it is an interesting topic to study the coupled system (1)-(3) mathematically. For the modified Zakharov equations with a quantum correction, a series of works have been devoted to the mathematical analysis. For example, we studied the exact solutions and obtained kinds of exact traveling wave solutions [6]; and we also investigated the existence of weak solutions for another form of Zakharov equations under Dirichlet boundary condition [7]. Long time behavior of the solutions for the dissipative modified Zakharov equations is also studied in our previous work [8, 9]. For some other classical results, see [4, 10] and the references therein.
System (1)-(3) with the absence of \(H^{2}\nabla\Delta n\) has been studied in recent years. Guo and Yuan [11] proved the existence and uniqueness of global smooth solutions via the so-called continuous method and delicate a priori estimates and studied the asymptotic behavior of the solutions. Gan et al. discussed the instability of standing wave, global existence and blowup for the Klein-Gordon-Zakharov system [12]. Some exact solutions were also obtained by various methods, see [13, 14] and the references therein. Furthermore, when system (1)-(3) with the absence of \(H^{2}\nabla\Delta n\) and \(- \vert u \vert ^{2}u\), it is more classic and describes the interaction between Langmuir waves and ion sound waves in a plasma [1, 3]. Some existence and uniqueness of global solutions under different conditions were investigated by many researchers [2, 15–17], as well as the exact explicit traveling wave solutions [18, 19].
However, for the modified Klein-Gordon-Zakharov equations (1)-(3) with \(H^{2}\nabla\Delta n\) and \(- \vert u \vert ^{2}u\), which is more suitable to describe the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves in a plasma considering the quantum effects, no more research results in mathematical studies have been obtained as we know, even the study on the well-posedness for the equations. Thus, in the present paper, we are going to do the research tentatively and investigate the existence and blowup for the modified Klein-Gordon-Zakharov equations (1)-(5). More specifically, we adopt a priori estimates and the Galerkin method to study the existence and uniqueness of local smooth solutions for the periodic initial value problems (1)-(5) under the condition that the initial data are sufficiently regular. What follows and what is more, by introducing two auxiliary functionals and two invariant manifolds, we study and derive a sharp threshold for the global existence and blowup of the solutions by applying potential well argument [20] and the concavity method [21]. Moreover, two more specific conditions of how small the initial data are for the solutions to exist globally are investigated by some suitable dilation transformation.
The rest of paper is organized as follows. In Section 2, we briefly give some notations and preliminaries. In Section 3, we establish a prior estimates for the solutions of the periodic initial value problem (1)-(5) and obtain the existence and uniqueness for the local solutions. In Section 4, a sharp threshold for global existence and blowup of the solutions are derived by utilizing two invariant manifolds, applying potential well argument and the concavity method. In the last section we make some conclusions.
2 Notations and preliminaries
We shall use the following conventional notations throughout the paper. Let \(L_{\mathrm{per}}^{k}\) and \(H_{\mathrm{per}}^{k}\), \(k=1, 2, \ldots \) , denote the Hilbert and Sobolev spaces of L-periodic, complex-valued functions endowed with the usual \(L^{2}\) inner product \((u,v)=\int_{\Omega}u(x)\overline{v}(x)\,dx\) and the norms
Here vÌ… denotes the complex conjugate of v. For brevity, we write \(\Vert u \Vert = \Vert u \Vert _{L^{2}}\) and denote the \(L^{p}\)-norm by \(\Vert u \Vert _{p}= (\int_{\Omega} \vert u \vert ^{p}\,dx )^{1/p}\). Without any ambiguity, we denote a generic positive constant by C which may vary from line to line.
In the following sections, we frequently use the following inequalities.
Lemma 1
Gagliardo-Nirenberg inequality [22]
Let Ω be a bounded domain with ∂Ω in \(C^{m}\), and let u be any function in \(W^{m,r}(\Omega)\cap L^{q}(\Omega)\), \(1\leq q, r\leq\infty\). For any integer j, \(0\leq j< m\), and for any number a in the interval \(j/m\leq a\leq1\), set
If \(m-j-N/r\) is not a nonnegative integer, then
If \(m-j-N/r\) is a nonnegative integer, then (7) holds for \(a=j/m\). The constant C depends only on Ω, r, q, j, a.
