where $\mathcal{P}(x; \varphi, \alpha)$ is bounded by $|\alpha| + |\varphi|$, and $\mathcal{Q}$ is composed of second or higher order terms and is given by

$$ \begin{aligned} & \mathcal{Q}(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x) \\ & = \mathcal{U}_2(x; \varphi, \alpha) \left( -\tilde{\mathcal{L}}_{\gamma} \varphi(t, x) + a(x) \varphi_t(t, x) + a(x) \dot{\alpha} \gamma_x(x + \alpha) \right) \\ & \quad + (I_n + \mathcal{U}_2(x; \varphi, \alpha)) \left( \gamma_{xx}(x + \alpha) |\dot{\alpha}|^2 - \mathcal{Q}_1(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x) \right. \\ & \qquad \left. - \mathcal{Q}_2(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x) + \mathcal{Q}_0(x; \alpha, \dot{\alpha}, \psi, \psi_t, \psi_x) \right) \end{aligned} $$

Indeed, concerning $\mathcal{Q}_0(x; \alpha, \dot{\alpha}, \psi, \psi_t, \psi_x)$, direct calculation yields

$$ \begin{aligned} & |\mathcal{Q}_0(x; \alpha, \dot{\alpha}, \psi, \psi_t, \psi_x)| \\ & \lesssim \sum_{j,k} |\partial_{\nu} \psi^j \partial^{\nu} \psi^k| + |\psi_t| |\dot{\alpha}| + |\psi_x| (|\psi| + |\alpha|) + |\dot{\alpha}|^2 + |\psi| |\alpha| \\ & \quad (8.38) \qquad \lesssim \sum_{j,k} |\partial_{\nu} \varphi^j \partial^{\nu} \varphi^k| + |\varphi_t| |\dot{\alpha}| + |\dot{\alpha}|^2 + (|\varphi| + |\alpha|)(|\varphi| + |\varphi_x| + |\dot{\alpha}|). \end{aligned} $$

By combining (8.34)–(8.38), we obtain estimates on the nonlinear term $\mathcal{Q}$:

$$ \begin{aligned} & |\mathcal{Q}(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x)| \\ & \lesssim \sum_{j,k} |\partial_{\nu} \varphi^j \partial^{\nu} \varphi^k| + |\dot{\alpha}|^2 + |\varphi_t| |\dot{\alpha}| + (|\varphi| + |\alpha|)(|\varphi| + |\varphi_x| + |\varphi_t| + |\dot{\alpha}|). \end{aligned} \quad (8.39) $$

provided that $|\varphi|, |\alpha| \ll 1$.

Next, we turn to the equation for $\alpha$. Recall that $\alpha(t)$ is chosen such that the orthogonality condition holds:

$$ (8.40) \qquad \langle \varphi(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} = 0 \quad \forall t \in [0, T]. $$

Differentiating twice equation (8.40) we obtain

$$ (8.41) \qquad \langle \varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} = \langle \varphi_{tt}(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} = 0 \quad \forall t \in [0, T], $$

Thanks to Lemma 2.8, we have

$$ (8.42) \qquad \langle \mathcal{L}_{\gamma} \varphi(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} = \langle \varphi(t, \cdot), \mathcal{L}_{\gamma} \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} = 0 \quad \forall t \in [0, T]. $$

By integrating equation (8.37) against $\gamma_x$, we obtain in light of the definition of $L$ and $l$ in (8.21)

$$ \begin{aligned} & \underbrace{\ddot{\alpha} \langle \mathcal{P}(x; \varphi(x), \alpha) + \gamma_x(x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}}_{\approx L + |\alpha| + \|\varphi\|_{L^1}} \\ & = - \langle a(x) \dot{\alpha} \gamma_x(x + \alpha) + a(x) \varphi_t(t, x) + \mathcal{Q}(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} \\ & = - \dot{\alpha} \underbrace{\langle a(x) \gamma_x(x + \alpha), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}}_{\approx l + |\alpha|} \\ & \quad - \underbrace{\langle a(x) \varphi_t(t, x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}}_{\text{first order term } \|\varphi_t\|_{L^1}} \end{aligned} $$
$$ -\underbrace{\langle Q(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)}}_{\text{higher order terms}} $$

Hence $\alpha$ satisfies

$$ (8.43) \qquad \ddot{\alpha} + \frac{l}{L} \dot{\alpha} = -\frac{1}{L} \langle a(\cdot)\varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} + \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) $$

with

$$ \begin{aligned} \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \\ = \dot{\alpha} \left( \frac{\langle a(x)\gamma_x(x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}}{\langle \gamma_x(x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}} - \frac{\langle a(x)\gamma_x(x+\alpha), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}}{\langle \mathcal{P}(x; \varphi(x), \alpha) + \gamma_x(x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}} \right) \\ + \langle a(x)\varphi_t(t, x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)} \left( \frac{1}{L} - \frac{1}{\langle \mathcal{P}(x; \varphi(x), \alpha) + \gamma_x(x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}} \right) \\ - \frac{\langle Q(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}}{\langle \mathcal{P}(x; \varphi(x), \alpha) + \gamma_x(x), \gamma_x(x) \rangle_{L^2(\mathbb{T}^1)}} \end{aligned} $$

One easily checks that estimate (8.25) holds for $\mathcal{O}$.

Now, by substituting (8.43) into equation (8.37) for $\varphi$, we obtain

$$ (8.44) \qquad -\varphi_{tt}(t, x) + \mathcal{L}_{\gamma}\varphi(t, x) - a(x)\varphi_t(t, x) = \left(a(x) - \frac{l}{L}\right) \dot{\alpha}\gamma_x(x) - \frac{1}{L}\langle a(\cdot)\varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)}\gamma_x(x) \\ \qquad \qquad \qquad \qquad + \mathcal{M}_1(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)). $$

Here the high order term is given by

$$ \begin{aligned} \mathcal{M}_1(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \\ = - \left( \frac{l}{L} \dot{\alpha} + \frac{1}{L} \langle a(\cdot)\varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} \right) \mathcal{P}(x; \varphi, \alpha) \\ + (\mathcal{P}(x; \varphi, \alpha) + \gamma_x(x)) \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \\ + a(x) \dot{\alpha} (\gamma_x(x + \alpha) - \gamma_x(x)) + \mathcal{Q}(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x), \end{aligned} \quad (8.45) $$

and satisfies estimate (8.24).

The coupled system (8.43)-(8.44) for $(\varphi, \alpha)$ encapsulates the evolution of $\phi(t, \cdot)$. However, the linear coupling terms on the right cannot be treated as a perturbation. Thus we introduce a new function

$$ (8.46) \qquad \Psi(t, x) := \varphi(t, x) + \alpha(t)\gamma_x(t, x). $$

We observe that this is governed by a simpler equation:

$$ \begin{aligned} & -\Psi_{tt}(t, x) + \mathcal{L}_{\gamma}\Psi(t, x) - a(x)\Psi_t(t, x) \\ &= -\varphi_{tt}(t, x) + \mathcal{L}_{\gamma}\varphi(t, x) - a(x)\varphi_t(t, x) - \ddot{\alpha}(t)\gamma_x(x) + \alpha(t)\mathcal{L}_{\gamma}\gamma_x - \dot{\alpha}(t)a(x)\gamma_x(x) \\ &= a(x)\dot{\alpha}(\gamma_x(x + \alpha) - \gamma_x(x)) + \ddot{\alpha}\mathcal{P}(x; \varphi, \alpha) + \mathcal{Q}(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x) \\ & \quad = \mathcal{M}(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)), \end{aligned} \quad (8.47) $$
where, by plugging (8.43) into above equation, the nonlinear term is given by

$$ (8.48) \qquad \begin{aligned} & \mathcal{M}(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \\ & = a(x)\dot{\alpha}(\gamma_x(x + \alpha) - \gamma_x(x)) + \mathcal{Q}(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x) \\ & \quad - \left( \frac{l}{L}\dot{\alpha} + \frac{1}{L}\langle a(\cdot)\varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} - \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \right) \mathcal{P}(x; \varphi, \alpha), \end{aligned} $$

and satisfies estimate (8.24). This concludes the proof of Proposition 8.7.

**8.3. Coercive estimates around closed geodesics.** Recall from Lemma 2.8 that the Jacobian operator $\mathcal{L}_\gamma$ is self-adjoint on functions in $\gamma^*(TN)$. Under the negative curvature assumption, we further demonstrate it is positive definite in a codimension one submanifold, more precisely, for functions $\varphi \in \gamma^*(TN)$ satisfying the condition (P3).

**Proposition 8.8.** Let $\gamma$ be a closed geodesic on $\mathcal{N}$. Assume that sectional curvature is strictly negative on $\gamma$. Then the operator $\mathcal{L}_\gamma$ given in Definition 2.6 is coercive in the following sense: there exists a constant $c_{co} > 0$ such that for any $\varphi \in H^1(\mathbb{T}^1; \gamma^*(TN))$ satisfying the rigidity condition (P3), we have the lower bound

$$ -\langle \mathcal{L}_{\gamma\varphi}, \varphi \rangle_{L^2(\mathbb{T}^1)} \geq c_{co} \|\varphi\|_{H^1(\mathbb{T}^1)}^2. $$

Before presenting its proof, we offer some comments.

**Remark 8.9.** Both the rigidity condition (P3) and the negative curvature assumptions are essential for this proposition. Indeed, if the first condition is dropped, we can take $\varphi(\cdot) = \gamma_x(\cdot)$ and get

$$ -\langle \mathcal{L}_{\gamma\varphi}, \varphi \rangle_{L^2(\mathbb{T}^1)} = \int_{\mathbb{T}^1} (\|\nabla_{\gamma_x}\varphi\|^2 - \langle R(\gamma_x, \varphi)\gamma_x, \varphi \rangle) dx = 0. $$

If the curvature is positive, then assumption may also fail. For example, let the target manifold be $\mathbb{S}^k \subset \mathbb{R}^{k+1}$ and suppose that $\gamma$ lies entirely in the plane $\mathbb{R}^k \times \{0\}$, then we can select $\varphi$ as $(0, 0, ..., 1)$ and yields the same degeneracy.

