Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes

TL;DR

This paper studies the stability and asymptotic behavior of even solutions near the soliton for the 1D nonlinear Klein-Gordon equation with certain nonlinearities, using virial estimates due to the absence of internal modes.

Contribution

It establishes asymptotic stability for solutions near the soliton in the absence of internal modes and resonance, employing virial estimates instead of dispersive tools.

Findings

Stability in energy space implies asymptotic stability in local energy norm.

Existence of a Lipschitz graph of initial data leading to stable trajectories.

Applicable for nonlinearities with lpha > 1, where no internal modes are present.

Abstract

We consider the dynamics of even solutions of the one-dimensional nonlinear Klein-Gordon equation $\partial_t^2 \phi - \partial_x^2 \phi + \phi - |\phi|^{2\alpha} \phi =0$ for $\alpha>1$, in the vicinity of the unstable soliton $Q$. Our main result is that stability in the energy space $H^1(\mathbb R)\times L^2(\mathbb R)$ implies asymptotic stability in a local energy norm. In particular, there exists a Lipschitz graph of initial data leading to stable and asymptotically stable trajectories. The condition $\alpha>1$ corresponds to cases where the linearized operator around $Q$ has no resonance and no internal mode. Recall that the case $\alpha>2$ is treated in Krieger-Nakanishi-Schlag using Strichartz and other local dispersive estimates. Since these tools are not available for low power nonlinearities, our approach is based on virial type estimates and the particular structure of the linearized operator observed in Chang-Gustafson-Nakanishi-Tsai.

Equations

$${\partial }_t^2\varphi -{\partial }_x^2\varphi +\varphi -f(\varphi )=0,(t,x)\in \mathbb{R}\times \mathbb{R},f(\varphi )=|\varphi {|}^{2\alpha }\varphi ,$$

$$\{\begin{array}{rl}& {\dot{\varphi }}_1={\varphi }_2\\ & {\dot{\varphi }}_2={\partial }_x^2{\varphi }_1-{\varphi }_1+f({\varphi }_1).\end{array}$$

$$E(\varphi ,{\partial }_t\varphi )=\frac{1}{2}\int \left\{({\partial }_t\varphi {)}^2+({\partial }_x\varphi {)}^2+{\varphi }^2-2F(\varphi )\right\}=E(\varphi (0),{\partial }_t\varphi (0)).$$

$$Q(x)=\frac{(\alpha+1)^{\frac{1}{2\alpha}}}{\cosh^{\frac{1}{\alpha}}(\alpha x)},\qquad Q^{\prime\prime}-Q+Q^{2\alpha+1}=0\quad\mbox{on $\mathbb{R}$.}$$

$$L=-{\partial }_x^2+1-(2\alpha +1)Q^{2\alpha }=-{\partial }_x^2+1-\frac{(2\alpha +1)(\alpha +1)}{{\cosh }^2(\alpha x)}.$$

$$Y_0(x)=c_0{\left(\cosh (\alpha x)\right)}^{-(1+\frac{1}{\alpha })},⟨Y_0,Y_0⟩=1,L{Y}_0=-{\nu }_0^2{Y}_0$$

$$\boldsymbol{Y}_{\pm}=\left(\begin{array}[]{c}Y_{0}\\ \pm\nu_{0}Y_{0}\end{array}\right),\quad\boldsymbol{Z}_{\pm}=\left(\begin{array}[]{c}Y_{0}\\ \pm\nu_{0}^{-1}Y_{0}\end{array}\right).$$

$$\{\begin{array}{rl}& {\dot{u}}_1=u_2\\ & {\dot{u}}_2=-L{u}_2\end{array}$$

$$\mbox{for all $t\geq 0$,}\quad\|\boldsymbol{\phi}(t)-(Q,0)\|_{H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})}<\delta$$

$$\underset{t\to +\infty }{\lim }\Vert \mathbf{\varphi }(t)-(Q,0){\Vert }_{H^1(I)\times L^2(I)}=0.$$

$$\mathcal{A}_{0}=\left\{\boldsymbol{\varepsilon}\in H^{1}(\mathbb{R})\times L^{2}(\mathbb{R})\mbox{ such that $\boldsymbol{\varepsilon}$ is even, }\|\boldsymbol{\varepsilon}\,\|_{H^{1}\times L^{2}}<\delta_{0}\mbox{ and }\langle\boldsymbol{\varepsilon},\boldsymbol{Z}_{+}\rangle=0\right\}.$$

$$\mathcal{M}=\left\{(Q,0)+\mathbf{\epsilon }+h(\mathbf{\epsilon }){\mathbf{Y}}_{+}\text{mbox}with\mathbf{\epsilon }\in {\mathcal{A}}_0\right\}$$

$$\Vert \mathbf{\varphi }(t)-(Q,0){\Vert }_{H^1(\mathbb{R})\times L^2(\mathbb{R})}\le C\Vert {\mathbf{\varphi }}_0-(Q,0){\Vert }_{H^1(\mathbb{R})\times L^2(\mathbb{R})}.$$

$$\Vert \mathbf{\varphi }(t)-(Q,0){\Vert }_{H^1(\mathbb{R})\times L^2(\mathbb{R})}<\frac{1}{2}{\delta }_0,$$

$${\int }_0^{+\infty }\Vert \mathbf{\varphi }(t)-(Q,0){\Vert }_{H^1(I)\times L^2(I)}^{2}dt<\infty .$$

$$Y_2(x)=c_2{Y}_0(x)\left(1-\frac{2}{\alpha }{\sinh }^2(\alpha x)\right),{\lambda }_2=\alpha (2-\alpha ),{\nu }_2={\lambda }_2^{\frac{1}{2}}.$$

$$\mathbf{u}(t)=(\cos ({\nu }_{2}t)Y_2,-{\nu }_2\sin ({\gamma }_{2}t)Y_2)$$

$$\{\begin{array}{rl}& \varphi (t,x)=Q(x)+a_1(t)Y_0(x)+u_1(t,x)\\ & {\partial }_t\varphi (t,x)=a_2(t){\nu }_0{Y}_0(x)+u_2(t,x)\end{array}$$

$$a_1(t)=⟨\varphi (t)-Q,Y_0⟩,a_2(t)=\frac{1}{{\nu }_0}⟨{\partial }_t\varphi (t),Y_0⟩,$$

$$⟨u_1(t),Y_0⟩=⟨u_2(t),Y_0⟩=0.$$

$$b_{+}=\frac{1}{2}(a_1+a_2),b_{-}=\frac{1}{2}(a_1-a_2),$$

$$\mathbf{\varphi }=(Q,0)+\mathbf{u}+b_{-}{\mathbf{Y}}_{-}+b_{+}{\mathbf{Y}}_{+},\mathbf{u}=(u_1,u_2).$$

