Global solution in a weak energy class for Klein-Gordon-Schrödinger system

1.   Introduction and results

In this paper we consider the global existence and uniqueness of solutions in the energy class for the one dimensional Klein-Gordon-Schrödinger(KGS) equations with partial initial data nonvanishing at $x = \pm \infty$:

where $\partial_{t} = \partial/\partial t, \partial_{x} = \partial/\partial x$ and $i = \sqrt{-1}$. $\bar{\psi}$ denotes the complex conjugate of $\psi$. This system describes the interaction between nucleons and mesons. Here $\psi$ is a complex scalar nucleon field while $\varphi$ is a real scalar meson one. In order to obtain more physical details, one can refer to ^[1,2]. Formally, the solution $(\phi, \psi)$ of (1.1)–(1.3) satisfies the following conservation laws:

There is a lot of literature concerning the wellposedness for the classical KGS system. I. Fukuda and M. Tsutsumi^[3] established the global wellposedness in energy space for the cauchy problem by the projection method of the Galerkin's type and the compactness argument. They also investigated the existence and uniqueness of the solutions for the initial-boundary value problem in ^[4,5]. The authors of ^[6,7] proved the existence of global strong solutions to the initial value problem. B. Wang^[8] studied the classical solutions for this equations with nonlinear terms and proved the global existence of small solutions. Moreover, some low regularity wellposedness results were reported. J. Colliander, J. Holmer and N. Tzirakis ^[9] pointed out the 3d KGS system is global wellposed in $L^{2} \times H^{-1/2}$. H. Pecher ^[10] developed this result in $H^{ > -{1/4}} \times H^{ > -{1/2}}$. In addition, some works associated with asymptotic limit were invoked in ^[11,12].

However, we note that most works are devoted to the high dimensional case, and they may not have so much interest in one dimensional case. In one space dimension, it does not seem to be extremely necessary outcome that the following boundary condition at $x = \pm\infty$ is imposed on the solution $\varphi$ of (1.1): $\varphi(t, x)\rightarrow 0 (|x|\rightarrow \infty)$. Indeed, without the loss of generality, we assume $x > 0$ and consider the stationary state

Integrating (1.1) from 0 to $x$ yields

which implies $\varphi(x)$ may increase at the level of $e^{x}$ with respect to the spatial variable. The regime $x < 0$ can be treated analogously.

This fact inspired us to consider KGS system (1.1)–(1.3) with restriction (1.4). To our knowledge, this issue has never been investigated in previous literature.

In order to study the existence of this kind of solutions satisfying (1.4), we introduce a function $M(x)$, which is a real-valued function in $C^{\infty}(\mathbb{R})$ and satisfies for some $R > 0$:

Moreover, we put $\phi(t, x) = \varphi(t, x)-M(x)$, $\phi_{0}(x) = \varphi _{0}(x)-M(x)$ and $\phi_{1}(x) = \varphi_{1}(x)$. Then (1.1)–(1.4) can be rewritten as follows:

The solution $(\phi, \psi)$ of (1.7)–(1.10) formally conserves the $L^{2}$-norm $\mathcal{Q}$ and the energy $\mathcal{E}$ along the flow as well:

where

Generally, we call $(\phi, \psi)$ the energy solution, if it is the solution of (1.7)–(1.10) and belongs to the class $H^{1}(\mathbb{R})\times L^2(\mathbb{R})\times H^{1}(\mathbb{R})$. However, when we try to prove the global existence and the uniqueness of the solutions (1.7)–(1.10), due to the boundlessness of $M(x)$, $(\phi, \psi)$ can not be bounded uniformly in this class, which leads that the compactness argument can not be applied to the system (1.7)–(1.10). For this, we consider a new weak topology $H^{1}(\mathbb{R})\times L^2(\mathbb{R}) \times(H^{1}(\mathbb{R})\cap L^{2}(\mathbb{R}; e^{|x|}dx))$ as energy class, indeed, it is the weakest function space in which the energy identity (1.11) makes sense, in which by utilizing the method in ^[3], we also can establish the local wellposedness. In addition, with the help of Gronwall type estimates, the nonlinear term $\int_{\mathbb{R}}M(x)|\psi(x)|^{2}dx$ can be bounded by a function depending on time, which leads to the possibility of global existence.