**Remark 8.10.** When the sectional curvature is strictly negative on $\gamma$, then there exists $C_{co} > 0$ such that

$$ C_{co}^{-1} \|(f,g)\|_{\mathcal{H}} \le (\mathcal{E}_{\gamma_p}(f,g))^{1/2} \le C_{co} \|(f,g)\|_{\mathcal{H}}, $$ 
for every $p \in [0, 2\pi)$ and for every $(f,g) \in \mathcal{H}_{\gamma_p, 0}$.

**Remark 8.11.** Thanks to Lemma 8.6 and the above proposition, the quantity $\mathcal{E}_\gamma(\Psi[t])$ can characterize the distance between $(\phi, \phi_t)$ and the rotation family of closed geodesics $\{(\gamma_p, 0)\}_{p \in [0, 2\pi)}$.

The rest part of this section is devoted to

**Proof of Proposition 8.8.** Without loss of generality, we assume that $|\gamma_x(x)| = 1$ for every $x \in \mathbb{T}^1$. Define the inner product

$$ \mathcal{S}(\varphi) := -\langle \mathcal{L}_{\gamma\varphi}, \varphi \rangle_{L^2(\mathbb{T}^1)}. $$

It equals up to a constant the second variation of the energy with respect to a variation $h(\cdot, \varphi): (-\delta, \delta) \times \mathbb{T}^1 \to \mathcal{N}$ with $h_s(0, x) = \varphi(x)$ and $h(0, x) = \gamma(x) \forall x \in \mathbb{T}^1$. Due to Lemma 2.8, this can also be expressed in the intrinsic coordinates

$$ \mathcal{S}(\varphi) = \int_{\mathbb{T}^1} (|\nabla_{\gamma_x}(\varphi)(x)|^2 - \langle R(\gamma_x, \varphi)\gamma_x, \varphi \rangle(x)) dx. $$
The negative curvature assumption implies that function $S(\varphi)$ is non-negative for any $\varphi \in H^1(\mathbb{T}^1; \gamma^*(TN))$. To prove the lower bound, we proceed by contradiction. Assume there exists a sequence $\{\varphi\}_{n \ge 1} \subset H^1(\mathbb{T}^1; \gamma^*(TM))$ with

$$ \|\varphi_n\|_{H^1(\mathbb{T}^1)} = 1, \langle \varphi_n, \gamma_x \rangle_{L^2(\mathbb{T}^1)} = 0 $$

and such that

$$ \lim_{n \to \infty} S(\varphi_n) = 0 $$

Using the Banach-Alaoglu lemma as well as the Rellich compactness lemma we can select a subsequence and $\varphi_* \in H^1(\mathbb{T}^1; \gamma^*(TN))$, which we label in the same way, such that

$$ (8.49) \qquad \varphi_{n,x} \rightharpoonup \varphi_{*,x} \text{ weakly in } L^2(\mathbb{T}^1; \mathbb{R}^N), $$

$$ (8.50) \qquad \varphi_n \to \varphi_* \text{ in } C^0(\mathbb{T}^1; \mathbb{R}^N). $$

We also immediately know from the limit of $S(\varphi_n)$ that

$$ (8.51) \qquad \int_{\mathbb{T}^1} |\nabla_{\gamma_x}(x) \varphi_n(x)|^2 dx \to 0, $$

$$ (8.52) \qquad \langle R(\gamma_x, \varphi_*) \gamma_x, \varphi_* \rangle(x) = 0, \forall x \in \mathbb{T}^1. $$

The preceding equality shows that for any $x \in \mathbb{T}^1$, the vector $\varphi_*$ is parallel to $\gamma_x$.

Next, we first show that the weak convergence of $\{\varphi_{n,x}\}$ yields the weak convergence of $\nabla_{\gamma_x}(x)\varphi_n$,

$$ (8.53) \qquad \nabla_{\gamma_x}(x)\varphi_n \to \nabla_{\dot{\gamma}_x}(x)\varphi_* \text{ weakly in } L^2(\mathbb{T}^1; \mathbb{R}^N). $$

Indeed, by the definition of $\nabla_{\gamma_x}(x)\varphi_n(x)$ we have

$$ \varphi_{n,x}(x) = \nabla_{\gamma_x}(x)\varphi_n(x) + g_n(x) \text{ with } g_n(x) \in T_{\gamma_x}^{\perp} \mathcal{N} \quad \forall x \in \mathbb{T}^1, $$

$$ \varphi_{*,x}(x) = \nabla_{\gamma_x}(x)\varphi_*(x) + g_*(x) \text{ with } g_*(x) \in T_{\gamma_x}^{\perp} \mathcal{N} \quad \forall x \in \mathbb{T}^1. $$

Thanks to the weak convergence (8.49) of $\varphi_{n,x}$ in $L^2(\mathbb{T}^1; \mathbb{R}^N)$, by choosing the test function $f \in L^2(\mathbb{T}^1; \gamma^*(TN)) \subset L^2(\mathbb{T}^1; \mathbb{R}^N)$, one has

$$ \langle \varphi_{n,x}, f \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)} \to \langle \varphi_{*,x}, f \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)}. $$

Thus,

$$ \langle \nabla_{\gamma_x}(x) \varphi_n, f \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)} \to \langle \nabla_{\gamma_x}(x) \varphi_*, f \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)} \quad \forall f \in L^2(\mathbb{T}^1; \gamma^*(TN)). $$

For any $f \in L^2(\mathbb{T}^1; \mathbb{R}^N)$, we define $f_1(x)$ as the projection of $f(x)$ on the tangent space $T_{\gamma_x}(x)\mathcal{N}$, then

$$ \langle \nabla_{\gamma_x}(x) \varphi_n, f \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)} = \langle \nabla_{\gamma_x}(x) \varphi_n, f_1 \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)} \\ \to \langle \nabla_{\gamma_x}(x) \varphi_*, f_1 \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)} = \langle \nabla_{\gamma_x}(x) \varphi_*, f \rangle_{L^2(\mathbb{T}^1; \mathbb{R}^N)}. $$

This finishes the proof of (8.53). From the lower semi-continuity, the weak convergence (8.53) yields

$$ (8.54) \qquad \|\nabla_{\gamma_x} \varphi_*\|_{L^2(\mathbb{T}^1)} \le \liminf_n \|\nabla_{\gamma_x} \varphi_n\|_{L^2(\mathbb{T}^1)}. $$

This combined with (8.51) yield

$$ (8.55) \qquad \nabla_{\gamma_x} \varphi_*(x) = 0 \quad \forall x \in \mathbb{T}^1. $$
Using the fact that $\gamma_x(x)$ and $\varphi_*(x)$ belong to $T_{\gamma_x}(x)\mathcal{N}$, we conclude that

$$ \partial_x(\varphi_* \cdot \gamma_x) = \partial_x \varphi_* \cdot \gamma_x + \varphi_* \cdot \partial_x \gamma_x = \nabla_{\gamma_x} \varphi_* \cdot \gamma_x + \varphi_* \cdot \nabla_{\gamma_x} \gamma_x = 0, $$

and so we obtain that

$$ \varphi_* \cdot \gamma_x \equiv \text{const} \quad \forall x \in \mathbb{T}^1. $$

Observing from the (P3) condition that

$$ \langle \varphi_*, \gamma_x \rangle_{L^2(\mathbb{T}^1)} = \lim_{n \to \infty} \langle \varphi_n, \gamma_x \rangle_{L^2(\mathbb{T}^1)} = 0, $$

we then conclude that $\varphi_* \cdot \gamma_x = 0$ for every $x$.

The negative sectional curvature assumption implies that

$$ \begin{aligned} 0 &= \lim_{n} \int_{\mathbb{T}^1} -\langle R(\gamma_x, \varphi_n)\gamma_x, \varphi_n \rangle(x) dx \\ &= - \int_{\mathbb{T}^1} (\langle R(\gamma_x, \varphi_*)\gamma_x, \varphi_* \rangle(x)) dx \\ &\geq c \int_{\mathbb{T}^1} (\langle \gamma_x, \gamma_x \rangle \langle \varphi_*, \varphi_* \rangle - (\langle \gamma_x, \varphi_* \rangle)^2)(x) dx \\ &= c \int_{\mathbb{T}^1} \langle \varphi_*, \varphi_* \rangle(x) dx. \end{aligned} $$

This implies that $\varphi_* = 0$. Therefore,

$$ \|\varphi_{n,x}\|_{L^2(\mathbb{T}^1)} \to 1. $$

To arrive at a contradiction, recall the orthogonal tangent frame $\{\mathbf{f}_j(x)\}_{j=1}^L$ of $T_{\gamma_x}(x)\mathcal{N}$ defined in Section 5.1, we write

$$ \varphi_n(x) = \sum_{j=1}^{L} a_{j,n}(x) \cdot \mathbf{f}_j(x). $$

Then the fact that $\varphi_* = 0$ yields

$$ \|a_{j,n}\|_{L^2(\mathbb{T}^1)} \to 0 \quad \forall j = 1, 2, \dots, L. $$

Next, writing

$$ \partial_x \varphi_n(x) = \sum_{j=1}^{L} \partial_x (a_{j,n}) \cdot \mathbf{f}_j(x) + \sum_{j=1}^{L} a_{j,n}(x) \cdot \partial_x \mathbf{f}_j(x), $$

$$ \nabla_{\dot{\gamma}} \varphi_n(x) = \sum_{j=1}^{L} \partial_x (a_{j,n}) \cdot \mathbf{f}_j(x) + \sum_{j=1}^{L} a_{j,n}(x) \cdot \nabla_{\gamma_x} \mathbf{f}_j(x), $$

the convergence of $\{a_{j,n}\}$ leads to

$$ \|\partial_x \varphi_n - \nabla_{\gamma_x} \varphi_n\|_{L^2(\mathbb{T}^1)} \to 0. $$

Thus

$$ \|\nabla_{\gamma_x} \varphi_n\|_{L^2(\mathbb{T}^1)} \to 1. $$

This is in contradiction with (8.51), and ends the proof of Proposition 8.8. □
8.4. Exponential stability of the linearized equation on Ψ based on propagation of smallness and coercive estimates. We prove Proposition 8.12 concerning the equation:

$$ (8.56) \qquad -\Psi_{tt} + \mathcal{L}_{\gamma}\Psi - a(x)\Psi_t = g \quad \text{with } \Psi[t] \in \mathcal{H}_{\gamma}. $$