As the specific cases for \(N=2\), there holds
3 The local existence and uniqueness of solutions
In this section, we will obtain existence and uniqueness of local solutions for problem (1)-(5). Firstly we derive some a priori estimates for the solutions.
Lemma 2
Assume \(u_{0}(x)\in H_{\mathrm{per}}^{1}(\Omega)\), \(u_{1}(x)\in L_{\mathrm{per}}^{2}(\Omega)\), \(n_{0}(x)\in H^{1}_{\mathrm{per}}(\Omega)\), and \(V_{0}(x)\in L_{\mathrm{per}}^{2}(\Omega)\). Then, for the solutions of problem (1)-(5), we have
where C is a constant depending only on \(\Vert u_{0} \Vert _{H_{\mathrm{per}}^{1}(\Omega)}\), \(\Vert u_{1} \Vert _{L_{\mathrm{per}}^{2}(\Omega)}\), \(\Vert n_{0} \Vert _{H_{\mathrm{per}}^{1}(\Omega)}\) and \(\Vert V_{0} \Vert _{L_{\mathrm{per}}^{2}(\Omega)}\).
Proof
Multiplying (1) by \(\overline{u}_{t}\), integrating with respect to x over Ω and taking the real part, we obtain
Using equations (2) and (3) repeatedly and noticing the periodicity, we have
and
Combining (12),(13) and (14), we have
which implies
where \(E_{0}\) is a positive constant depending only on \(\Vert u_{0} \Vert _{H_{\mathrm{per}}^{1}(\Omega)}\), \(\Vert u_{1} \Vert _{L_{\mathrm{per}}^{2}(\Omega)}\), \(\Vert n_{0} \Vert _{H_{\mathrm{per}}^{1}(\Omega)}\) and \(\Vert V_{0} \Vert _{L_{\mathrm{per}}^{2}(\Omega)}\). Since
Thus we have
By the Gagliardo-Nirenberg inequality (10) and (18), we have
Thus Lemma 2 is completed from (18) and (19). □
Lemma 3
Assume \(u_{0}(x)\in H_{\mathrm{per}}^{2}(\Omega)\), \(u_{1}(x)\in H_{\mathrm{per}}^{1}(\Omega)\), \(n_{0}(x)\in H^{2}_{\mathrm{per}}(\Omega)\), and \(V_{0}(x)\in H_{\mathrm{per}}^{1}(\Omega)\). Then, for the solutions of problem (1)-(5), there holds
where C is a constant depending only on \(\Vert u_{0} \Vert _{H_{\mathrm{per}}^{2}(\Omega)}\), \(\Vert u_{1} \Vert _{H_{\mathrm{per}}^{1}(\Omega)}\), \(\Vert n_{0} \Vert _{H_{\mathrm{per}}^{2}(\Omega)}\), \(\Vert V_{0} \Vert _{H_{\mathrm{per}}^{1}(\Omega)}\) and T.
Proof
Taking the inner product of (1) with \(-\Delta\overline{u}_{t}\) in Ω and using the integration by parts, we obtain
From Gagliardo-Nirenberg inequality (8) and Lemma 2, we have
and
Combining (21),(22) and (23), we get
where C is a positive constant.
On the other hand, equations (2) and (3) are equivalent to
Then one multiplies (25) by \(n_{t}\) and integrates with respect to x over Ω, and combines the result with (2) to have
By the Gagliardo-Nirenberg inequality (8) and Lemma 2, we can obtain
From (24),(26) and (27), it follows
By Gronwall’s inequality, we can complete the proof of Lemma 3. □
Lemma 4
Assume \(u_{0}(x)\in H_{\mathrm{per}}^{3}(\Omega)\), \(u_{1}(x)\in H_{\mathrm{per}}^{2}(\Omega)\), \(n_{0}(x)\in H^{3}_{\mathrm{per}}(\Omega)\), and \(V_{0}(x)\in H_{\mathrm{per}}^{2}(\Omega)\). Then, for the solutions of problem (1)-(5), there holds
where C is a constant depending only on \(\Vert u_{0} \Vert _{H_{\mathrm{per}}^{3}(\Omega)}\), \(\Vert u_{1} \Vert _{H_{\mathrm{per}}^{2}(\Omega)}\), \(\Vert n_{0} \Vert _{H_{\mathrm{per}}^{3}(\Omega)}\), \(\Vert V_{0} \Vert _{H_{\mathrm{per}}^{2}(\Omega)}\) and T.