Before stating results, we first introduce several basic notations used throughout this paper. For $m\in \mathbb{N}$, we denote the standard $L^{2}$ Sobolev space by $H^{m}(\mathbb{R})$ and its dual space by $H^{-m}(\mathbb{R})$, where we put $H^{0}(\mathbb{R}) = L^{2}(\mathbb{R})$. For $s\in\mathbb{R}$, we denote the function space $L^{2}(\mathbb{R}; e^{s|x|}dx)$ by $\Pi(s)$. Let $\mathcal{H}^{1}$ denote the space $H^{1}(\mathbb{R})\cap \Pi(\frac{1}{2})$, and $\mathcal{H}^{-1}$ denote its dual space. For $m\in\mathbb{N}$, $1\leq p\leq\infty$, $I\subset\mathbb{R}$ denote an interval and $X$ denote a Banach space, we define the Banach space $W^{m, p}(I; X)$ by

with the norm

Now we state our main results in this paper.

Theorem 1.1. Assume that $(\phi_{0}, \phi_{1})\in H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})$, where $\phi_{0}$ and $\phi_{1}$ are real-valued, $\psi_{0}\in \mathcal{H}^{1}$ and $M$ is real-valued smoothing function satisfying $(A)$. Let $I$ be a bounded open interval in $\mathbb{R}$ with $0\in I$. Then, $(1.7)$–$(1.10)$ is uniquely solvable in the following class:

Remark 1. If the function $(\phi, \psi)$ satisfies (1.12), (1.13) and (1.7), (1.8) in the distribution sense, then we have

Here, $C_{w}^{m}(I; X)$ denotes the set of all $m$-time weakly continuously differentiable functions from $I$ to $X$.

Moreover, we can actually construct the energy solution globally in time. We have the following stronger wellposedness result in the weak energy class $H^{1}(\mathbb{R})\times L^2(\mathbb{R}) \times\mathcal{H}^{1}(\mathbb{R})$.

Theorem 1.2. Under the assumptions of Theorem 1.1. There exists a unique solution $(\phi, \psi)$ of $(1.7)$–$(1.10)$ such that

Remark 2. From $L^2$ conservation and stationary equation, it only can yield $\varphi$ in systems (1.1)–(1.4) admits the singularity, i.e., the exponential spatial growth form, this property cannot be extended to $\psi$ in Schrödinger equation. It turns out that $\psi$ decays exponentially, while $\varphi$ increases exponentially.

The rest of this paper is organized as follows. In Section 2, we prepare some basic lemmas which are used to prove our main theorems. Section 3 is devoted to proving the existence and the uniqueness for the weak solutions.

2.   Preliminaries

In this section, we state two lemmas for the proof of Theorems 1.1 and 1.2. We begin with the estimates for the linear perturbed Schrödinger equation.

Lemma 2.1. We put $H(t) = (\partial_{x}+it(\partial_{x}M))^{2}$. Let $M$ satisfy $(A)$. Then, the operator $iH(t)$ generates the evolution operator $U(t, s)$, $-\infty < s\leq t < +\infty$ associated with the linear Schrödinger equation:

The evolution operator $U(t, s)$ satisfies the following Strichartz estimates for any $T > 0$.

(i) Let $I = (-T, T)$. Assume that $f\in L^{1}(I; L^{2})$, $q$ and $r$ are positive constants such that $2\leq q\leq\infty$ and $(\frac{1}{2}-\frac{1}{q})r = 2$. Then,

where $C$ depends only on $q$ and $T$.

(ii) Let $I = (-T, T)$, $q$ and $r$ be two positive constants such that $1\leq q\leq 2$ and $\frac{1}{q}+\frac{2}{r} = \frac{5}{2}$. Assume that $f\in L^{r}(I; L^{q})$. Then,

where $C$ depends only on $q$ and $T$.

Proof. The Lemma 2.1 is variant of the Strichartz estimates. For the proof, one can see ^{[13,14,15,16]}.

Next, let us recall the Gagliardo-Nirenberg inequality ^[17] in $\mathbb{R}$, which will be used to treat the nonlinear terms.