The key point shall be to understand how the friction term $a(x)\Psi_t$ causes the exponential decay of a suitable energy functional, namely $\mathcal{E}_{\gamma}(\Psi[t])$ which we defined in Definition 8.5. Recall that

$$ \Psi(t, x) \in T_{\gamma(x)}\mathcal{N} \quad \forall x \in \mathbb{T}^1, $$

and the orthogonal tangent frame $\{\mathbf{f}_j(\mathbf{x})\}_{j=1}^L$ of $T_{\gamma(x)}\mathcal{N}$ defined in Section 5.1. Set

$$ \Psi(t, x) = \sum_{j=1}^{L} a_j(t, x) \cdot \mathbf{f}_j(x), \quad \forall x \in \mathbb{T}^1. $$

Then $\Psi_t(t, x)$ stays in $T_{\gamma(x)}\mathcal{N}$:

$$ \Psi_t(t, x) = \sum_{j=1}^{n} \partial_t a_j(t, x) \cdot \mathbf{f}_j(x) \in T_{\gamma(x)}\mathcal{N} \quad \forall x \in \mathbb{T}^1. $$

Thus

$$ \partial_t(\nabla_{\dot{\gamma}}\Psi) = \nabla_{\dot{\gamma}}(\partial_t\Psi) \quad \text{and} \quad \partial_t(\mathcal{L}_{\gamma}\Psi) = \mathcal{L}_{\gamma}\Psi_t. $$

Using the fact that $\mathcal{L}_{\gamma}$ is self-adjoint for functions in $H^1(\mathbb{T}^1; \gamma^*(\mathcal{T}\mathcal{N}))$, namely Lemma 2.8,

$$ \begin{aligned} -\frac{d}{dt} \langle \mathcal{L}_{\gamma} \Psi, \Psi \rangle_{L^2(\mathbb{T}^1)} &= -\langle \partial_t(\mathcal{L}_{\gamma} \Psi), \Psi \rangle_{L^2(\mathbb{T}^1)} - \langle \mathcal{L}_{\gamma} \Psi, \partial_t \Psi \rangle_{L^2(\mathbb{T}^1)} \\ &= -\langle \mathcal{L}_{\gamma} \Psi_t, \Psi \rangle_{L^2(\mathbb{T}^1)} - \langle \mathcal{L}_{\gamma} \Psi, \Psi_t \rangle_{L^2(\mathbb{T}^1)} \\ &= -2 \langle \mathcal{L}_{\gamma} \Psi, \Psi_t \rangle_{L^2(\mathbb{T}^1)}. \end{aligned} $$

Hence the variation of the energy functional $\mathcal{E}_{\gamma}(\Psi[t])$ reads

$$ \begin{aligned} \frac{d}{dt} \mathcal{E}_{\gamma}(\Psi[t]) &= \langle \Psi_{tt}, \Psi_t \rangle_{L^2(\mathbb{T}^1)} - \langle \mathcal{L}_{\gamma} \Psi, \Psi_t \rangle_{L^2(\mathbb{T}^1)} \\ &= \langle \mathcal{L}_{\gamma} \Psi - a(x) \Psi_t - g, \Psi_t \rangle_{L^2(\mathbb{T}^1)} - \langle \mathcal{L}_{\gamma} \Psi, \Psi_t \rangle_{L^2(\mathbb{T}^1)} \\ &= -\langle a(x) \Psi_t, \Psi_t \rangle_{L^2(\mathbb{T}^1)} - \langle g, \Psi_t \rangle_{L^2(\mathbb{T}^1)}. \end{aligned} \quad (8.57) $$

Heuristically, if during some period $t \in (0, T)$ the following two hold simultanesly

$$ \begin{aligned} \|g\|_{L^2(0,T;L^2(\mathbb{T}^1))} &\ll \mathcal{E}_{\gamma}^{\frac{1}{2}}(\Psi[0]), \\ \int_{0}^{T} \langle a(x)\Psi_t, \Psi_t \rangle_{L^2(\mathbb{T}^1)} dt &\geq c \mathcal{E}_{\gamma}(\Psi[0]), \end{aligned} $$

then $\mathcal{E}_{\gamma}(\Psi[t])$ decays.

In light of the preceding discussion, by combining the propagation of the smallness property (Corollary 3.2) and the coercive estimate Proposition 8.8, we deduce the following result concerning the asymptotic stability of the component $\Psi[t]$:

**Proposition 8.12.** There exist positive constants $\delta_{li}$ and $c_{li}$ such that for any pair $\Psi$ and $g$ satisfying the following properties

$$ (8.58) \qquad -\Psi_{tt} + \mathcal{L}_{\gamma}\Psi - a(x)\Psi_t = g, $$
$$ (8.59) \qquad \Psi[t] \in \mathcal{H}_{\gamma}, \forall t \in [0, 32\pi], $$

$$ (8.60) \qquad \|g\|_{L^2(0,32\pi; L^2(\mathbb{T}^1))}^2 \le \delta_{li} \mathcal{E}_{\gamma}(\Psi[0]), $$

one has

$$ (8.61) \qquad \int_0^{32\pi} \langle a(x)\Psi_t, \Psi_t \rangle_{L^2(\mathbb{T}^1)} dt \ge c_{li} \mathcal{E}_{\gamma}(\Psi[0]), $$

and

$$ (8.62) \qquad \mathcal{E}_{\gamma}(\Psi[32\pi]) \le (1 - c_{li})\mathcal{E}_{\gamma}(\Psi[0]), $$

$$ (8.63) \qquad \mathcal{E}_{\gamma}(\Psi[t]) \le 2\mathcal{E}_{\gamma}(\Psi[0]), \forall t \in [0, 32\pi]. $$

In the sequel, we reduce the proposition to the Lemma 8.13 and then establish this auxiliary result.

### 8.4.1. A reduction step: Lemma 8.13 implies Proposition 8.12.

**Lemma 8.13.** There exist constants $\tilde{\delta}_{li} > 0$ and $\tilde{c}_{li} > 0$ such that for any pair $\Psi$ and $g$ satisfying the following properties

$$ (8.64) \qquad -\Psi_{tt} + \mathcal{L}_{\gamma}\Psi - a(x)\Psi_t = g, $$

$$ (8.65) \qquad \Psi[t] \in \mathcal{H}_{\gamma}, \forall t \in [0, 32\pi], $$

$$ (8.66) \qquad \Psi[16\pi] \in \mathcal{H}_{\gamma,0}, $$

$$ (8.67) \qquad \|g\|_{L^2(0,32\pi; L^2(\mathbb{T}^1))}^2 \le \tilde{\delta}_{li} \|\Psi[16\pi]\|_{\mathcal{H}}^2, $$

one has

$$ (8.68) \qquad \int_0^{32\pi} \langle a(x)\Psi_t, \Psi_t \rangle_{L^2(\mathbb{T}^1)} dt \ge \tilde{c}_{li} \|\Psi[16\pi]\|_{\mathcal{H}}^2. $$

We first present the following basic energy estimate:

**Lemma 8.14.** There exist $\delta > 0$ and $C > 0$ such that for every $s \in [0, 32\pi]$, and for every pair $\Psi$ and $g$ satisfying the following properties

$$ -\Psi_{tt} + \mathcal{L}_{\gamma}\Psi - a(x)\Psi_t = g, $$

$$ \Psi[t] \in \mathcal{H}_{\gamma}, \forall t \in [0, 32\pi], $$

$$ \|g\|_{L^2(0,32\pi; L^2(\mathbb{T}^1))}^2 \le \delta \mathcal{E}_{\gamma}(\Psi[s]), $$

one has

$$ C^{-1}\mathcal{E}_{\gamma}(\Psi[t_2]) \le \mathcal{E}_{\gamma}(\Psi[t_1]) \le C\mathcal{E}_{\gamma}(\Psi[t_2]) \quad \forall t_1, t_2 \in [0, 32\pi]. $$

and

$$ \mathcal{E}_{\gamma}(\Psi[t_2]) \le 2\mathcal{E}_{\gamma}(\Psi[t_1]) \quad \forall 0 \le t_1 \le t_2 \le 32\pi. $$

*Proof.* To prove the first inequality, it suffices to show that

$$ C^{-1}\mathcal{E}_{\gamma}(\Psi[s]) \le \mathcal{E}_{\gamma}(\Psi[t]) \le C\mathcal{E}_{\gamma}(\Psi[s]) \quad \forall t \in [0, 32\pi]. $$

Thanks to (8.57), there exists $C$ such that

$$ \mathcal{E}_{\gamma}(\Psi[s_1]) \le C \left( \mathcal{E}_{\gamma}(\Psi[s_2]) + \|g\|_{L^2_{t,x}}^2 \right) \quad \forall s_1, s_2 \in [0, 32\pi]. $$
On the one hand, we have

$$ \mathcal{E}_{\gamma}(\Psi[t]) \le C \left( \mathcal{E}_{\gamma}(\Psi[s]) + \|g\|_{L_{t,x}^2}^2 \right) \le (C + \delta)\mathcal{E}_{\gamma}(\Psi[s]), $$

On the other hand, by selecting $\delta$ small enough, we obtain

$$ \mathcal{E}_{\gamma}(\Psi[s]) \le C \left( \mathcal{E}_{\gamma}(\Psi[t]) + \|g\|_{L_{t,x}^2}^2 \right) \le C \left( \mathcal{E}_{\gamma}(\Psi[t]) + \delta \mathcal{E}_{\gamma}(\Psi[s]) \right) \le 2C \mathcal{E}_{\gamma}(\Psi[t]). $$

Next, we show the second inequality. Thanks to (8.57),

$$ \begin{aligned} \mathcal{E}_{\gamma}(\Psi[t_2]) &\le \frac{3}{2}\mathcal{E}_{\gamma}(\Psi[t_1]) + C\|g\|_{L_{t,x}^2}^2 \\ &\le \frac{3}{2}\mathcal{E}_{\gamma}(\Psi[t_1]) + C\delta\mathcal{E}_{\gamma}(\Psi[s]) \le 2\mathcal{E}_{\gamma}(\Psi[t_1]), \end{aligned} $$

provided that $\delta$ is sufficiently small.