$$\Vert u_1(t){\Vert }_{H^1}+\Vert u_2(t){\Vert }_{L^2}+|a_1(t)|+|a_2(t)|+|b_{+}(t)|+|b_{-}(t)|\le C_0\delta .$$

$$\{\begin{array}{rl}& {\dot{a}}_1={\nu }_0{a}_2\\ & {\dot{a}}_2={\nu }_0{a}_1+\frac{N_0}{{\nu }_0}\end{array}\text{mbox}equivalently\{\begin{array}{rl}& {\dot{b}}_{+}={\nu }_0{b}_{+}+\frac{N_0}{2{\nu }_0}\\ & {\dot{b}}_{-}=-{\nu }_0{b}_{-}-\frac{N_0}{2{\nu }_0}\end{array}$$

$$\{\begin{array}{rl}& {\dot{u}}_1=u_2\\ & {\dot{u}}_2=-L{u}_1+N^{\perp }\end{array}$$

$$N=f(Q+a_1{Y}_0+u_1)-f(Q)-f^{\prime }(Q)a_1{Y}_0-f^{\prime }(Q)u_1,$$

$$N_0=⟨N,Y_0⟩,N^{\perp }=N-N_0{Y}_0.$$

$$\rho (x)=\mathrm{sech}\left(\frac{x}{10}\right).$$

$$\Vert w{\Vert }_{\rho }={\left[\int \left(({\partial }_{x}w{)}^2+\rho w^2\right)\right]}^{\frac{1}{2}}.$$

$$\chi =1$$

$${\zeta }_A(x)=\exp \left(-\frac{1}{A}(1-\chi (x))|x|\right),{\phi }_A(x)={\int }_0^x{\zeta }_A^2(y)dy,x\in \mathbb{R}.$$

Full text

The condition $\alpha>1$ corresponds to cases where the linearized operator around $Q$ has no resonance and no internal mode. Recall that the case $\alpha>2$ is treated in [22] using Strichartz and other local dispersive estimates. Since these tools are not available for low power nonlinearities, our approach is based on virial type estimates and the particular structure of the linearized operator observed in [6].

M.K. was partially funded by Chilean research grants FONDECYT 1170164. C.M. was partially funded by Chilean research grants FONDECYT 1150202. M.K. and C.M. were partially funded by project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. Part of this work was done while C.M. and M.K. were visiting the CMLS at École Polytechnique, France. Part of this work was done while C.M. was visiting the Departamento de Matemáticas Aplicadas de Granada, UGR, Spain.

1. Introduction

1.1. Main results

Consider the one-dimensional focusing nonlinear Klein-Gordon equation

[TABLE]

where $\alpha>0$. This equation also rewrites as a first order system in time for the function $\boldsymbol{\phi}=(\phi,\partial_{t}\phi)=(\phi_{1},\phi_{2})$,

[TABLE]

Let $F(\phi)=\int_{0}^{\phi}f(s)ds=\frac{1}{2\alpha+2}|\phi|^{2\alpha+2}$. Note that (1) is Hamiltonian. The conservation of energy of a solution $(\phi,\partial_{t}\phi)$ of (1) writes

[TABLE]

For initial data in the energy space $H^{1}\times L^{2}$, local well-posedness, as well as global well-posedness for small solutions, is well-known (see for example [5], Theorem 6.2.2 and Proposition 6.3.3).

Denote by $Q$ the standing wave solution of (1), also called soliton, explicitly given by

[TABLE]

The linearized operator $L$ around $Q$ writes

[TABLE]

For any $\alpha>0$, the first eigenvalue of $L$ is $\lambda_{0}=-\alpha(\alpha+2)=-\nu_{0}^{2}$ ($\nu_{0}>0$) with corresponding normalized eigenfunction

[TABLE]

(we denote $\langle A,B\rangle=\int A\cdot B$). The second eigenvalue of $L$ is [math] with eigenfunction $Y_{1}=c_{1}Q^{\prime}$. In the case $\alpha>1$, there is no other eigenvalue in $[0,1)$, which means that there is no internal mode for the model (see Section 1.3).

Let

[TABLE]

The functions $\boldsymbol{u}_{\pm}(t,x)=e^{\pm\nu_{0}t}\boldsymbol{Y}_{\pm}(x)$ are solutions of the linearized problem

[TABLE]

illustrating the presence of exponentially stable and unstable modes both relevant in the dynamics of solutions in the vicinity of a soliton.

In this paper, by global solution of (1), we mean a function $\boldsymbol{\phi}\in\mathcal{C}([0,\infty),H^{1}\times L^{2})$ satisfying (1) for all $t\geq 0$. We only consider solutions with even symmetry.

Our main result is the following conditional asymptotic stability theorem.

Theorem 1**.**

Let $\alpha>1$. There exists $\delta>0$ such that if a global even solution $\boldsymbol{\phi}=(\phi,\partial_{t}\phi)$ of (1) satisfies

[TABLE]

then, for any interval $I$ of $\mathbb{R}$,

[TABLE]

For the sake of completeness, we provide a description of the set of initial data leading to global solutions satisfying the stability assumption (6) (see also Theorem 4.1 in [2]).

For $\delta_{0}>0$, let

[TABLE]

Theorem 2**.**

Let $\alpha>1$. There exist $C,\delta_{0}>0$ and a Lipschitz function $h\colon\mathcal{A}_{0}\to\mathbb{R}$ with $h(0)=0$ and $|h(\boldsymbol{\varepsilon})|\leq C\|\boldsymbol{\varepsilon}\|_{H^{1}\times L^{2}}^{3/2}$ such that denoting

[TABLE]

the following holds

1. (i)

If $\boldsymbol{\phi}_{0}\in\mathcal{M}$ then the solution $\boldsymbol{\phi}$ of (1) with initial data $\boldsymbol{\phi}_{0}$ is global and satisfies, for all $t\geq 0$,

[TABLE] 2. (ii)

If a global even solution $\boldsymbol{\phi}$ of (1) satisfies, for all $t\geq 0$,

[TABLE]

then for all $t\geq 0$, $\boldsymbol{\phi}(t)\in\mathcal{M}$.

1.2. Related results and comments on the proof

First, we comment on two articles devoted to soliton dynamics for the one-dimensional nonlinear Klein-Gordon equation (1).