Proof
Multiplying (1) by \(\Delta^{2}\overline{u}_{t}\), integrating with respect to x over Ω and taking the real part, we have
Meanwhile, one takes the inner product of (25) with \(-\Delta n_{t}\) in Ω, and combines it with (2) to obtain
Adding (30) and (31) together, we obtain
Now we need some estimates for the terms on the right-hand side of (32). First from Gagliardo-Nirenberg inequality (8), Lemma 2 and Lemma 3, we have
and
Similarly, we can compute that
Combining (32),(33), (34),and (35), we get
where C is a positive constant. Applying Gronwall’s inequality, the proof of Lemma 4 is completed. □
Generally, based on the results of the previous lemmas and the mathematical deduction, we have the following lemma for problem (1)-(5).
Lemma 5
Assume \(u_{0}(x)\in H_{\mathrm{per}}^{k+1}(\Omega)\), \(u_{1}(x)\in H_{\mathrm{per}}^{k}(\Omega )\), \(n_{0}(x)\in H^{k+1}_{\mathrm{per}}(\Omega)\), and \(V_{0}(x)\in H_{\mathrm{per}}^{k}(\Omega )\) (\(k\geq0\)). Then, for the solutions of problem (1)-(5), we have the following estimates:
where C is a positive constant depending only on \(\Vert u_{0} \Vert _{H_{\mathrm{per}}^{k+1}(\Omega)}\), \(\Vert u_{1} \Vert _{H_{\mathrm{per}}^{k}(\Omega)}\), \(\Vert n_{0} \Vert _{H_{\mathrm{per}}^{k+1}(\Omega)}\), \(\Vert V_{0} \Vert _{H_{\mathrm{per}}^{k}(\Omega)}\) and T.
Based on the estimates in Lemmas 2-5, we can employ the Galerkin method to obtain the existence of local smooth solutions for problem (1)-(5). Similarly, the uniqueness of the smooth solutions can also be obtained by the usual method of energy estimates. We omit the detailed proof here. Thus we have the following existence and uniqueness theorems for the local smooth solutions of problem (1)-(5).
Theorem 1
Existence and uniqueness for local smooth solutions
Suppose that \(u_{0}(x)\in H_{\mathrm{per}}^{k+1}(\Omega)\), \(u_{1}(x)\in H_{\mathrm{per}}^{k}(\Omega)\), \(n_{0}(x)\in H^{k+1}_{\mathrm{per}}(\Omega)\), and \(V_{0}(x)\in H_{\mathrm{per}}^{k}(\Omega)\) (\(k\geq2\)). Then there exists unique local smooth solutions \(u(x,t)\), \(n(x,t)\) and \(V(x,t)\), which satisfy
4 Global existence and blowup
In the previous section, when we employ the Gronwall’s inequality to deduce the estimates for higher regularity, it holds only with \(C=C(T)\). That is why we just obtain the local existence for the solutions. However, we found in the proof of Lemma 2 that the boundedness of the estimates is independent of the time T. Thus we need to ask, if the initial values \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\), does problem (1)-(5) admit a global unique solution? If it does, what are the conditions? In this section, we will answer these questions.
First, from the results in Lemma 2 and previous section theorem, we know that for any \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega )\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega )\), problem (1)-(5) admits a unique solution
where \(t\in[0, T_{\mathrm{max}})\) (\(0< T_{\mathrm{max}}<\infty\)). And from (16), the following conserved energy holds:
where \(E(u,u_{t},n,V)\) is defined as
As we have noticed, system (1)-(3) includes derivative nonlinearity and different-degree nonlinearities, thus we need to imply some proper techniques to handle these terms. To deal with the derivative nonlinearity, we first introduce a homogeneous Sobolev space \(\dot{H}_{\mathrm{per}}^{-1}(\Omega)\) defined by
Second, we make the assumption that there exists a real vector-valued function \(g(x,t)\in L_{\mathrm{per}}^{2}(\Omega)\) such that
For the different-degree nonlinearities, we introduce some functionals and manifolds to handle it. That is to say, for any \((\phi,\psi)\in H_{\mathrm{per}}^{1}(\Omega)\times H_{\mathrm{per}}^{1}(\Omega)\), we define
and
where \(\lambda>1\) is a constant.