<figure><img src="image_13.png" /></figure>

Similarly, based on the well-posedness property Lemma 2.12, we also obtain the following result without giving its proof:

**Lemma 8.15.** There exist $\delta > 0$ and $C > 0$ such that for any $s \in [0, 32\pi]$, and for any pair $\Psi$ and $g$ satisfying equation (8.64) and

$$ \|g\|_{L^2(0,32\pi; L^2(\mathbb{T}^1))}^2 \le \delta \|\Psi[s]\|_{\mathcal{H}}^2, $$

one has

$$ C^{-1}\|\Psi[t_2]\|_{\mathcal{H}}^2 \le \|\Psi[t_1]\|_{\mathcal{H}}^2 \le C\|\Psi[t_2]\|_{\mathcal{H}}^2 \quad \forall t_1, t_2 \in [0, 32\pi]. $$

Now, assume that Lemma 8.13 holds with some $\tilde{\delta}_{li}$ and $\tilde{c}_{li}$, we will find the constants $\delta_{li}$ and $c_{li}$ such that Proposition 8.12 is fulfilled. Suppose that $(\Psi, g)$ satisfy (8.58)–(8.60), and define

$$ \tilde{\Psi}(t, x) := \Psi(t, x) - b\dot{\gamma}(x) \quad \forall t \in [0, 32\pi] \quad \forall x \in \mathbb{T}^1, $$

where $b$ is a constant chosen such that $\tilde{\Psi}[16\pi] \in \mathcal{H}_{\gamma,0}$. Due to the observation (2.9),

$$ (8.69) \qquad \mathcal{E}_{\gamma}(\tilde{\Psi}[t]) = \mathcal{E}_{\gamma}(\Psi[t]) \quad \forall t \in [0, 32\pi]. $$

Since $\mathcal{L}_{\gamma}\dot{\gamma} = 0$ and $\Psi_t = \tilde{\Psi}_t$, the new function $\tilde{\Psi}$ also satisfies

$$ -\tilde{\Psi}_{tt} + \mathcal{L}_{\gamma}\tilde{\Psi} - a(x)\tilde{\Psi}_t = g. $$

Moreover, by Lemma 8.14, assuming $\delta_{li}$ small enough we have

$$ \|g\|_{L^2(0,32\pi; L^2(\mathbb{T}^1))}^2 \le \delta_{li}\mathcal{E}_{\gamma}(\Psi[0]) \le C\delta_{li}\mathcal{E}_{\gamma}(\Psi[16\pi]) = C\delta_{li}\mathcal{E}_{\gamma}(\tilde{\Psi}[16\pi]) $$

Recall from Remark 8.10 that

$$ C^{-1}\|(f,g)\|_{\mathcal{H}}^2 \le \mathcal{E}_{\gamma}(f,g) \le C\|(f,g)\|_{\mathcal{H}}^2 \quad \forall (f,g) \in \mathcal{H}_{\gamma,0}. $$

Because $\tilde{\Psi}[16\pi]$ belongs to $\mathcal{H}_{\gamma,0}$, we get

$$ \|g\|_{L^2(0,32\pi; L^2(\mathbb{T}^1))}^2 \le C_0\delta_{li}\|\tilde{\Psi}[16\pi]\|_{\mathcal{H}}^2. $$

Assuming that $C_0\delta_{li} \le \tilde{\delta}_{li}$, then $(\tilde{\Psi}, g)$ satisfies the conditions given in Lemma 8.13. Thus

$$ \int_0^{32\pi} \langle a(x)\tilde{\Psi}_t, \tilde{\Psi}_t \rangle_{L^2(\mathbb{T}^1)} dt \ge \tilde{c}_{li}\|\tilde{\Psi}[16\pi]\|_{\mathcal{H}}^2. $$
Therefore,

$$ \int_{0}^{32\pi} \langle a(x)\Psi_t, \Psi_t \rangle_{L^2(\mathbb{T}^1)} dt \geq \tilde{c}_{li}C^{-1}\mathcal{E}_{\gamma}(\tilde{\Psi}[16\pi]) = \tilde{c}_{li}C^{-1}\mathcal{E}_{\gamma}(\Psi[16\pi]) \geq \tilde{c}_{li}C^{-1}\mathcal{E}_{\gamma}(\Psi[0]). $$

This is exactly inequality (8.61) by selecting $c_{li}$ as $\tilde{c}_{li}C^{-1}$.

We also notice that (8.62) is a direct consequence of (8.61) by decreasing the values of $c_{li}$ and $\delta_{li}$ if necessary. Indeed,

$$ \begin{aligned} \mathcal{E}_{\gamma}(\Psi[0]) - \mathcal{E}_{\gamma}(\Psi[32\pi]) &= \int_{0}^{32\pi} \langle a(x)\Psi_t, \Psi_t \rangle_{L^2(\mathbb{T}^1)} + \langle g, \Psi_t \rangle_{L^2(\mathbb{T}^1)} dt \\ &\geq c_{li}\mathcal{E}_{\gamma}(\Psi[0]) - C\delta_{li}^{1/2}\mathcal{E}_{\gamma}(\Psi[0]). \end{aligned} $$

Finally, inequality (8.63) is already shown in Lemma 8.14. This finishes the proof of this reduction procedure.

### 8.4.2. The proof of Lemma 8.13.
This proof is based on two aspects.

1) The propagation of smallness of this linear system with source terms. In this step we do not use the specific structure of the solution, namely that the solution belongs to $\mathcal{H}_{\gamma}$ and that the geodesic $\gamma$ has negative curvature.

2) If some solution satisfies $\Psi_t$ small in a time period $J$, then $\langle \mathcal{L}_{\gamma}\Psi, \Psi \rangle$ must be small in a smaller interval inside $J$. This leads to a contradiction with the coercive estimate; Proposition 8.8.

**Step 1.** Assume that for some $\tilde{\delta}_{li}$ and $\tilde{c}_{li}$ small, there is some non-trivial pair $\Psi, g$ satisfying

$$ (8.70) \qquad -\Psi_{tt} + \mathcal{L}_{\gamma}\Psi - a(x)\Psi_t = g, $$

$$ (8.71) \qquad \Psi[t] \in \mathcal{H}_{\gamma} \quad \forall t \in [0, 32\pi], $$

$$ (8.72) \qquad \Psi[16\pi] \in \mathcal{H}_{\gamma,0}, $$

$$ (8.73) \qquad \|g\|_{L^2(0,32\pi;L^2(\mathbb{T}^1))}^2 \leq \tilde{\delta}_{li} \|\Psi[16\pi]\|_{\mathcal{H}}^2, $$

and

$$ (8.74) \qquad \int_{0}^{32\pi} \langle a(x)\Psi_t, \Psi_t \rangle_{L^2(\mathbb{T}^1)} dt \leq \tilde{c}_{li} \|\Psi[16\pi]\|_{\mathcal{H}}^2. $$

Then, the propagation of smallness property, Corollary 3.2, implies that

$$ (8.75) \qquad \|\Psi_t\|_{L_x^\infty(\mathbb{T}^1; L_t^2(13\pi, 19\pi))}^2 \leq C_q (\tilde{\delta}_{li}^{1/q} + \tilde{c}_{li}^{1/q}) \|\Psi[16\pi]\|_{\mathcal{H}}^2. $$

Note from Lemma 8.15 that

$$ C^{-1} \|\Psi[t_2]\|_{\mathcal{H}}^2 \leq \|\Psi[t_1]\|_{\mathcal{H}}^2 \leq C \|\Psi[t_2]\|_{\mathcal{H}}^2 \quad \forall t_1, t_2 \in [0, 32\pi]. $$

Set for every $t \in [0, 32\pi]$,

$$ \Psi(t, x) = \tilde{\Psi}(t, x) + b(t)\dot{\gamma}(x) \quad \text{with} $$

$$ \tilde{\Psi}[t] \in \mathcal{H}_{\gamma,0} \text{ and } b(t) = \langle \Psi(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)}. $$

Notice that in particular $b(16\pi) = 0$. Furthermore, for any $t \in [13\pi, 19\pi]$ one has

$$ \Psi(t, x) = \Psi(16\pi, x) + \int_{16\pi}^{t} \Psi_t(s, x)ds, $$
hence

$$
\begin{align*}
b(t) &= \langle \Psi(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} \\
&= \left\langle \Psi(16\pi, \cdot) + \int_{16\pi}^{t} \Psi_t(s, \cdot) ds, \gamma_x(\cdot) \right\rangle_{L^2(\mathbb{T}^1)} \\
&= \left\langle \int_{16\pi}^{t} \Psi_t(s, \cdot) ds, \gamma_x(\cdot) \right\rangle_{L^2(\mathbb{T}^1)}.
\end{align*}
$$

This, combined with inequality (8.75), implies that for every $t \in [13\pi, 19\pi]$

$$
(8.76) \qquad |b(t)|^2 \le \|\Psi_t\|_{L_{t,x}^1((13\pi,19\pi)^2 \times \mathbb{T}^1)}^2 \lesssim (\tilde{\delta}_{li}^{1/q} + \tilde{c}_{li}^{1/q}) \|\Psi[16\pi]\|_{\mathcal{H}}^2.
$$

Step 2. Keep in mind the preceding estimate on $\|\Psi_t\|_{L_x^\infty(\mathbb{T}^1; L_t^2(13\pi,19\pi))}$, we then come back to (8.64), which we integrate against $\Psi$. The idea is to use the smallness of $\Psi_t$ on the time interval $(13\pi, 19\pi)$. We define a non-negative smooth truncated function $\eta$ such that

$$
\eta \operatorname{supp} (13\pi, 19\pi), \eta = 1 \text{ on } [14\pi, 18\pi].
$$

Integrating equation (8.58) against ηΨ, we obtain

$$
\int_{\mathbb{R}} \langle \mathcal{L}_{\gamma} \Psi, \eta \Psi \rangle_{L^2(\mathbb{T}^1)} dt = \int_{\mathbb{R}} \int_{\mathbb{T}^1} (\Psi_{tt} + a(x)\Psi_t + g) \cdot \eta \Psi \,dx\,dt.
$$

Concerning the right hand side, we have

$$
\begin{align*}
\left| \int_{\mathbb{T}^1} \int_{\mathbb{R}} \Psi_{tt} \cdot \eta \Psi \, dt \, dx \right| &= \left| \int_{\mathbb{T}^1} \int_{\mathbb{R}} \Psi_t \cdot (\eta_t \Psi + \eta \Psi_t) \, dt \, dx \right| \\
&= \left| \int_{13\pi}^{19\pi} \langle \Psi_t, \eta_t \Psi + \eta \Psi_t \rangle_{L^2(\mathbb{T}^1)} \, dt \right| \\
&\lesssim \|\Psi_t\|_{L^2(13\pi, 19\pi; L^2(\mathbb{T}^1))} \|\Psi[t]\|_{L^2(13\pi, 19\pi; \mathcal{H})} \\
&\lesssim (\tilde{\delta}_{li}^{1/2q} + \tilde{c}_{li}^{1/2q}) \|\Psi[16\pi]\|_{\mathcal{H}}^2.
\end{align*}
$$

Similarly,

$$
\left| \int_{\mathbb{T}^1} \int_{\mathbb{R}} (a(x) \Psi_t + g) \cdot \eta \Psi \, dt \, dx \right| \lesssim (\tilde{\delta}_{li}^{1/2q} + \tilde{c}_{li}^{1/2q}) \|\Psi[16\pi]\|_{\mathcal{H}}^2.
$$

Thus

$$
(8.77) \qquad \int_{\mathbb{R}} \langle \mathcal{L}_{\gamma} \Psi, \eta \Psi \rangle_{L^2(\mathbb{T}^1)} dt \le \tilde{C} (\tilde{\delta}_{li}^{1/2q} + \tilde{c}_{li}^{1/2q}) \|\Psi[16\pi]\|_{\mathcal{H}}^2,
$$

where the constant $\tilde{C}$ is independent of the choice of $\tilde{\delta}_{li}$ and $\tilde{c}_{li}$.