Using techniques based on Strichartz and other local dispersive estimates, Krieger et al. [22] have completely treated the case $\alpha>2$ in the case of even data. Indeed, they classify all solutions whose energy does not exceed too much that of the ground state $Q$. This includes the construction, by the fixed point argument, of a $\mathcal{C}^{1}$ center-stable manifold around the soliton and the proof of asymptotic stability and scattering (linear behavior) around the ground state for solutions on the manifold. The method seems limited to $\alpha\geq 2$ because of the use of Strichartz estimates to control the nonlinear term, see comment in Section 3.4 of [22].

By formal and numerical methods, Bizoń et al. [4] have shown that for even solutions trapped by the soliton, the convergence rate to $Q$ heavily depends on the power $\alpha$ of the nonlinearity. In the $L^{\infty}$ sense, they conjecture the following trichotomy: (a) fast dispersive decay for $\alpha>1$; (b) slow decay for $\alpha=1$; (c) very slow decay for $0<\alpha<1$. The threshold value $\alpha=1$ corresponds to the emergence of a resonance at the linear level, while $\alpha<1$ leads to one or several internal modes (see Section 1.3). Following these observations, unifying the case $\alpha>1$ was the main motivation of the present work.

Our method does not give an explicit decay rate as $t\to+\infty$, but we notice as a by-product of the proof of Theorem 1 that, for any interval $I$ of $\mathbb{R}$, it holds

[TABLE]

This is to be compared with the results obtained in [18] on the (local) asymptotic stability of the kink for the $\phi^{4}$ model under small odd perturbations. Indeed, in the latter case, the presence of an internal mode leads to a lower convergence rate since the component $z(t)$ of the solution along the internal mode only satisfies the weaker estimate $\int_{0}^{+\infty}|z(t)|^{4}dt<\infty$ (see Theorem 1.2 in [18]). Although we do not claim optimality of such results, in the case of (1) with $0<\alpha\leq 1$, we do not expect estimates such as in (10) to hold.

The proof of Theorem 1 is mainly based on localized virial type arguments similar to that used in [18, 25, 27], for example. Unlike in these works, we avoid numerical computations of certain constants related to the coercivity of the virial functional by using factorization properties of the linearized operator described in [6] (see also references [29, 37], cited in [6]). A formal presentation of this approach is given in Section 4.1. We point out that the same structure was crucially used in the construction of blow-up solutions for the wave maps, Yang-Mills and $O(3)$ $\sigma$-models in [30, 31]. Note that in the present paper, we compensate the loss of two derivatives due to the change of variables to still work in the energy space.

We refer to [1, 16, 17, 19, 20, 23, 35, 36] for various results of asymptotic stability for the nonlinear Klein-Gordon equation and $\phi^{4}$ equation or variants of these models.

Several other conditional asymptotic stability results or classifications in a neighborhood of the ground state for the nonlinear Klein-Gordon in higher dimensions and for the nonlinear Schrödinger equation were also obtained in [10, 11, 32, 34], for example. We also mention [21] where for the mass supercritical Schrödinger equation in one dimension, a finite co-dimensional manifold of initial data trapped by the soliton was constructed.

Concerning the generalized Korteweg-de Vries equation and related models, studies of the dynamics of the solutions close to the soliton are presented in [9, 14, 15, 24, 26, 27, 28, 33], in blow-up contexts or for bounded solutions. Note that the method introduced in [24, 26], using the special structure of a transformed linearized problem, also has some analogy with our proof.

For global existence results in the case of semilinear and quasilinear wave equations, we refer to [12, 13].

Finally, we refer to [2, 3] and references therein for refined descriptions of dynamics of solutions in various settings.

1.3. Resonances and internal modes

As mentioned before, the absence of any other eigenvalue in $[0,1)$ for the operator $L$ when $\alpha>1$ is important in our proof. For $0<\alpha\leq 1$, we continue the description of the spectrum of $L$. For $\alpha=1$, there is an even resonance at $1$. For any $0<\alpha<1$, there is a third eigenvalue associated to an even eigenfunction

[TABLE]

In particular, for any $0<\alpha<1$, the function

[TABLE]

is solution of (5). These solutions are typical of the notion of internal modes and show that asymptotic stability (even up to the exponential instable mode) cannot be true at the linear level for such value of $\alpha$. An important issue is the nature of the interaction of such internal mode with the nonlinearity. We recall that such an internal mode was treated in the context of the $\phi^{4}$ equation in  [18]. Pioneering results on internal modes were obtained in [36]. See other references in [18].

For $\alpha\in(\frac{1}{2},1)$, there are no other eigenvalue on $[0,1)$. For $\alpha=\frac{1}{2}$, there is an odd resonance at $1$. For $\alpha\in(\frac{1}{3},\frac{1}{2})$, there is a fourth eigenvalue, associated to an odd eigenfunction. For $\alpha\in(\frac{1}{4},\frac{1}{3})$, there are five eigenvalues, three of them being associated to even eigenfunctions. In particular, there are two even internal modes. This procedure can be continued for all $\alpha>0$, showing the emergence of arbitrarily many internal modes (and sometimes resonances) as $\alpha\to 0^{+}$.

The above information is taken from Section 3 of [6].

2. Preliminaries

2.1. Decomposition of a solution in a vicinity of the soliton

Let $\boldsymbol{\phi}=(\phi,\partial_{t}\phi)$ be a solution of (1) satisfying (6) for some small $\delta>0$. We decompose $(\phi,\partial_{t}\phi)$ as follows

[TABLE]

where

[TABLE]

so that

[TABLE]

Setting

[TABLE]

we observe that $\boldsymbol{\phi}$ also writes as

[TABLE]

From (6), for all $t\in[0,\infty)$, it holds

[TABLE]

Moreover, using $Q^{\prime\prime}-Q+f(Q)=0$, $LY_{0}=-\nu_{0}^{2}Y_{0}$ and (12), the systems of equations of $(a_{1},a_{2})$ and $(u_{1},u_{2})$ write

[TABLE]

and

[TABLE]

where

[TABLE]

2.2. Notation for virial arguments

Let $\rho$ be the following weight function

[TABLE]

For any function $w\in H^{1}$, consider the norm

[TABLE]

We consider a smooth even function $\chi:\mathbb{R}\to\mathbb{R}$ satisfying

[TABLE]

For $A>0$, we define the functions $\zeta_{A}$ and $\varphi_{A}$ as follows

[TABLE]

For $B>0$, we also define

[TABLE]

and we consider the function $\psi$ defined as

[TABLE]

The notation $X\lesssim Y$ means $X\leq CY$ for a constant independent of $A$ and $B$.