Lemma 2.2. Let $1\leq p, q, r\leq\infty$ and let $j, m$ be two integers, $0\leq j < m$. If

for some $a\in[\frac{j}{m}, 1]$ ($a < 1$ if $r > 1$ and $m-j-\frac{1}{r} = 0$), then there exists $C(m, j, a, q, r)$ such that

for every $u\in\mathcal{D}(\mathbb{R})$.

3.   Proof of the main results

In this part, we will prove Theorems 1.1 and 1.2. We start with the uniqueness of the weak energy solutions.

Proof of Theorem 1.1. We assume that $(\phi, \psi)$ is the solution of (1.7)–(1.10) satisfying (1.12)–(1.13), and put

By (1.19) and (A), for a.e., $t\in I$, we note that $u\in L^{\infty}(I; H^{1})$ and $\partial_{t}u\in L^{\infty}(I; H^{-1})$. Then, the function $u$ satisfies

Since all the terms except $i\partial_{t}u$ belong to $H^{-1}$ in (3.3), we can conclude that $\partial_{t}u\in H^{-1}$ and (3.3) holds true in $H^{-1}$. Let $U(t, s)$ be the evolution operator generated by $i(\partial_{x}+2it(\partial_{x}M))^{2}$ (see Lemma 2.1). By the Duhamel principle, (3.2) and (3.3), we have

Let $(\phi, \psi)$ and $(\tilde{\phi}, \tilde{\psi})$ be the two different solutions to (1.7)–(1.10) with the same initial data such that $\phi$ and $\tilde{\phi}$ are real-valued, and $(\phi, \psi)$ and $(\tilde{\phi}, \tilde{\psi})$ satisfy (1.12)–(1.13). And let $v$ be defined as in (3.1) for $\tilde{\psi}$. Then, $u$ and $v$ satisfy (3.4). Set

Then, by (3.4), we have

Let $T$ be a positive constant with $[0, T]\subset I$ to be determined later. Denote $I_{T} = [0, T]$, and take the $L^{\infty}(I_{T}; L^{2})$ norm of (3.5), by Lemma 2.1, (1.12) and (1.13), we can obtain

where $C$ is a positive constant. Moreover, from the definition of $\alpha$, it follows that $\alpha(t)$ satisfies

If we put $\omega = (1-\partial_{x}^{2})^{\frac{1}{2}}$, by (3.7), we then have

Using Sobolev embedding $L^{1}\hookrightarrow H^{-1}$, we can deduce,

And, among them

Accordingly, we conclude with the estimates (3.6) and (3.10)–(3.11)

where $C(T)$ is a positive constant depending increasingly on $T$. If we choose $T > 0$ so small that $C(T)\leq \frac{1}{2}$, then (3.12) implies that

We repeat the above procedure to obtain (3.13) for all $t\in I$. This shows Theorem 1.1.

Next we prove Theorem 1.2 by using the results stated in Theorem 1.1.

Proof of Theorem 1.2. Let $\{(\phi_{0n}, \phi_{1n})\}$, $\{\psi_{0n}\}$ and $\{v_{n}\}$ be the sequences in $C_{0}^{\infty}(\mathbb{R})$ such that $\phi_{0n}\rightarrow \phi_{0}$ in $H^{1}$, $\phi_{1n}\rightarrow \phi_{1}$ in $L^{2}$ and $\psi_{0n}\rightarrow \psi_{0}$ in $\mathcal{H}^{1}$. Actually, we can choose these sequences as follows. Let $\chi$ be a function in $C_{0}^{\infty}(\mathbb{R})$ with compact support on $[0, 2]$ such that $\chi(x) = 1$ for $|x|\leq 1$, and let $\rho$ be a function in $C_{0}^{\infty}(\mathbb{R})$ such that $\rho\geq 0$ and $\int_{\mathbb{R}}\rho(x)dx = 1$. For $\eta, \epsilon > 0$, we define $\chi_{\eta}(x) = \chi(\eta x)$ and $\rho_{\epsilon}(x) = \epsilon^{-1}\rho(x/\epsilon)$. Moreover, we define

where $\ast$ denotes the convolution with respect to the spatial variable $x$. With the suitable choice of $\eta, \varepsilon > 0$, we may obtain the desired sequences.