On the other hand, thanks to the coercive estimate provided in Proposition 8.8 and the
observation (2.9), we have

$$
\begin{align*}
\int_{\mathbb{R}} \langle \mathcal{L}_{\gamma} \Psi, \eta \Psi \rangle_{L^2(\mathbb{T}^1)} dt &= \int_{13\pi}^{19\pi} \eta \langle \mathcal{L}_{\gamma} \Psi, \Psi \rangle_{L^2(\mathbb{T}^1)} dt \\
&= \int_{13\pi}^{19\pi} \eta \langle \mathcal{L}_{\gamma} \tilde{\Psi}, \tilde{\Psi} \rangle_{L^2(\mathbb{T}^1)} dt \\
&\geq c \int_{14\pi}^{18\pi} \| \tilde{\Psi}(t) \|_{H^1(\mathbb{T}^1)}^2 dt
\end{align*}
$$
$$ 
\begin{aligned}
&\geq \frac{c}{2} \int_{14\pi}^{18\pi} \|\Psi(t)\|_{H^1(\mathbb{T}^1)}^2 dt - C \int_{14\pi}^{18\pi} |b(t)|^2 dt \\
&= \frac{c}{2} \int_{14\pi}^{18\pi} \left( \|\Psi[t]\|_{\mathcal{H}}^2 - \|\Psi_t(t)\|_{L^2(\mathbb{T}^1)}^2 \right) dt - C \int_{14\pi}^{18\pi} |b(t)|^2 dt.
\end{aligned}
$$

Due to the smallness of $\Psi_t$ and $b(t)$, namely inequalities (8.75) and (8.76), by assuming $\tilde{\delta}_{li}$ and $\tilde{c}_{li}$ sufficiently small we have

$$ \int_{\mathbb{R}} \langle \mathcal{L}_{\gamma} \Psi, \eta \Psi \rangle_{L^2(\mathbb{T}^1)} dt \geq \tilde{c} \|\Psi[16\pi]\|_{\mathcal{H}}^2, $$

where the constant $\tilde{c}$ is independent of the choice of $\tilde{\delta}_{li}$ and $\tilde{c}_{li}$.

Hence any non-trivial solution satisfying (8.70)–(8.74) must implies

$$ \tilde{\delta}_{li}^{1/2q} + \tilde{c}_{li}^{1/2q} \geq \tilde{c} \tilde{C}^{-1}. $$

This leads to the proof of the estimate (8.68) by selecting $\tilde{\delta}_{li}$ and $\tilde{c}_{li}$ sufficiently small such that the preceding inequality fails. Thus we finish the proof of Lemma 8.13.

### 8.5. Exponential stability of the full system on $(\varphi, \alpha)$.

#### 8.5.1. Fixed time stability.
This section is devoted to the following result. Its proof is based on the following ingredients: Proposition 8.12 concerning the stability of the linearized equation on $\Psi$, the observation Lemma 8.6 on $\mathcal{E}_{\gamma}(\Psi[t])$, the structure of the nonlinear term (namely the lack of the quadratic term $\alpha^2$), and the standard bootstrap argument.

**Proposition 8.16.** Let $\gamma$ be a closed geodesic on $\mathcal{N}$ along which the sectional curvature is negative. Let $c_{li}$ be the constant given in Proposition 8.12. There exist constants $\delta_{fi} > 0$ and $M > 1$ such that for every initial state $(\varphi, \varphi_t, \alpha, \dot{\alpha})(0) \in \mathcal{H}_{\gamma,0,0} \times \mathbb{R}^2$ satisfying

$$ (8.78) \qquad \mathcal{E}_{\gamma}(\varphi[0]) + \dot{\alpha}^2(0) + \alpha^2(0) \leq \delta_{fi}^2, $$

the system (8.20) admits a unique solution in period $[0, 32\pi]$. The function $\varphi[t]$ belongs to $\mathcal{H}_{\gamma,0,0}$ for every $t \in [0, 32\pi]$, the function $\Psi$ given as (8.22) satisfies (8.23). Moreover,

$$ (8.79) \qquad \mathcal{E}_{\gamma}(\Psi[32\pi]) \leq (1 - c_{li})\mathcal{E}_{\gamma}(\Psi[0]), $$

$$ (8.80) \qquad \mathcal{E}_{\gamma}(\Psi[t]) \leq 2\mathcal{E}_{\gamma}(\Psi[0]), \quad \forall t \in [0, 32\pi], $$

$$ (8.81) \qquad |\alpha(t)| \leq |\alpha(0)| + M \mathcal{E}_{\gamma}^{1/2}(\Psi[0]), \quad \forall t \in [0, 32\pi]. $$

**Proof of Proposition 8.16.** Define $D := [0, 32\pi] \times \mathbb{T}^1$. Recall the definition of space $\mathcal{W} = W_{32\pi}$ given in Definition 2.9, the constant $L$ in (8.21), the definition of constants $c_{li}$ and $\delta_{li}$ in Proposition 8.12, and the constant $C_{co} > 1$ from Remark 8.10 such that

$$ C_{co}^{-1} \|(f,g)\|_{\mathcal{H}} \leq (\mathcal{E}_{\gamma}(f,g))^{1/2} \leq C_{co} \|(f,g)\|_{\mathcal{H}} \quad \forall (f,g) \in \mathcal{H}_{\gamma,0}, $$

and the identity from Lemma 8.6,

$$ (8.82) \qquad \mathcal{E}_{\gamma}(\Psi[t]) = \mathcal{E}_{\gamma}(\varphi[t]) + \frac{L}{2} \dot{\alpha}^2(t). $$

We first present the following lemma concerning some useful estimates for the coupled system.
**Lemma 8.17.** There exist constants $\delta_2 > 0$ and $C_2 > 0$ such that, for every initial state $(\varphi, \varphi_t, \alpha, \dot{\alpha})(0) \in \mathcal{H}_{\gamma,0,0} \times \mathbb{R}^2$ satisfying

$$ (8.83) \qquad \mathcal{E}_{\gamma}(\varphi[0]) + \dot{\alpha}^2(0) + \alpha^2(0) \le \delta_2^2, $$

the system (8.20) admits a unique solution in period $[0, 32\pi]$. The function $\varphi[t]$ belongs to $\mathcal{H}_{\gamma,0,0}$ for every $t \in [0, 32\pi]$, the function $\Psi$ given as (8.22) satisfies (8.23). Moreover,

$$ (8.84) \qquad \|\varphi\|_{W_{32\pi}} + \|\alpha\|_{C^1(0,32\pi)} \le C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\alpha(0)| + |\dot{\alpha}(0)|), $$

$$ (8.85) \qquad \|\varphi\|_{W_{32\pi}} + \|\dot{\alpha}\|_{C(0,32\pi)} \le C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|), $$

and the nonlinear terms on the right-hand side of (8.20) and (8.23) satisfy

$$ (8.86) \qquad \| (M, M_1)(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \|_{L^2_{t,x}(D)} + \| \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \|_{L^2(0,32\pi)} \\ \le C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\alpha(0)| + |\dot{\alpha}(0)|) (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|). $$

**Proof of Lemma 8.17.** The small data well-posedness of this coupled system on $(\varphi, \alpha)$ is a consequence of the well-posedness of the system on $\phi$ and the equivalence between the flows of $\phi$ and $(\varphi, \alpha)$. Indeed, thanks to the decomposition results Lemma 8.3 and Remark 8.4, any small initial state $(\varphi, \varphi_t, \alpha, \dot{\alpha})(0) \in \mathcal{H}_{\gamma,0,0} \times \mathbb{R}^2$ corresponds to an initial state $(\phi, \phi_t)(0)$ close to $(\gamma, 0)$. According to Lemma 2.10, the wave maps equation admits a unique solution $\phi[t]$ on the time interval $(0, 32\pi)$. Thus due to the rule of decomposition we have that $\varphi[t]$ belongs to $\mathcal{H}_{\gamma,0,0}$ for every $t \in [0, 32\pi]$, and according to Proposition 8.7 the function $\Psi$ given as (8.22) satisfies (8.23).