These functions $\zeta_{A}$, $\varphi_{A}$, $\zeta_{B}$, $\varphi_{B}$ and $\psi_{B}$ will be used in two distinct virial arguments with different scales

[TABLE]

3. Virial argument in $u$

Set

[TABLE]

and

[TABLE]

We refer to [18] for the use of such virial argument in a similar context. Here, $w$ represents a localized version of $u_{1}$, in the scale $A$ (see (24)). We shall prove the following result.

Proposition 1**.**

There exist $C_{1}>0$ and $\delta_{1}>0$ such that for any $0<\delta\leq\delta_{1}$, the following holds. Fix $A=\delta^{-1}$. Assume that for all $t\geq 0$, (15) holds. Then, for all $t\geq 0$,

[TABLE]

Remark 1*.*

Note that estimate (27) does not involve any type of spectral analysis. Its purpose is to give a simple control of $\int(\partial_{x}w)^{2}$ in terms of $\int\operatorname{sech}(\frac{x}{2})w^{2}$ and $|a_{1}|^{4}$.

The rest of this section is devoted to the proof of Proposition 1. We compute from (25)

[TABLE]

Replacing $\dot{u}_{1}$ by $u_{2}$ and integrating by parts, the first integral in the right-hand side vanishes. The expression of $\dot{u}_{2}$ in (17) rewrites

[TABLE]

and so

[TABLE]

To treat the first line in the expression of $\dot{\mathcal{I}}$, we claim the following.

Lemma 1**.**

It holds

[TABLE]

Moreover

[TABLE]

and

[TABLE]

Proof.

Proof of (28). By integration by parts

[TABLE]

We rewrite the above expression using the auxiliary function $w$. Indeed,

[TABLE]

and so

[TABLE]

Next,

[TABLE]

Identity (28) follows.

Proof of (29)-(30). By elementary computations, we have

[TABLE]

[TABLE]

which proves (29). Estimate (30) then follows from the definition of $\chi$. ∎

To treat the second line in the expression of $\dot{\mathcal{I}}$, we claim the following.

Lemma 2**.**

[TABLE]

Proof.

First, we treat the term $-\int\left(\varphi_{A}\partial_{x}u_{1}+\frac{1}{2}\varphi_{A}^{\prime}u_{1}\right)N_{0}Y_{0}$. By Taylor’s expansion, one has

[TABLE]

and thus, by decay estimates on $Q$ and $Y_{0}$, and by (15), $|a_{1}|\lesssim 1$, $\|u_{1}\|_{L^{\infty}}\lesssim\|u_{1}\|_{H^{1}}\lesssim 1$, $A\geq 4$, it holds

[TABLE]

Using integration by parts,

[TABLE]

Note that for all $x\in\mathbb{R}$, $|\varphi_{A}^{\prime}(x)|\leq 1$ and $|\varphi_{A}(x)|\leq|x|$, and so

[TABLE]

for an implicit constant independent of $A$. Thus, by the Cauchy-Schwarz inequality,

[TABLE]

Second, we decompose

[TABLE]

We rewrite $I_{1}$, $I_{2}$, $I_{3}$ and $I_{4}$ as follows

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

To control the two terms that are purely nonlinear in $u_{1}$, we need the following claim.

Claim 1**.**

It holds

[TABLE]

Proof of Claim 1.

The first equality in (36) corresponds to the definition of $w$ in (26). Next, by integration by parts and standard estimates, we have

[TABLE]

Thus,

[TABLE]

which implies (36). ∎

In particular, (36) implies that

[TABLE]

which takes care of the last terms in $I_{1}$ and $I_{4}$.

By Taylor expansion, $\alpha\geq 1$, $|a_{1}|\lesssim 1$ and $\|u_{1}\|_{L^{\infty}}\lesssim 1$, we have

[TABLE]

Similarly, using also (35) and $A\geq 4$, we find the following estimates

[TABLE]

[TABLE]

and

[TABLE]

Moreover, again by Taylor expansion and (35) (with $A>8$), we have

[TABLE]

Collecting these estimates, (32) is proved.

Taking $\|u_{1}\|_{L^{\infty}}\leq\delta_{A}$, for $\delta_{A}$ small enough, we have proved

[TABLE]

Using $A=\delta^{-1}$ and $\|u_{1}\|_{L^{\infty}}^{2\alpha}\lesssim\delta^{2\alpha}$ (from (15)), for $\delta_{1}$ small enough, we obtain (27). ∎

4. Virial argument for the transformed problem

4.1. Heuristic

We recall results from [6], pages 1086-1087. Let

[TABLE]

and

[TABLE]

(The above notation means $Uf=Y_{0}(Y_{0}^{-1}f)^{\prime}$.) Then, the operators $L$ and $L_{-}$ rewrite as $L=U^{\star}U+\lambda_{0}$, $L_{-}=UU^{\star}+\lambda_{0}$ and it follows that

[TABLE]

Now, let

[TABLE]

and

[TABLE]

A similar structure $L_{-}=S^{\star}S$, $L_{0}=SS^{\star}$, leads to

[TABLE]

In particular, let $(u_{1},u_{2})$ be a solution of (5), and set $\tilde{u}_{1}=Uu_{1}$, $\tilde{u}_{2}=Uu_{2}$. Then,

[TABLE]

Next, set

[TABLE]

Then, $(v_{1},v_{2})$ satisfies the following transformed problem:

[TABLE]

The key point for our analysis is that for $\alpha>1$, the potential in $L_{0}$ is positive. This property happens to be the only spectral information needed for the proof of Theorem 1.

Observe that $UY_{0}=0$, $UQ^{\prime}=-\alpha Q$ and $SQ=0$, which means that the prior decomposition of the solution $(\phi,\partial_{t}\phi)$ as in Section 2.1 and a coercivity argument as in Section 5 are necessary to avoid loosing information through the transformation. (Here, we work with even functions and so only the direction $Y_{0}$ is relevant.)

4.2. Transformed problem

With respect to the above heuristic, we need to localize and regularize the functions involved. For $\gamma>0$ small to be defined later, set

[TABLE]

where $\chi_{B}$ is defined in (23). We refer to Section 5 for coercivity results relating $u_{1}$ and $v_{1}$. The introduction of the operator $(1-\gamma\partial_{x}^{2})^{-1}$ with a small constant $\gamma$ is needed to compensate the loss of two derivatives due to the operator $SU$, without destroying the special algebra described heuristically. Now, we explain the role of the localization term $\chi_{B}$ in the definitions of $v_{1}$ and $v_{2}$. Note that Proposition 1 provides an estimate on the function $w$, which is a localized version of $u$ (see (26)). To use this information, the functions $v_{1}$ and $v_{2}$ also need to contain a certain localization.