Additionally, we define two constrained variational problems
According to the definitions, it is easy to find that the energy functional E can be rewritten as
or
For the properties of \(F(\phi,\psi)\) and \(d_{\Phi}\), we have the following results.
Proposition 4.1
\(F(\phi,\psi)\) is bounded below on Φ, \(F(\phi,\psi)>0\) for all \((\phi,\psi)\in\Phi \) and \(d_{\Phi}>0\).
Proof
First, from (43) and (44), we have
Thus on Φ there holds \(G(\phi,\psi)=0\) and
On the other hand, from (44) we have
which implies
That is also to say
Combining (52) and (55), we can complete the proof of Proposition 4.1. □
For the functional \(P(\phi,\psi)\) and \(d_{\Phi^{-}}\), we have the following results.
Proposition 4.2
If \(G(\phi,\psi)\leq0\) and \((\phi, \psi)\neq(0,0)\), then \(P(\theta ^{\frac{1}{2}}\phi,\theta\psi)\) is an increasing function of \(\theta\in (0,\infty)\). And
Furthermore, if \(G(\phi,\psi)<0\), then
Proof
The first result that \(P(\theta^{\frac{1}{2}}\phi,\theta\psi)\) is an increasing function of θ can be proved directly by computing the derivation with respect to θ. Here we prove (56).
First, from definition (44), we have
Then, for \(\theta=1\), \(G(\phi,\psi)\leq0\), and for \(\theta>0\) close to zero, \(G(\theta^{\frac{1}{2}}\phi,\theta\psi)>0\). Thus from the continuity there exists \(\theta_{0}\in(0,1)\) such that \(G(\theta_{0}^{\frac {1}{2}}\phi,\theta_{0}\psi)=0\), which also implies \((\theta_{0}^{\frac {1}{2}}\phi,\theta_{0}\psi)\in\Phi\). Noticing \(\Phi\subset\Phi^{-}\), and from (48), there holds
Meanwhile, we have
Thus one combines (59) and (60) to get (56).
Finally, since \(G(\phi,\psi)<0\), then there also exists \(\theta\in (0,1)\) such that \(G(\theta^{\frac{1}{2}}\phi,\theta\psi)=0\) and \((\phi, \psi)\neq(0,0)\). Thus there holds
which leads to (57). The proof of Proposition 4.2 is completed. □
Since \(d_{\Phi}>0\), then we define a set \(\mathcal{S}\) as
and introduce two invariant sets as
For the sets \(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\), we have the following proposition.
Proposition 4.3
\(\mathcal{S}_{1}\) and \(\mathcal{S}_{2}\) are invariant sets under the solution flow generated by the periodic initial value problem (1)-(5).
Proof
We prove the proposition by contradiction. First we prove that \(\mathcal {S}_{1}\) is an invariant set. Let \((u_{0},u_{1},n_{0},V_{0})\in\mathcal{S}_{1}\), and suppose that there exists a time \(t_{1}\) such that \((u(t_{1}),u_{t}(t_{1}),n(t_{1}),V(t_{1}))\notin\mathcal{S}_{1}\), then \(G(u(t_{1}),n(t_{1}))\leq0\) and \((u(t_{1}),n(t_{1}))\neq(0,0)\), which implies \((u(t_{1}),n(t_{1}))\in\Phi^{-}\). Set
then \(G(u(t),n(t))\geq0\) for all \(0\leq t< s\). Let \(\{s_{k}\}\) be the minimizing sequence for problem (62), then \((u(s_{k}),n(s_{k}))\in\Phi^{-}\). By the weak lower semi-continuity of \(G(u(\cdot),n(\cdot))\), we have
On the other hand, from (45), (49) and (56) in Proposition 4.2, we obtain
which contradicts definition (48). So \(\mathcal{S}_{1}\) is invariant.