Notice that $(\gamma, 0)$ is a steady state of the damped wave maps equation, thus using the continuous dependence result Lemma 2.11,

$$ \|(w_x, w_t, w)\|_{L^\infty_t L^2_x(D)} + \|w_u\|_{L^2_u L^\infty_v \cap L^\infty_v L^2_u(D)} + \|w_v\|_{L^2_v L^\infty_u \cap L^\infty_u L^2_v(D)} \le C \|w[0]\|_{\mathcal{H}}, $$

where $w(t, x) := \phi(t, x) - \gamma(x)$. This implies the unique solution $\phi[t]$ is close enough to the geodesic $(\gamma, 0)$ during this period. Using again Lemma 8.3 and Remark 8.4, the projection $(\varphi, \varphi_t, \alpha, \dot{\alpha})(t) \in \mathcal{H}_{\gamma,0,0} \times \mathbb{R}^2$ is the unique solution of (8.20) with given data at time $t = 0$ and is small: for every $t \in [0, 32\pi]$,

$$ (8.87) \qquad |\alpha(t)| + |\dot{\alpha}(t)| + \|\varphi[t]\|_{\mathcal{H}} \le C_{tr} \|\phi[t] - (\gamma, 0)\|_{\mathcal{H}} \\ \le C C_{tr} \|w[0]\|_{\mathcal{H}} \le C C_{tr}^2 (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}). $$

To demonstrate estimates (8.84)–(8.86), we first estimate from the upper bound of $M, M_1, \mathcal{O}$ in (8.24)–(8.25) that

$$ \begin{aligned} & \left\| (M, M_1)(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \right\|_{L^2_{t,x}(D)} + \left\| \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \right\|_{L^2(0,32\pi)} \\ & \lesssim \left\| \sum_{j,k} \left| \partial_{\nu} \varphi^j \partial^\nu \varphi^k \right| + |\dot{\alpha}|^2 + |\varphi_t| |\dot{\alpha}| + (|\varphi| + |\alpha|)(|\varphi| + |\varphi_x| + |\varphi_t| + |\dot{\alpha}|) \right\|_{L^2_{t,x}(D)} \\ & \qquad + \left\| \|\varphi_t\|_{L^1(\mathbb{T}^1)} (|\alpha| + |\varphi|(x) + \|\varphi\|_{L^1(\mathbb{T}^1)}) \right\|_{L^2_{t,x}(D)} \\ & \lesssim \|\varphi\|_{W_{32\pi}}^2 + \|\dot{\alpha}\|_{C(0,32\pi)}^2 + \|\dot{\alpha}\|_{C(0,32\pi)} \|\varphi\|_{W_{32\pi}} + (\|\varphi\|_{W_{32\pi}} + \|\alpha\|_{C(0,32\pi)}) (\|\varphi\|_{W_{32\pi}} + \|\dot{\alpha}\|_{C(0,32\pi)}) \end{aligned} \\ (8.88) \qquad \lesssim (\|\varphi\|_{W_{32\pi}} + \|\alpha\|_{C^1(0,32\pi)}) (\|\varphi\|_{W_{32\pi}} + \|\dot{\alpha}\|_{C(0,32\pi)}). \end{aligned} $$
We first prove (8.84). The function $\varphi$ is governed by (8.44). We regard this equation as a linear wave equation on $\varphi$ with the right-hand side as a given source term. Thus by the well-posedness result of the linear wave equation Lemma 2.12, as well as estimates (8.87)–(8.88), we obtain

$$
\begin{aligned}
\|\varphi\|_{W_{32\pi}} & \lesssim \|\varphi[0]\|_{\mathcal{H}} + \|\mathcal{M}_3(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x)\|_{L^2_{t,x}(D)} \\
& \qquad + \left\| \left(a(x) - \frac{l}{L}\right) \dot{\alpha}\gamma_x(x) - \frac{1}{L} \langle a(\cdot)\varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} \gamma_x(x) \right\|_{L^2_{t,x}(D)} \\
& \lesssim (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) + (\|\varphi\|_{W_{32\pi}} + \|\alpha\|_{C^{1}(0,32\pi)}) (\|\varphi\|_{W_{32\pi}} + \|\dot{\alpha}\|_{C(0,32\pi)}).
\end{aligned}
$$

Recall that $\alpha$ is bounded by

$$ \|\alpha\|_{C^1(0,32\pi)} \lesssim |\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}. $$

Thus, estimate (8.84) follows by combining the preceding two estimates and by assuming $\delta_2 = \delta_{2,0}$ sufficiently small.

Next, we turn to the proof of inequality (8.85). This time we shall first prove the desired estimate on a small time interval $(0, T)$, and then iterate the procedure to $(0, 32\pi)$. Let the value of $T$ to be chosen later on. We consider the equation on $D_T = [0, T] \times \mathbb{T}^1$ and work with the $W_T$ norm.

In analogy to the estimate (8.88), and using estimate (8.84),

$$
\begin{aligned}
& \left\| (\mathcal{M}, \mathcal{M}_1)(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \right\|_{L^2_{t,x}(D_T)} + \left\| \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \right\|_{L^2(0,T)} \\
& \lesssim (\|\varphi\|_{W_T} + \|\alpha\|_{C^1(0,T)})(\|\varphi\|_{W_T} + \|\dot{\alpha}\|_{C(0,T)}) \\
(8.89) \qquad & \lesssim (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}})(\|\varphi\|_{W_T} + \|\dot{\alpha}\|_{C(0,T)})
\end{aligned}
$$

Again, thanks to the well-posedness result of the linear wave equation Lemma 2.12, as well as estimates (8.84) and (8.89) we obtain

$$
\begin{aligned}
\|\varphi\|_{W_T} & \lesssim \|\varphi[0]\|_{\mathcal{H}} + T^{1/2} \|\mathcal{M}_1(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x)\|_{L^2_{t,x}(D_T)} \\
& \qquad + T^{1/2} \left\| \left(a(x) - \frac{l}{L}\right) \dot{\alpha}\gamma_x(x) - \frac{1}{L} \langle a(\cdot)\varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} \gamma_x(x) \right\|_{L^2_{t,x}(D_T)} \\
& \lesssim \|\varphi[0]\|_{\mathcal{H}} + T^{1/2} (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) (\|\varphi\|_{W_T} + \|\dot{\alpha}\|_{C(0,T)}) \\
& \qquad + T \|\dot{\alpha}\|_{C(0,T)} + T \|\varphi\|_{W_T}
\end{aligned}
$$

Next we turn to the equation governing $\alpha$ in (8.20). Notice that it can be regarded as a first order equation on $\dot{\alpha}$, and

$$
\begin{aligned}
\|\dot{\alpha}\|_{C(0,T)} & \lesssim |\dot{\alpha}(0)| + \left\| \frac{1}{L} \langle a(\cdot)\varphi_t(t, \cdot), \gamma_x(\cdot) \rangle_{L^2(\mathbb{T}^1)} + \mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)) \right\|_{L^1(0,T)} \\
& \lesssim |\dot{\alpha}(0)| + \|\varphi_t\|_{L^1_{t,x}(D_T)} + T^{1/2} \|\mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot))\|_{L^2(0,T)} \\
& \lesssim |\dot{\alpha}(0)| + T \|\varphi\|_{W_T} + T^{1/2} (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) (\|\varphi\|_{W_T} + \|\dot{\alpha}\|_{C(0,T)}).
\end{aligned}
$$

By combining the above two estimates, we find effective computable $T > 0$ and $C > 0$ such that for any initial state satisfying (8.83) with $\delta_{2,0}$, we infer

$$ \|\varphi\|_{W_T} + \|\dot{\alpha}\|_{C(T)} \le C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|), $$
and, in particular

$$ \|\varphi\|_{C(0,T;\mathcal{H})} + \|\dot{\alpha}\|_{C(T)} \le C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|). $$

Thanks to the bound (8.84), by reducing the value of $\delta_2$ we deduce that

$$ \mathcal{E}_{\gamma}(\varphi[t]) + \dot{\alpha}^2(t) + \alpha^2(t) \le \delta_{2,0}^2 $$

holds for every $t \in [0, 32\pi]$ and for every initial state satisfying (8.83). Therefore, by repeating the procedure on $\{[kT, (k+1T)): k = 0, 1, ..., [32\pi/T]\}$, we conclude the estimate (8.85).

Finally, the inequality (8.86) is a direct consequence of (8.88) together with the bounds (8.84)-(8.85). This finishes the proof of Lemma 8.17. □

Now we come back to the proof of Proposition 8.16. Fix the value of $\delta_{fi}$ and $M$ as

$$ (8.90) \qquad \delta_{fi} := \min \left\{ \frac{\delta_{li}^{1/2}}{8C_{co}C_2(C_{co} + 2/\sqrt{L})}, \delta_2 \right\}, \quad M := 32\pi C_2 (C_{co} + 2/\sqrt{L}). $$

The main issue is to obtain the estimate (8.79). Since $\delta_{fi}$ is smaller than $\delta_2$, thanks to Lemma 8.17, the unique solution $(\varphi, \varphi_t, \alpha, \dot{\alpha})$ and the nonlinear terms $\mathcal{M}(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x)$, $\mathcal{M}_1(x; \alpha, \dot{\alpha}, \varphi, \varphi_t, \varphi_x)$, and $\mathcal{O}(\alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot))$ satisfy

$$ \begin{aligned} \|\varphi\|_{W} + \|\alpha\|_{C^{1}(0,32\pi)} &\le C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\alpha(0)| + |\dot{\alpha}(0)|) \le 2C_{co}C_2\delta_{fi}, \\ \|\varphi\|_{W} + \|\dot{\alpha}\|_{C(0,32\pi)} &\le C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|), \\ \|\mathcal{M}\|_{L_{t,x}^2(D)} + \|\mathcal{M}_1\|_{L_{t,x}^2(D)} + \|\mathcal{O}\|_{L^2(0,32\pi)} &\le 2C_{co}C_2\delta (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|). \end{aligned} $$

The idea is to use Proposition 8.12. Clearly, by the definition of $\Psi$ in (8.22) and equation (8.23), the function $\Psi$ satisfies conditions (8.58)-(8.59) with

$$ g = \mathcal{M}(x; \alpha, \dot{\alpha}, \varphi(\cdot), \varphi_t(\cdot), \varphi_x(\cdot)). $$

Furthermore, one has from the definition of $\delta_{fi}$ that

$$ \begin{aligned} \|g\|_{L_{t,x}^2(D)}^2 &= \|\mathcal{M}\|_{L_{t,x}^2(D)}^2 \le 4C_{co}^2 C_2^2 \delta_{fi}^2 (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|)^2 \\ &\le 8C_{co}^2 C_2^2 \delta_{fi}^2 \left(C_{co}^2 + \frac{2}{L}\right) \mathcal{E}_{\gamma}(\Psi[0]) \le \frac{\delta_{li}}{2} \mathcal{E}_{\gamma}(\Psi[0]), \end{aligned} $$

namely, the condition (8.60) also holds. Thus, by applying Proposition 8.12, we obtain estimates (8.79)-(8.80). Finally, the inequality (8.81) directly follows from

$$ \begin{aligned} |\alpha(t)| &\le |\alpha(0)| + \int_0^t |\dot{\alpha}(s)|ds \\ &\le |\alpha(0)| + 32\pi C_2 (\|\varphi[0]\|_{\mathcal{H}} + |\dot{\alpha}(0)|) \\ &\le |\alpha(0)| + 32\pi C_2 (C_{co} + 2/\sqrt{L}) \mathcal{E}_{\gamma}^{1/2}(\Psi[0]) \end{aligned} $$

This finishes the proof of Proposition 8.16. □

### 8.5.2. The proof of Theorem 1.3 Part (ii).
Armed with the fixed time stability result Proposition 8.16, we are now in a position to present the proof of Theorem 1.3.