We consider the Cauchy problem of (1.7)–(1.10) with $\partial_{x}^{2}M-M$ replaced by $m_{n}$ in (1.8). Then, for each pair of the initial data $(\phi_{0n}, \phi_{1n}, \psi_{0n})$, we have the unique local solutions $(\phi_n, \psi_{n})$ of the initial value problem of (1.7)–(1.10) belonging to

with $\phi_{n}$ real-valued, where the existence time $T > 0$ depends only on $\|\phi_{0n}\|_{H^{3}}$, $\|\phi_{1n}\|_{H^{2}}$ and $\|\psi_{0n}\|_{H^{3}\cap\Pi(1/2)}$. For the local existence of smooth solutions, one can also refer to Theorem 2.1 in ^[18].

We multiply (1.7) by $\partial_{t}\phi_{n}$ and multiply (1.8) by $\partial_{t}\bar{\psi}_{n}$, respectively, take the real part to obtain the energy identity:

where the energy functional $\mathcal{E}$ is defined as in (1.11). Moreover, multiplying (1.8) by $\bar{{\psi}}_{n}$, integrating the resulting equation in $x$ over $\mathbb{R}$ and taking the imaginary part, we can obtain

Next we estimate $\int_{\mathbb{R}}|\psi(x)|^{2}\phi(x)dx$. Applying Cauchy inequality, Hölder inequality and Lemma 2.2, we get

which implies

where $C(R)$ is a constant which is only depending on $R$ given in $(A)$. Therefore,

Inequality (3.18) implies that

Then, we estimate $\int_{\mathbb{R}}M(x)|\psi(x)|^{2}dx$. We multiply (1.8) by $M\bar{\psi}_{n}$, integrate the resulting equation in $x$ over $\mathbb{R}$ and take the imaginary part to get

Using Gronwall inequality and (3.20), we get

Therefore,

where $C$ does not depend on $n$ and $T$. (3.18), (3.21), (3.22) and the linear hyperbolic theory show that

where $C$ does not depend on $n$ and $T$.

Similarly, it is not difficult to obtain the a priori estimates (dependent on $n$) for the higher order derivatives of $(\phi_{n}, \psi_{n})$, by which we can extend the above local solutions $(\phi_{n}, \psi_{n})$ globally in time for each $n\geq 1$. Accordingly, (3.15)–(3.23) hold true for any time $t\in \mathbb{R}$. Thereafter, applying the standard compactness analysis, one can derive that there exists at least one global energy solution $(\phi(t), \psi(t))$ of (1.7)–(1.10) satisfying (1.18)–(1.19) and

Moreover, the equation (1.7) and the theory of linear hyperbolic equations, together with (1.13), imply that

With that, we claim

Since

it follows

Notice that $\psi(t)\in C(\mathbb{R}; L^{2})\cap C_{w}(\mathbb{R}; H^{1})\cap C_{w}(\mathbb{R}; \Pi(1/2))$(see Remark 1), by (3.24) and (3.25), we conclude the claim. Moreover, Let $t_{0}$ be an arbitrary real constant with $t_{0}\neq 0$. In view of the uniqueness in Theorem 1.1, we then have

A replacement of the initial time and the initial data by $t_{0}$ and $(\phi(t_{0}), \partial_{t}\phi(t_{0}), \psi(t_{0}))$ will follows the strong continuity of solution in energy class $H^{1}\times L^{2}\times \mathcal{H}^{1}$ at any time. This closes the proof.

Remark 3. Due to the chaotic structure inside the atom, the generation of this singular solution is not difficult to understand. It should be pointed out that we can also consider the existence of the solutions of this system in the higher-dimensional space $\mathbb{R}^{n+1}(n\geq 2)$ by compactness argument. However, it does not guarantee that the solution increases exponentially in all spatial directions. In addition, due to the breakdown of the Sobolev embedding $H^1\hookrightarrow L^\infty$, we can not obtain the uniqueness of this kind of solutions.

Acknowledgments

The authors are grateful to the anonymous referees for valuable suggestions. The authors are supported by NNSF of China (Nos. 12061040, 11701244) and NSF of Gansu Province(CN) (Nos. 20JR5RA460, 21JR7RA217).

Conflict of interest

The authors have no conflicts to disclose.