We deduce the following system for $(v_{1},v_{2})$ from the one for $(u_{1},u_{2})$ in (17)

[TABLE]

First, we note that

[TABLE]

Moreover, since $SUL=L_{0}SU$, it holds

[TABLE]

Since

[TABLE]

we obtain

[TABLE]

Therefore, we have obtained the following system for $(v_{1},v_{2})$

[TABLE]

For this transformed system we construct a second virial functional, where the spectral analysis reduces to the fact that the potential in $L_{0}$ is positive.

4.3. Virial functional for the transformed problem

We set

[TABLE]

and (see (22) and (23))

[TABLE]

Here, $z$ represents a localized version of the function $v_{1}$. The scale of localization $B$ is intermediate between the one involved in the definition of $w$ from $u_{1}$ (see (24) and (26)) and the weight function $\rho$ defined in (19) (similar to a localization at the soliton scale).

Proposition 2**.**

There exist $C_{2}>0$ and $\delta_{2}>0$ such that for $\gamma$ small enough and for any $0<\delta\leq\delta_{2}$, the following holds. Fix $B=\delta^{-\frac{1}{4}}$. Assume that for all $t\geq 0$, (15) holds. Then, for all $t\geq 0$,

[TABLE]

Remark 2*.*

The objective of estimate (41) is to control the local norm $\|z\|_{\rho}^{2}$ up to small error in terms of $\|w\|_{\rho}^{2}$ and $|a_{1}|^{3}$.

The rest of this section is devoted to the proof of Proposition 2. As in the computation of $\dot{\mathcal{I}}$ in the proof of Proposition 1, we have from (39),

[TABLE]

First, using the definition of $L_{0}$ in (37) and integrating by parts, we have

[TABLE]

From (23), we note that $\psi_{B}^{\prime}=\chi_{B}^{2}\zeta_{B}^{2}+(\chi_{B}^{2})^{\prime}\varphi_{B}$ and

[TABLE]

Thus,

[TABLE]

By the definition of $z$ in (40), proceeding as in the proof of (31) in Lemma 1, we have

[TABLE]

and

[TABLE]

Thus,

[TABLE]

where we have set

[TABLE]

Recalling (40), (23), (22) and integrating by parts,

[TABLE]

Therefore, setting

[TABLE]

we have obtained

[TABLE]

Lemma 3**.**

There exists $B_{0}>0$ such that for all $B\geq B_{0}$, $V\geq 0$ on $\mathbb{R}$. More precisely,

[TABLE]

Proof.

First, from (30) (with $A$ replaced by $B$), it holds

[TABLE]

Second, since for $x\in[0,+\infty)\mapsto\zeta_{B}(x)$ is non-increasing, we have for $x\geq 0$,

[TABLE]

Since $Q^{\prime}(x)\leq 0$ for $x\geq 0$, we obtain, for a constant $C>0$,

[TABLE]

choosing $B_{0}$ large enough. By parity, this estimate holds for any $x\in\mathbb{R}$. ∎

Using this lemma, and the above computations for $J_{1}$, we conclude

[TABLE]

To control the terms $\widetilde{J}_{1}$, $J_{2}$, $J_{3}$ and $J_{4}$, we need some technical estimates.

4.4. Technical estimates

Lemma 4**.**

1. (i)

Estimates on $w$.

[TABLE]

[TABLE] 2. (ii)

Estimates on $z$.

[TABLE]

[TABLE] 3. (iii)

Estimates on $v_{1}$.

[TABLE]

[TABLE]

Proof.

Proof of (44) and (45). For any $x,y\in\mathbb{R}$, using $w(x)=w(y)+\int_{y}^{x}\partial_{x}w$ and the inequality $(a+b)^{2}\leq 2a^{2}+2b^{2}$, we have

[TABLE]

Integrating (50) in $x\in[-2B^{2},2B^{2}]$ and $y\in[-1,1]$, we find (44). Multiplying (50) by \operatorname{sech}\bigl{(}\frac{x}{10}\bigr{)} and integrating in $x\in\mathbb{R}$ and $y\in[-1,1]$, we find (45).

Proof of (46) and (47). The proof is similar. For any $x\in\mathbb{R}$ and $y\in\mathbb{R}$, we have

[TABLE]

We multiply by \operatorname{sech}\bigl{(}\frac{x}{10}\bigr{)} and $V_{0}(y)\geq 0$ and integrate in $x\in\mathbb{R}$ and $y\in\mathbb{R}$. Since $\int V_{0}>0$ and $\int|y|V_{0}(y)dy<\infty$ from (42), we obtain (46).

We multiply by $\zeta_{B}(x)$ and $V_{0}(y)$ and integrate in $x\in\mathbb{R}$ and $y\in\mathbb{R}$. Since

[TABLE]

we obtain (47).

Proof of (48) and (49). Note by direct computations that

[TABLE]

Thus,

[TABLE]

Moreover, using Fourier analysis,

[TABLE]

As a consequence, it holds

[TABLE]

Using (51), the definition of $v_{1}$ in (38), the definition of $w$ in (26) and $A\gg B^{2}$, we obtain

[TABLE]

and then (44) implies (48).

Moreover, by direct computation

[TABLE]

Thus, similarly,

[TABLE]

Using (52), we obtain

[TABLE]

By the definition of $w$, $A\gg B^{2}$ and the definition of $\chi_{B}$ and $\zeta_{A}$, we have

[TABLE]

and $\|\chi_{B}u_{1}\operatorname{sech}(x)\|_{L^{2}}\lesssim\|w\operatorname{sech}(x)\|_{L^{2}}$. Thus, estimate (44) imply (49). ∎

Lemma 5**.**

For any $0<K\leq 1$ and $\gamma>0$ small enough, for any $f\in L^{2}$,

[TABLE]

where the implicit constant is independent of $\gamma$ and $K$.

Proof.

We set $g=\operatorname{sech}(Kx)(1-\gamma\partial_{x}^{2})^{-1}f$ and $k=\operatorname{sech}(Kx)f$. We have

[TABLE]

Thus,

[TABLE]

For $0<K\leq 1$ and $\gamma\leq\frac{1}{2}$, we apply the operator $\left[(1-\gamma K^{2})-\gamma\partial_{x}^{2}\right]^{-1}$, to obtain

[TABLE]

For $0<K\leq 1$ and $\gamma\leq\frac{1}{2}$, one has

[TABLE]

Thus, $\|\left[(1-\gamma K^{2})-\gamma\partial_{x}^{2}\right]^{-1}k\|_{L^{2}}\lesssim\|k\|_{L^{2}},$ and

[TABLE]

We deduce, for a constant $C$ independent of $\gamma$,

[TABLE]

which implies (53) for $\gamma$ small enough. ∎

4.5. Control of error terms

Now, we are in a position to control the error terms in (43).