Recall the following constants: $\delta_{tr}$ and $C_{tr}$ in Remark 8.4, $C_{co}$ in Remark 8.10, $c_{li}$ in Proposition 8.12, $L$ in (8.21), $\delta_{fi}$ and $M$ in Proposition 8.16. Let us fix the value of $\varepsilon$ as

$$ (8.91) \qquad \varepsilon := \frac{\delta_{fi}}{2C_{tr}L(C_{co} + \sqrt{L}/2)(1 + 2/\sqrt{L} + 2M/c_{li})}. $$
Thanks to Proposition 8.16, Remark 8.10, the identity (8.82), using a simple induction argument we obtain the following: for any initial state $\phi[0]$ satisfying the condition,

$$ (8.92) \qquad \|\phi[0] - (\gamma, 0)\|_{\mathcal{H}} \le \varepsilon, $$

the system admits a unique solution in period $(0, +\infty)$. Moreover, under the decomposition $(\varphi, \varphi_t, \alpha, \dot{\alpha}) \in \mathcal{H}_{\gamma,0,0} \times \mathbb{R}^2$, we have that for every $n \in \mathbb{N}$,

$$ (8.93) \qquad \mathcal{E}_{\gamma}(\Psi[32\pi(n+1)]) \le (1-c_{li})\mathcal{E}_{\gamma}(\Psi[32\pi n]), $$

$$ (8.94) \qquad \mathcal{E}_{\gamma}(\Psi[t]) \le 2\mathcal{E}_{\gamma}(\Psi[32\pi n]) \quad \forall t \in [32\pi n, 32\pi(n+1)], $$

$$ (8.95) \qquad |\alpha(t)| \le |\alpha(32\pi n)| + M\mathcal{E}_{\gamma}^{1/2}(\Psi[32\pi n]) \quad \forall t \in [32\pi n, 32\pi(n+1)], $$

$$ (8.96) \qquad \mathcal{E}_{\gamma}(\varphi[32\pi(n+1)]) + \dot{\alpha}^2(32\pi(n+1)) + \alpha^2(32\pi(n+1)) < \delta_{fi}^2. $$

The case $n$ equals 0 is a direct consequence of Proposition 8.16. Suppose that the argument is correct for every $n \in \{0, ..., k\}$. In particular, inequality (8.96) for the case $n = k$ is exactly the smallness condition of the initial state at time $32\pi(k+1)$ for Proposition 8.16. Then, for the case $n$ equals to $k+1$, the existence of a unique solution as well as the estimates (8.93)–(8.95) are guaranteed by Proposition 8.16. It suffices to show inequality (8.96) to conclude the induction argument.

Indeed, thanks to the inequality (8.93) for the cases $n \in \{0, ..., k+1\}$,

$$ \begin{aligned} \mathcal{E}_{\gamma}(\Psi[32\pi(k+2)]) &\le (1-c_{li})\mathcal{E}_{\gamma}(\Psi[32\pi(k+1)]) \\ &\le (1-c_{li})^2\mathcal{E}_{\gamma}(\Psi[32\pi k]) \le (1-c_{li})^{k+2}\mathcal{E}_{\gamma}(\Psi[0]) \end{aligned} $$

These bounds, to be combined with the inequality (8.95) for the cases $n \in \{0, ..., k+1\}$, yield

$$ \begin{aligned} |\alpha(32\pi(k+2))| &\le |\alpha(32\pi(k+1))| + M\mathcal{E}_{\gamma}^{1/2}(\Psi[32\pi(k+1)]) \\ &\le |\alpha(32\pi k)| + M\left(\mathcal{E}_{\gamma}^{1/2}(\Psi[32\pi(k+1)]) + \mathcal{E}_{\gamma}^{1/2}(\Psi[32\pi k])\right) \\ &\le |\alpha(0)| + M\sum_{n=0}^{k+1} \mathcal{E}_{\gamma}^{1/2}(\Psi[32\pi n]) \\ &\le |\alpha(0)| + M\sum_{n=0}^{k+1} (1-c_{li})^{n/2} \mathcal{E}_{\gamma}^{1/2}(\Psi[0]) \\ &\le |\alpha(0)| + \frac{2M}{c_{li}} \mathcal{E}_{\gamma}^{1/2}(\Psi[0]). \end{aligned} $$

Hence,

$$ \begin{aligned} & \mathcal{E}_{\gamma}(\varphi[32\pi(k+2)]) + \dot{\alpha}^2(32\pi(k+2)) + \alpha^2(32\pi(k+2)) \\ &\le \left(1 + \frac{2}{L} + \frac{4M^2}{c_{li}^2}\right) \mathcal{E}_{\gamma}(\Psi[0]) + 2|\alpha(0)|^2 \\ &= \left(1 + \frac{2}{L} + \frac{4M^2}{c_{li}^2}\right) \left(C_{co}^2 \|\varphi[0]\|_{\mathcal{H}}^2 + \frac{L}{2}\dot{\alpha}^2(0)\right) + 2|\alpha(0)|^2 \\ &\le \left(1 + \frac{2}{L} + \frac{4M^2}{c_{li}^2}\right) \left(C_{co}^2 + \frac{L}{2}\right) (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}})^2 \\ &\le \left(1 + \frac{2}{L} + \frac{4M^2}{c_{li}^2}\right) \left(C_{co}^2 + \frac{L}{2}\right) C_{tr}^2 \varepsilon^2 < \delta_{fi}^2. \end{aligned} $$
This concludes the induction procedure.

Now, we are in a position to prove the exponential stability property. First, we observe that $|\dot{\alpha}(t)| + \|\varphi[t]\|_{\mathcal{H}}$ decays exponentially: for every $t \in (0, +\infty)$,

$$|\dot{\alpha}(t)| + \|\varphi[t]\|_{\mathcal{H}} \lesssim \mathcal{E}_{\gamma}^{1/2}(\Psi[t]) \lesssim e^{-rt}\mathcal{E}_{\gamma}^{1/2}(\Psi[0]) \lesssim e^{-rt}(|\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}).$$

Next, we fix the value of rotation $p$, which is indeed the limit of $\alpha(t)$. Due to the estimates (8.93) and (8.96), and the Cauchy convergence principle, the sequence $\{\alpha(t)\}_{t \in (0, +\infty)}$ admits a limit:

$$|p| \le C (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}), \\ |\alpha(t) - p| \le Ce^{-rt} (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) \quad \forall t \in (0, +\infty).$$

Recall from Lemma 8.3 concerning the decomposition of $\phi$:

$$\phi(t, x) = \gamma(x + \alpha(t)) + \varphi(t, x) + \mathcal{F}(x; \varphi(t, x), \alpha(t)) \quad \forall t \in (0, +\infty) \quad \forall x \in \mathbb{T}^1,$$

where the nonlinear mapping $\mathcal{F}$ satisfies (8.11). Thus,

$$\begin{aligned} \|\phi(t) - \gamma_p\|_{L^2(\mathbb{T}^1)} &\lesssim \|\gamma_{\alpha(t)} - \gamma_p\|_{L^2(\mathbb{T}^1)} + \|\varphi(t)\|_{L^2(\mathbb{T}^1)} + \|\mathcal{F}(x; \varphi(t, x), \alpha(t))\|_{L^2(\mathbb{T}^1)} \\ &\lesssim |\alpha(t) - p| + \|\varphi(t)\|_{L^2(\mathbb{T}^1)} \\ &\lesssim e^{-rt} (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) \quad \forall t \in (0, +\infty). \end{aligned}$$

And, similarly,

$$\begin{aligned} & \|(\phi(t) - \gamma_p)_x\|_{L^2(\mathbb{T}^1)} \\ &\lesssim \|(\gamma_{\alpha(t)} - \gamma_p)_x\|_{L^2(\mathbb{T}^1)} + \|\varphi_x(t)\|_{L^2(\mathbb{T}^1)} + \|\mathcal{F}_x(x; \varphi(t, x), \alpha(t))\|_{L^2(\mathbb{T}^1)} + \|\mathcal{F}_{\varphi}(x; \varphi(t, x), \alpha(t))\varphi_x(t, x)\|_{L^2(\mathbb{T}^1)} \\ &\lesssim \|(\gamma_{\alpha(t)} - \gamma_p)_x\|_{L^2(\mathbb{T}^1)} + \|\varphi_x(t)\|_{L^2(\mathbb{T}^1)} + \|\varphi\|^2 + |\varphi||\alpha||_{L^2(\mathbb{T}^1)} + \|(\varphi| + |\alpha|)\varphi_x\|_{L^2(\mathbb{T}^1)} \\ &\lesssim |\alpha(t) - p| + \|\varphi(t)\|_{H^1(\mathbb{T}^1)} \\ &\lesssim e^{-rt} (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) \quad \forall t \in (0, +\infty). \end{aligned}$$

as well as

$$\begin{aligned} & \|\phi_t(t)\|_{L^2(\mathbb{T}^1)} \\ &\lesssim \|(\gamma_{\alpha(t)})_x \dot{\alpha}\|_{L^2(\mathbb{T}^1)} + \|\varphi_t(t)\|_{L^2(\mathbb{T}^1)} + \|\mathcal{F}_{\varphi}(x; \varphi(t, x), \alpha(t))\varphi_t(t, x)\|_{L^2(\mathbb{T}^1)} + \|\mathcal{F}_{\alpha}(x; \varphi(t, x), \alpha(t))\dot{\alpha}\|_{L^2(\mathbb{T}^1)} \\ &\lesssim |\dot{\alpha}(t)| + \|\varphi_t(t)\|_{L^2(\mathbb{T}^1)} \\ &\lesssim e^{-rt} (|\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) \quad \forall t \in (0, +\infty). \end{aligned}$$

By combining the above three inequalities, we get the exponential stability result:

$$\|\phi[t] - (\gamma_p, 0)\|_{\mathcal{H}} \lesssim e^{-rt} (|\alpha(0)| + |\dot{\alpha}(0)| + \|\varphi[0]\|_{\mathcal{H}}) \quad \forall t \in (0, +\infty),$$

provided that the initial condition (8.92) is verified.