Now we turn to proving that \(\mathcal{S}_{2}\) is also invariant. Similarly, let \((u_{0},u_{1},n_{0},V_{0})\in\mathcal{S}_{2}\), and assume that there exists \(t_{2}\) such that \((u(t_{2}),u_{t}(t_{2}),n(t_{2}),V(t_{2}))\notin \mathcal{S}_{2}\), which implies \(G(u(t_{2}),n(t_{2}))\geq0\). From (45) and (49), we have
Let
then \(G(u(t),n(t))>d_{\Phi}\) for all \(0\leq t< s\). However, from (45), there holds
From Proposition 4.2, we know that if \(G(u(s),n(s))<0\), then \(G(u(s),n(s))>d_{\Phi}\), which makes contradiction. Thus \(\mathcal{S}_{2}\) is also invariant. The proof of Proposition 4.3 is completed. □
Based on the results from previous propositions, we can derive a sharp threshold of global existence and blowup for the solution \((u(x, t),n(x, t), V(x, t))\) to the periodic initial value problem (1)-(5) in terms of the relationship between the initial energy \(E(u_{0}, u_{1},n_{0}, V_{0})\) and \(d_{\Phi}>0\). Here we state the main results as follows.
Theorem 2
Global existence and blowup
Suppose that \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\) and satisfy
Then
(1) If
then the solution \((u(x,t), n(x,t), V(x,t))\) of the periodic initial value problem (1)-(5) blows up in finite time. That is, there exists \(T>0\) such that
(2) If
then the solution \((u(x,t), n(x,t), V(x,t))\) of the periodic initial value problem (1)-(5) exists globally on \(t\in[0,\infty )\) and satisfies
or
where \(C^{\star}\) is a positive constant which satisfies the Gagliardo-Nirenberg inequality.
Proof
First we prove (1) of Theorem 2. From (39), (49) and condition (68), we have
(1) When \(G(u_{0}, n_{0})<0\), as the initial values \((u_{0}, u_{1},n_{0},V_{0})\in \mathcal{S}_{2}\), and it follows that \((u(x,t), u_{t}(x,t), n(x,t), V(x,t)) \in\mathcal{S}_{2}\) by Proposition 4.3. Thus
And from (49) and (73), there holds
On the other hand, since \((u(x,t), u_{t}(x,t),n(x,t),V(x,t))\) is a solution of periodic initial value problem (1)-(5), under assumption (42), we set
Thus we get
where uÌ… is the complex conjugate of u. By computing \(F^{\prime\prime}(t)\), one can obtain
Combining (74), it yields that
Before continuing to complete the proof, we give some property about the function \(Y(t)\) defined in (76).
Proposition 4.4
The function \(Y^{\prime}(t)\) in (77) is positive for some t. That is, there exists \(t_{1}>0\) such that \(Y^{\prime}(t)>0\) for all \(t>t_{1}\).
Proof
We prove it by contradiction. Suppose that, for all \(t>0\), there holds
Combining (79), \(Y(t)\) must tend to a finite and nonnegative limit \(Y_{0}\) as \(t\rightarrow\infty\). From Proposition 4.3, it concludes \(Y_{0}>0\). And as \(t\rightarrow\infty\), there holds \(Y(t)\rightarrow Y_{0}\), \(Y^{\prime}(t)\rightarrow0\) and \(Y^{\prime\prime}(t)\rightarrow0\). Then, from (78) and \(G(u,n)<0\), it follows that
and
Now, for any fixed \(t>0\), and since \(G(u,n)<0\), there exists \(0<\theta <1\) such that \(G(\theta^{\frac{1}{2}}u,\theta n)=0\) and \((\theta^{\frac{1}{2}}u,\theta n)\neq0\). Thus from (44) we can get
and
Thus by (48) we get
From (43) and (83) it follows that
Thus
On the other hand, from definition (44) and (82), we can obtain as \(t\rightarrow\infty\),
Together with
there holds
and
which shows
Since \(0<\theta<1\), we have as \(t\rightarrow\infty\),
Combining (87) and (93), we get
In all, from (82), (85) and (94), we can conclude that as \(t\rightarrow\infty\),
which contradicts \(F(u, n)< d_{\Phi}\) from (75). So supposition (80) is not true. Thus \(Y^{\prime}(t)>0\) for some t. Thus the proof of Proposition 4.4 is completed. □
Corollary 1
Under the conditions of Proposition 4.4, the function \(Y(t)\) defined in (76) and \(Z(t)= \Vert u(x,t) \Vert ^{2}\) are both increasing for all \(t>t_{1}\).