Control of $\widetilde{J}_{1}$. By the definition of $\zeta_{B}$, it holds

[TABLE]

Thus, using the properties of $\chi$ in (21), we have

[TABLE]

Next, since $|\varphi_{B}|\lesssim B$ and $|(\chi_{B}^{2})^{\prime}|\lesssim B^{-2}$, $|(\chi_{B}^{2})^{\prime\prime\prime}|\lesssim B^{-6}$, we have

[TABLE]

Using (48)-(49), we conclude for this term

[TABLE]

Control of $J_{2}$. By the Cauchy-Schwarz inequality,

[TABLE]

First, we estimate using (53)

[TABLE]

From the definition of $z$ in (40), we have

[TABLE]

and so

[TABLE]

Using $|\chi^{\prime}|\lesssim 1$, the definitions of $\chi_{B}$ and $\zeta_{B}$ and again the definition of $z$

[TABLE]

and so

[TABLE]

Thus, using $|\psi_{B}|\lesssim|x|\chi_{B}^{2}$,

[TABLE]

It follows that

[TABLE]

Second, we also estimate using (53)

[TABLE]

We claim

[TABLE]

Indeed, using $|\chi_{B}^{\prime}|\lesssim B^{-2}$, $|\varphi_{B}|\lesssim|x|$, $\chi_{B}=0$ for $|x|\geq 2B^{2}$ and $\zeta_{B}\leq 1$,

[TABLE]

Using (56), we infer that $|(\psi_{B}^{\prime})^{2}v_{1}^{2}|\lesssim\chi_{B}^{2}v_{1}^{2}$, thus $|Q(\psi_{B}^{\prime})^{2}v_{1}^{2}|\lesssim z^{2}$, and so

[TABLE]

Now, we estimate $\|Q^{\alpha}v_{1}\|_{L^{2}}$ and $\|Q^{\alpha}\partial_{x}v_{1}\|_{L^{2}}$. From the definition of $z$ in (40), we have $e^{-|x|}v_{1}^{2}\chi_{B}^{2}\lesssim z^{2}$. Thus, from the definition of $\chi_{B}$,

[TABLE]

It follows using also (48) that

[TABLE]

Differentiating $z=\chi_{B}\zeta_{B}v_{1}$, we have

[TABLE]

Thus, as before,

[TABLE]

It follows using (48) and (49) that

[TABLE]

Collecting these estimates, we conlude

[TABLE]

Control of $J_{3}$. Using Cauchy-Schwarz inequality and (51), we have

[TABLE]

First, using $|\psi_{B}|\lesssim B$ (from its definition and $|\varphi_{B}|\lesssim B$) and (49),

[TABLE]

Then, since $|\varphi_{B}|\lesssim B$ and $\varphi_{B}^{\prime}=\zeta_{B}^{2}$,

[TABLE]

Thus, using the definition (40), $z=\chi_{B}\zeta_{B}v_{1}$ and then (48),

[TABLE]

In conclusion,

[TABLE]

Second, differentiating $w=\zeta_{A}u_{1}$, we have $\partial_{x}w=\zeta_{A}^{\prime}u_{1}+\zeta_{A}\partial_{x}u_{1}$, so that (using also the assumption $A\gg B^{2}$),

[TABLE]

Thus, using also (44),

[TABLE]

Next, by the definition of $\chi_{B}$ and (44),

[TABLE]

In conclusion,

[TABLE]

Collecting (58) and (59), we obtain

[TABLE]

Control of $J_{4}$. Using the Cauchy-Schwarz inequality, (51) and then $N^{\perp}=N-N_{0}Y_{0}$, we have

[TABLE]

By (33), $|a_{1}|\lesssim 1$, $\|u_{1}\|_{L^{\infty}}\lesssim 1$, and decay properties of $Y_{0}$ and $Q$, we have

[TABLE]

Using $\chi_{B}\lesssim\zeta_{A}$ (since $A\gg B^{2}$ in (24)) and (44), it holds

[TABLE]

Moreover, from (34),

[TABLE]

Therefore, using again (58), we obtain

[TABLE]

4.6. End of proof of Proposition 2

From (43), (46), (54), (57), (60) and (61), it follows that there exist $C_{2}>0$ and $C>0$ such that

[TABLE]

We fix $\gamma>0$ such that $C\gamma\leq 2C_{2}$ and also small enough to satisfy Lemma 5.

The value of $\gamma$ being now fixed, we do not mention anymore dependency in $\gamma$. Using standard inequalities and $B$ large enough, we obtain, for a possibly large constant $C>0$,

[TABLE]

Choosing (as specified in the statement of Proposition 2)

[TABLE]

and next using the assumption (15), we have

[TABLE]

Therefore, using again (15), for $\delta$ small enough (to absorb some constants), we obtain

[TABLE]

This estimate completes the proof of Proposition 2.

5. Coercivity and proof of Theorem 1

In this section, the constant $\gamma$ is fixed as in Proposition  2.

5.1. Coercivity results

Lemma 6**.**

Let $B>2$. Let $u$ and $v$ be Schwartz functions related by

[TABLE]

Assume

[TABLE]

It holds

[TABLE]

Proof.

Using the expression of $S$ and $U$, we rewrite (62) as

[TABLE]

and thus

[TABLE]

Integrating between [math] and $x>0$, this yields, for some constant $a$,

[TABLE]

which rewrites as

[TABLE]

Integrating on $[0,x]$, $x>0$, and multiplying by ${Y_{0}}$, it holds, for some constant $b$,

[TABLE]

where

[TABLE]

Let us now estimate $\int\tilde{u}^{2}\operatorname{sech}\left(\frac{x}{2}\right)$. First, by the Cauchy-Schwarz inequality,

[TABLE]

Second,

[TABLE]

Thus,

[TABLE]

Third, since $\frac{|Q^{\prime}|}{Q}\lesssim 1$, we obtain similarly,

[TABLE]

Collecting these estimates, we obtain, for all $x\geq 0$,

[TABLE]

The same holds for $x\leq 0$, and thus

[TABLE]

To complete the proof, we estimate the constants $a$ and $b$ in (65). Using (63) and parity property, projecting (65) on $Y_{0}$ yields

[TABLE]

Thus,

[TABLE]

Using (63), ${Y_{0}}\int_{0}^{x}\frac{Q}{Y_{0}}=-\alpha^{-1}Q^{\prime}$ and projecting (65) on $Q^{\prime}$ yields similarly

[TABLE]

We conclude the proof using again (65). ∎

The next result is a consequence of the previous general lemma, in the framework of the time-dependent functions introduced in (12), (26), (38) and (40).

Lemma 7**.**

For $B$ large enough, it holds

[TABLE]

and

[TABLE]

Proof.

Recall that the function $u_{1}$ is even so that it satisfies $\langle u_{1},Q^{\prime}\rangle=0$ in addition to the orthogonality (12). Therefore, applying (64),

[TABLE]

which implies by (26) and (20)

[TABLE]

By (40) and (55), it holds

[TABLE]

Thus, using (48)-(49),

[TABLE]

Using (45) and the definition of $\chi_{B}$ in (23), it holds

[TABLE]

Inserting these estimates into (68), it follows for $B$ large enough that

[TABLE]

The last two estimates imply (67).

Finally,

[TABLE]

and (66) follows. ∎

5.2. Proof of Theorem 1

Recall that the constants $\gamma>0$, $\delta_{1},\delta_{2}>0$ were defined in Propositions 1 and 2.

Proposition 3**.**

There exist $C_{3}>0$ and $0<\delta_{3}\leq\min(\delta_{1},\delta_{2})$ such that for any $0<\delta\leq\delta_{3}$, the following holds. Fix $A=\delta^{-1}$ and $B=\delta^{-\frac{1}{4}}$. Assume that for all $t\geq 0$, (15) holds.

Let

[TABLE]

Then, for all $t\geq 0$,

[TABLE]

Proof.

In the context of Propositions 1 and 2, observe that fixing $A=\delta^{-1}$ and $B=\delta^{-\frac{1}{4}}$, for $\delta>0$ small is consistent with the requirement $A\gg B^{2}\gg B\gg 1$ in (24).

Combining (41) with (67) and (27) with (66), for $\delta_{3}>0$ small enough and $0<\delta\leq\delta_{3}$, one obtains, for a constant $C>0$,

[TABLE]

Define $\mathcal{H}$ as in (69). It follows by combining the above estimates that

[TABLE]

Possibly choosing a smaller $\delta_{3}$, we obtain

[TABLE]

This estimate, together with (67), implies (70) for some $C_{3}>0$ (depending on $\delta_{3}$). ∎

We set

[TABLE]

Lemma 8**.**

There exist $C_{4}>0$ and $0<\delta_{4}\leq\delta_{3}$ such that for any $0<\delta\leq\delta_{4}$, the following holds. Fix $A=\delta^{-1}$. Assume that for all $t\geq 0$, (15) holds. Then, for all $t\geq 0$,

[TABLE]

and

[TABLE]

In particular,

[TABLE]

Proof.

From (34) and (13), it holds

[TABLE]

Estimates (71) and (72) then follow from (16). Last, estimate (73) is a consequence of (72) taking $\delta_{4}>0$ small enough. ∎

Combining (70) and (73), it holds

[TABLE]

and thus, for possibly smaller $\delta>0$,

[TABLE]

By the choice of $A=\delta^{-1}$, the bound $|\varphi_{A}|\lesssim A$, and (15), we have for all $t\geq 0$,

[TABLE]

Similarly, using also (51), it holds

[TABLE]

Estimate $|\mathcal{B}|\lesssim\delta^{2}$ is also clear from (15).

Therefore, integrating estimate (74) on $[0,t]$ and passing to the limit as $t\to+\infty$, it follows that

[TABLE]

Since $\int[(\partial_{x}u_{1})^{2}+u_{1}^{2}]\operatorname{sech}(x)\lesssim\|w\|_{\rho}^{2}$, this implies

[TABLE]

Using (75), we conclude the proof of Theorem 1 as in Section 5.2 of [18]. Let

[TABLE]

Using (17), we have

[TABLE]

We check that

[TABLE]

(See (33)-(34) in the proof of Lemma 2.) In particular, it follows that

[TABLE]

Using the bound $|\mathcal{K}|\lesssim\delta^{2}$ and (75), we deduce

[TABLE]

Similarly, we check that

[TABLE]

and so, as before

[TABLE]

By (76), there exists an increasing sequence $t_{n}\to+\infty$ such that

[TABLE]

For $t\geq 0$, integrating (77) on $[t,t_{n}]$, and passing to the limit as $n\to\infty$, we obtain

[TABLE]

By (76), we deduce that $\lim_{t\to\infty}\mathcal{G}(t)=0$.

Finally, by (16) and (34), we have

[TABLE]

and so as before, by integration on $[t,t_{n}]$ and $n\to\infty$,

[TABLE]

which proves $\lim_{t\to\infty}|a_{1}(t)|+|a_{2}(t)|=0$.

By the decomposition (11), this clearly implies (7). The proof of Theorem 1 is complete.

6. Proof of Theorem 2

6.1. Conservation of energy

Using (3) and (4) and performing a standard computation, we expand the conservation of energy (2) for a solution $(\phi,\partial_{t}\phi)$ written under the form (11) with the orthogonality conditions (12), to obtain

[TABLE]

Using the notation (13), we have

[TABLE]

Let $\delta_{0}>0$ be defined by

[TABLE]

Then, (78) applied at $t=0$ gives $|2\left\{E(\phi,\partial_{t}\phi)-E(Q,0)\right\}|\lesssim\delta_{0}^{2}$. Thus, by conservation of energy, estimate (78) at some $t>0$ gives

[TABLE]

Under the orthogonality conditions (12), the parity of $u_{1}$, from the spectral analysis recalled in the Introduction (see [6]), it follows that for some $\mu>0$,

[TABLE]

Thus, as long as $\|u_{1}\|_{H^{1}}+\|u_{2}\|_{L^{2}}+|b_{+}|+|b_{-}|\lesssim\delta_{0}^{1/2}$, the following energy estimate holds

[TABLE]

6.2. Construction of the graph

By the energy estimate (80), Lemma 8 and a standard contradiction argument, we construct initial data leading to global solutions close to the ground state $Q$.

Let $\boldsymbol{\varepsilon}=(\varepsilon_{1},\varepsilon_{2})\in\mathcal{A}_{0}$ (see (8)). Then, the condition $\langle\boldsymbol{\varepsilon},\boldsymbol{Z}_{+}\rangle=0$ rewrites

[TABLE]

Define $b_{-}(0)$ and $(u_{1}(0),u_{2}(0))$ such that

[TABLE]

and

[TABLE]

Then, it holds

[TABLE]

This means that the initial data in the statement of Theorem 2 decomposes as (see (14))

[TABLE]

Now, we prove that there exists at least a choice of $h(\boldsymbol{\varepsilon})=b_{+}(0)$ such that the corresponding solution $\boldsymbol{\phi}$ is global and satisfies (9).