Proof
We can compute that
and
Since \(G(u,n)<0\), we have
which implies that
Combining \(G(u,n)<0\), (96) and (97), we have \(Z^{\prime\prime}(t)>0\). So \(Z^{\prime}(t)\) is strictly increasing for all \(t>0\). Thus if we choose \((u_{0}, u_{1})\) properly such that \(Z^{\prime }(0)=\int_{\Omega}(u_{0}\overline{u_{1}}+u_{1}\overline{u_{0}})\,dx\geq0\), then for all \(t>0\), \(Z^{\prime}(t)>0\). Therefore, \(Z(t)= \Vert u(x,t) \Vert ^{2}\) is strictly increasing for all \(t>0\). Without loss of generality and for simplicity, we omit the condition in the present paper and assume that if \(Y(t)\) is increasing for all \(t>t_{1}\), then \(\Vert u \Vert ^{2}\) is increasing for all \(t>t_{1}\). □
Now we go back to the proof. First from (39), (40) and (78), one gets
and
Since \(E(u_{0},u_{1},n_{0},V_{0})\) is a fixed value, and by Corollary 1, the term \(2 \Vert u \Vert ^{2}-6E(u_{0},u_{1},n_{0},V_{0})\) will eventually become positive and still remain positive thereafter. Meanwhile, combining (97), we know that the quantity
will eventually become positive and will remain positive thereafter. Thus
From (76), (77) and (99), we obtain
and
which implies
On the other hand, we have
Thus from (102) we have
Therefore \(F^{-\frac{1}{4}}(t)\) is convex for sufficiently large t, and \(Y(t)\geq0\), thus there exists a finite time \(T^{\star}\) such that
which implies
Thus one gets \(T<\infty\) and
The proof (1) of Theorem 2 is completed.
(2) Now we turn to proving (2) of Theorem 2. When \(G(u_{0},n_{0})>0\), (73) and Proposition 4.3 imply that \((u(x,t), u_{t}(x,t),n(x,t),V(x,t))\in\mathcal{S}_{1}\) and \(E(u, u_{t}, n,V)< d_{\Phi}\). There will be two cases to be discussed: \(\int_{\Omega}n \vert u \vert ^{2}\,dx\geq0\) and \(\int_{\Omega}n \vert u \vert ^{2}\,dx<0\), respectively.
For case (i) \(\int_{\Omega}n \vert u \vert ^{2}\,dx\geq0\), from (39), (40) and (73) we have
Thus we established the a priori estimates of \(u_{t}\) in \(L^{2}(\Omega)\), u in \(H^{1}(\Omega)\), n in \(H^{1}(\Omega)\) and V in \(L^{2}(\Omega)\) for \(t\in[0, T)\). Thus it must be \(T=\infty\). Then the solution \((u,n,V)\) of the periodic initial value problem (1)-(5) exists globally on \(t\in[0,\infty)\). Furthermore, (107) implies estimate (71).
For case (ii) \(\int_{\Omega}n \vert u \vert ^{2}\,dx<0\). First, from Hölder’s inequality, we have
Thus we can get
which leads to
Meanwhile, from \(G(u,n)>0\) and \(F(u,n)< d_{\Phi}\), we obtain
and
Thus from (111) and (112) there holds
According to inequality (10) in Lemma 1 and (110), we have
where \(C^{\star}\) is a positive constant which satisfies the Gagliardo-Nirenberg inequality. Combining (113) and (114) yields
Therefore, by (110) and (115), we have
Similarly, we have established the a priori estimates of \(u_{t}\) in \(L^{2}(\Omega)\), u in \(H^{1}(\Omega)\), n in \(H^{1}(\Omega)\) and V in \(L^{2}(\Omega)\) for \(t\in[0, T)\). Thus it must be \(T=\infty\). Then the solution \((u,n,V)\) of the periodic initial value problem (1)-(5) exists globally on \(t\in[0,\infty)\). Furthermore, (116) implies estimate (72).
From the discussions of case (i) and case (ii), we complete the proof of (2) of Theorem 2. In sum, the proof of Theorem 2 is completed.  □
Based on the results in Theorem 2, we give two more specific conditions of how small the initial data are for the solutions to exist globally.
Theorem 3
Small initial values criterions
Suppose that \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\) and satisfy
or
Then the solution \((u(x,t), n(x,t), V(x,t))\) of the periodic initial value problem (1)-(5) exists globally.
Proof
(1) If \(\int_{\Omega}n_{0} \vert u_{0} \vert ^{2}<0\), and from (117), we have
Next we will prove \(G(u_{0},n_{0})>0\). If it is not true, there holds \(G(u_{0},n_{0})\leq0\). Similar to Proposition 4.2, there exists \(0<\theta\leq 1\) such that \(G(\theta^{\frac{1}{2}}u_{0},\theta n_{0})= 0\) and \((\theta ^{\frac{1}{2}}u_{0},\theta n_{0})\neq 0\). Since \((u_{0},n_{0})\neq(0,0)\), so \((\theta^{\frac{1}{2}}u_{0},\theta n_{0})\in\Phi\) and
Meanwhile, for \(0<\theta\leq1\), \((\theta^{\frac{1}{2}}u_{0},u_{1},\theta n_{0},V_{0})\) satisfy condition (117), so we arrive at
which is contradictory to (120). So there must be \(G(u_{0},n_{0})>0\). Combining (119) and Theorem 2, we obtain the first result.
(2) If \(\int_{\Omega}n_{0} \vert u_{0} \vert ^{2}>0\), and from (118), we have
While from (44), there holds
Thus combining (122), (123) and Theorem 2, we obtain the second result. In sum, the proof of Theorem 3 is completed. □
Remark 1
From Theorem 3, one could see that if \((u_{0}, u_{1}, n_{0},V_{0})\in H_{\mathrm{per}}^{1}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\times H^{1}_{\mathrm{per}}(\Omega)\times L_{\mathrm{per}}^{2}(\Omega)\), no matter \(\int_{\Omega}n_{0} \vert u_{0} \vert ^{2}<0\) or \(\int _{\Omega}n_{0} \vert u_{0} \vert ^{2}>0\), the solution \((u(x,t), n(x,t), V(x,t))\) of the periodic initial value problem (1)-(5) still exists globally only if
5 Conclusions
The modified Klein-Gordon-Zakharov equations combine the classical Klein-Gordon equation and the modified Zakharov equations for plasmas with a quantum correction, which considers the quantum effects. Thus it is better to describe the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves. In this paper, we mainly do the mathematical analysis for the modified Klein-Gordon-Zakharov equations with periodic initial conditions. First we obtained the existence and uniqueness for local smooth solutions for the periodic initial value problem (1)-(5) via the a priori estimates and the Galerkin method. Secondly, a sharp threshold for the global existence and blowup of the solutions was derived by introducing some auxiliary functionals and invariant manifolds and applying potential well argument and the concavity method. Furthermore, two more specific conditions of how small the initial data are were given out to ensure that the solutions of the periodic initial value problem (1)-(5) exist globally.
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (No. 11426069, 11271141), China Scholarship Council (No. 201708440509), and Natural Science Foundation of Guangdong Province, China (No. 2017A030310564, 2014A030313514).
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CG conceived of the study, performed the theoretical derivation and wrote this paper. SF revised and edited the paper. Both of the authors contributed equally in preparing this paper and approved this version. All authors read and approved the final manuscript.
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Guo, C., Fang, S. Existence and blowup of solutions for the modified Klein-Gordon-Zakharov equations for plasmas with a quantum correction. Adv Differ Equ 2017, 293 (2017). https://doi.org/10.1186/s13662-017-1357-1
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DOI: https://doi.org/10.1186/s13662-017-1357-1
Keywords
- existence and blowup
- modified Klein-Gordon-Zakharov equations
- quantum correction
- dilation transformation