Let $\delta_{0}>0$ small enough and $K>1$ large enough to be chosen. We introduce the following bootstrap estimates

[TABLE]

Given any $(u_{1}(0),u_{2}(0))$ and $b_{-}(0)$ such that

[TABLE]

and $b_{+}(0)$ satisfying

[TABLE]

we define

[TABLE]

Note that since $K>1$, $T$ is well-defined in $[0,+\infty]$. We aim at proving that there exists at least one value of $b_{+}(0)\in[-K^{5}\delta_{0}^{2},K^{5}\delta_{0}^{2}]$ such that $T=\infty$. We argue by contradiction, assuming that any $b_{+}(0)\in[-K^{5}\delta_{0}^{2},K^{5}\delta_{0}^{2}]$ leads to $T<\infty$.

First, we strictly improve the estimate on $(u_{1},u_{2})$ in (81). Indeed, by (80) and (82)-(83), it holds

[TABLE]

for some constant $C_{5}>0$. Thus, under the constraints

[TABLE]

it holds $\|u_{1}\|_{H^{1}}^{2}+\|u_{2}\|_{L^{2}}^{2}\leq\frac{3}{4}K^{4}\delta_{0}^{2}$, which strictly improves (81).

Second, we use (72) to control $b_{-}$. By (81)-(82)-(83), since $\|w\|_{\rho}\lesssim\|u_{1}\|_{H^{1}}$, it holds

[TABLE]

for some constant $C_{6}>0$. Thus, by integration on $[0,t]$ and using (84), we obtain

[TABLE]

Under the constaints

[TABLE]

it holds $b_{-}^{2}\leq\frac{3}{4}K^{2}\delta_{0}^{2}$ which strictly improves (82).

By the previous estimates (under the constraints (85)-(86)) and a continuity argument, we see that if $T<+\infty$, then $|b_{+}(T)|=K^{5}\delta_{0}^{2}$.

Third, we observe that if $t\in[0,T]$ is such that $|b_{+}(t)|=K^{5}\delta_{0}^{2}$, then it follows from (71) that

[TABLE]

for some constant $C_{7}>0$. Under the constraints

[TABLE]

the inequality

[TABLE]

holds. By standard arguments, such transversality condition implies that $T$ is the first time for which $|b_{+}(t)|=K^{5}\delta_{0}^{2}$ and moreover that $T$ is continuous in the variable $b_{+}(0)$ (see e.g. [7, 8] for a similar argument). Now, the image of the continuous map

[TABLE]

is exactly $\{-K^{5}\delta_{0}^{2},K^{5}\delta_{0}^{2}\}$ (since the image of $-K^{5}\delta_{0}^{2}$ is $-K^{5}\delta_{0}^{2}$ and the image of $K^{5}\delta_{0}^{2}$ is $K^{5}\delta_{0}^{2}$), which is a contradiction.

As a consequence, provided the constraints in (85)-(86)-(87) are all fullfilled, there exists at least one value of $b_{+}(0)\in(-K^{5}\delta_{0}^{2},K^{5}\delta_{0}^{2})$ such that $T=\infty$.

Finally, we easily see that to satisfy (85)-(86)-(87), it is sufficient first to fix $K>0$ large enough, depending only on $C_{5}$, $C_{6}$ and $C_{7}$, and then to choose $\delta_{0}>0$ small enough.

6.3. Uniqueness and Lipschitz regularity

The following proposition implies both the uniqueness of the choice of $h(\boldsymbol{\varepsilon})=b_{+}(0)$, for a given $\boldsymbol{\varepsilon}\in\mathcal{A}_{0}$, and the Lipschitz regularity of the graph $\mathcal{M}$ defined from the resulting map $\boldsymbol{\varepsilon}\in\mathcal{A}_{0}\mapsto h(\boldsymbol{\varepsilon})$. It is thus sufficient to complete the proof of Theorem 2.

Proposition 4**.**

There exist $C,\delta>0$ such if $\boldsymbol{\phi}$ and $\tilde{\boldsymbol{\phi}}$ are two even solutions of (1) satisfying

[TABLE]

then, decomposing

[TABLE]

with $\langle\boldsymbol{\varepsilon},\boldsymbol{Z}_{+}\rangle=\langle\tilde{\boldsymbol{\varepsilon}},\boldsymbol{Z}_{+}\rangle=0$, it holds

[TABLE]

Proof.

We use the decomposition and the notation of Section 2.1 for the two solutions $\boldsymbol{\phi}$ and $\tilde{\boldsymbol{\phi}}$ satisfying (88). In particular, from (15), there exists $C_{0}>0$ such that for all $t\geq 0$,

[TABLE]

We denote

[TABLE]

Then, from (16), (17), the equations of $(\check{u}_{1},\check{u}_{2},\check{b}_{+},\check{b}_{-})$ write

[TABLE]

We claim that

[TABLE]

Indeed, by Taylor formula, for any $v,\tilde{v}$, it holds (recall that $\alpha>1$)

[TABLE]

Using this inequality for $\check{N}=N-\tilde{N}$, where $N$ is defined in (18), and (90), we obtain

[TABLE]

Using the Cauchy-Schwarz inequality and again (90), we find $\|\check{N}\|_{L^{2}}\lesssim\delta(|\check{a}_{1}|+|\check{u}_{1}|)$ and estimate (92) follows.

Let

[TABLE]

By (91) and (92) (and the coercivity property (79) for $\check{u}_{1}$) we have, for some $K>0$,

[TABLE]

For the sake of contradiction, assume that the following holds

[TABLE]

We introduce the following boostrap estimate

[TABLE]

Define

[TABLE]

We work on the interval $[0,T]$. Note that from (93) and (95), it holds

[TABLE]

In particular, by standard arguments, $\beta_{+}$ is positive and increasing on $[0,T]$.

Next, by (93) and (95),

[TABLE]

and thus, by integration,

[TABLE]

Therefore, by (94), for $\delta$ small enough,

[TABLE]

Then, by (93) and (95),

[TABLE]

and so by integration and (94),

[TABLE]

Therefore, for $\delta$ small enough,

[TABLE]

Last, it is clear that for $\delta$ small, it holds $K\delta\beta_{+}\leq\frac{\nu_{0}}{5}\beta_{+}$.

Therefore, we have proved that, for all $t\in[0,T]$,

[TABLE]

By a continuity argument, this means that $T=+\infty$. By the exponential growth (96) and $\beta_{+}(0)>0$, we obtain a contradiction with the global bound (90) on $|b_{+}|$.

Since estimate (94) is contradicted, and since it holds

[TABLE]

we have proved (89). ∎