Thus, this finishes the proof of Theorem 1.3.

**Acknowledgments** Part of this work was accomplished during visits between the three authors, and they would like to thank Sorbonne Université, EPFL, and Peking University for the warm hospitality. Jean-Michel Coron would like to thank Xu Zhang for the kind invitation to
visit Sichuan University. This visit, during which part of this article was carried out, was supported by his contract with the New Cornerstone Science Foundation. Joachim Krieger would like to thank the hospitality of IHP in June 2025. Shengquan Xiang was supported by the NSF of China under Grant No. 12301562.

## APPENDIX A. PROOFS OF SOME GEOMETRIC LEMMAS

This section is devoted to the proofs of some geometric Lemmas 2.1, 2.8, and 5.1.

### A.1. On the extension of the second fundamental form.

*Proof of Lemma 2.1.* Since $\mathcal{N} \subset \mathbb{R}^N$ is a compact submanifold of dimension $R$, there exists an open tubular neighborhood $\mathcal{U}$ of $\mathcal{N}$ in $\mathbb{R}^N$ and a smooth projection $\mathcal{P}: \mathcal{N} \to \mathcal{N}$ such that every point $x \in \mathcal{U}$ can be uniquely written as

$$x = y + v \text{ with } y = \mathcal{P}(x) \in \mathcal{N}, v \in N_y \mathcal{N}.$$

We have the smooth orthogonal projections, for every $\phi \in \mathcal{N}$,

$$\Pi_T(\phi): \mathbb{R}^N \to T_{\phi} \mathcal{N},$$

$$\Pi_N(\phi): \mathbb{R}^N \to N_{\phi} \mathcal{N}.$$

Recall that the second fundamental form $\Pi$ is a smooth tensor field: $\Pi: TN \times TN \to NN$. Now we extend it to $\tilde{\Pi}$ as follows. For $(x, u, v) \in \mathcal{U} \times \mathbb{R}^N \times \mathbb{R}^N$, define

$$\tilde{\Pi}(x)(u, v) := \Pi(\mathcal{P}(x)) (\Pi_T(\mathcal{P}(x))u, \Pi_T(\mathcal{P}(x))v) \in N_{\mathcal{P}(x)}\mathcal{N} \subset \mathbb{R}^N.$$

This new map $\tilde{\Pi}$ is smooth on $\mathcal{U} \times \mathbb{R}^N \times \mathbb{R}^N$. For every $(\phi, u, v)$ with $\phi \in \mathcal{N}$ and $u, v \in T_{\phi} \mathcal{N}$ we recover the original map:

$$\tilde{\Pi}(\phi)(u, v) = \Pi(\phi)(u, v).$$

Next, we can define a map $\tilde{\Pi}$ from $\mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}^N$ to $\mathbb{R}^N$. Choose a smooth cutoff function $\chi: \mathbb{R}^N \to [0, 1]$ such that $\chi = 1$ on a smaller tube $U_0 \subset \mathcal{U}$ and that $\text{supp}\chi \subset \mathcal{U}$. Define, for every $(x, u, v) \in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}^N$,

$$\tilde{\Pi}(x)(u, v) := \begin{cases} \chi(x) \tilde{\Pi}(x)(u, v) & \text{if } x \in \mathcal{U}, \\ 0 & \text{otherwise.} \end{cases}$$

This new map is smooth and agrees with $\Pi$ for every $(\phi, u, v)$ with $\phi \in \mathcal{N}$ and $u, v \in T_{\phi} \mathcal{N}$. By the construction, this new map is clearly bilinear and symmetric.

Now we define the smooth functions $\{S_{jk}: \mathbb{R}^N \to \mathbb{R}^N\}$ as follows. Let the vectors $\{\iota_1, \dots, \iota_N\}$ be the standard orthonormal basis of $\mathbb{R}^N$. Set

$$S_{jk}(x) := -\tilde{\Pi}(x)(\iota_j, \iota_k) \in \mathbb{R}^N.$$

It is automatically symmetric due to the symmetry of $\tilde{\Pi}$. Moreover, by the definition of $\{S_{jk}\}$ and the bilinearity of $\tilde{\Pi}$, one has for every $x \in \mathbb{R}^N, \forall v, w \in \mathbb{R}^N$,

$$\tilde{\Pi}(x)(v, w) = \sum_{j,k=1}^{N} \tilde{\Pi}(x)(\iota_j, \iota_k) v^j w^k = - \sum_{j,k=1}^{N} S_{jk}(x) v^j w^k.$$

This finishes the proof of Lemma 2.1. □
**A.2. Proof of Lemma 2.8.** Let us now check the operator $\mathcal{L}_\gamma$ given in Definition 2.6 satisfies the identities in Lemma 2.8. Without loss of generality, we assume that $|\gamma_x(x)| = 1$. Let $\varphi$ be a vector field along the geodesic $\gamma$. Our first goal is to show that

$$ (A.1) \qquad -\langle \mathcal{L}_\gamma \varphi, \varphi \rangle_{L^2(\mathbb{T}^1)} = \int_{\mathbb{T}^1} (\|\nabla_{\gamma_x} \varphi\|^2 - \langle R(\gamma_x, \varphi)\gamma_x, \varphi \rangle) dx. $$

Construct a map

$$ \begin{aligned} c(\cdot, \cdot) : (-\varepsilon, \varepsilon) \times \mathbb{T}^1 &\to \mathcal{N}, \\ (s, x) &\mapsto \exp_{\gamma(x)} (s\varphi(x)). \end{aligned} $$

For any $s$, $c(s, \cdot)$ is a closed curve on $\mathcal{N}$ that is close to $\gamma$. By the construction of $c$ and the extension of exponential maps, we have

$$ c(s, x) = \gamma(x) + s\varphi(x) - \frac{s^2}{2}\Pi(\varphi(x), \varphi(x)) + O(s^3). $$

Define the energy for the curve $c(s, \cdot)$ as

$$ E(c(s, \cdot)) := \frac{1}{2} \int_{\mathbb{T}^1} \left| \frac{\partial}{\partial x} c(s, x) \right|^2 dx $$

On the one hand, thanks to Synge's second variation formula for the energy (see [54, Theorem 6.1.4]),

$$ \frac{d^2}{ds^2} E(c(s, \cdot))|_{s=0} = \int_{\mathbb{T}^1} (\|\nabla_{\gamma_x} \varphi\|^2 - \langle R(\gamma_x, \varphi)\gamma_x, \varphi \rangle) dx. $$

On the other hand, direct calculation yields

$$ 2E(c(s, \cdot)) = \int_{\mathbb{T}^1} \left| \gamma_x + s\varphi_x - \frac{s^2}{2} \partial_x (\Pi(\varphi(x), \varphi(x))) + O(s^3) \right|^2 dx. $$

Recall that

$$ \gamma_{xx} = \nabla_{\gamma_x} \gamma_x + \Pi(\gamma_x, \gamma_x). $$

Thus

$$ \begin{aligned} \frac{d^2}{ds^2} E(c(s, \cdot))|_{s=0} &= \int_{\mathbb{T}^1} |\varphi_x|^2 dx - \int_{\mathbb{T}^1} \langle \partial_x (\Pi(\varphi(x), \varphi(x))), \gamma_x(x) \rangle dx, \\ &= \int_{\mathbb{T}^1} |\varphi_x|^2 dx + \int_{\mathbb{T}^1} \langle \Pi(\varphi(x), \varphi(x)), \partial_x(\gamma_x)(x) \rangle dx, \\ &= \int_{\mathbb{T}^1} |\varphi_x|^2 dx + \int_{\mathbb{T}^1} \langle \Pi(\varphi, \varphi), \Pi(\gamma_x, \gamma_x) \rangle dx, \end{aligned} $$

Thus, to show (A.1) it suffices to prove

$$ \begin{aligned} (A.2) \qquad & \int_{\mathbb{T}^1} \langle \varphi^r \partial_r S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma^k + 2S_{jk}(\gamma) \partial_x \gamma^j \partial_x \varphi^k, \varphi \rangle dx \\ &= \int_{\mathbb{T}^1} \langle \partial_x (\Pi(\varphi(x), \varphi(x))), \gamma_x(x) \rangle dx \\ &= - \int_{\mathbb{T}^1} \langle \Pi(\varphi, \varphi), \Pi(\gamma_x, \gamma_x) \rangle dx. \end{aligned} $$
The equality between the first and second terms is not straightforward to establish. Thus we turn to show the equivalence between the first and third. Consider an auxiliary function

$$I(s) := \int_{\mathbb{T}^1} \left\langle -\Pi(c(s, x)) \left( \frac{\partial c}{\partial x}(s, x), \frac{\partial c}{\partial x}(s, x) \right), \frac{\partial c}{\partial s}(s, x) \right\rangle dx.$$

Note that the above equation is well-defined, since for every given $(s, x)$, the vectors $\frac{\partial c}{\partial x}(s, x)$ and $\frac{\partial c}{\partial s}(s, x)$ belong to $T_{c(s,x)}\mathcal{N}$. Recall the extrinsic formula of the second fundamental form, equation (2.2). Thus

$$I(s) := \int_{\mathbb{T}^1} \left\langle S_{jk}(c(s,x)) \left(\frac{\partial c}{\partial x}\right)^j(s,x) \left(\frac{\partial c}{\partial x}\right)^k(s,x), \frac{\partial c}{\partial s}(s,x) \right\rangle dx.$$

Note that the second derivatives such as $\frac{\partial^2 c}{\partial s \partial x}$ shall be understood in $\mathbb{R}^N$. Direct calculation yields

$$\begin{aligned} \frac{d}{ds} I(0) &= \int_{\mathbb{T}^1} \langle \varphi^r \partial_r S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma^k + 2S_{jk}(\gamma) \partial_x \gamma^j \partial_x \varphi^k, \varphi \rangle dx \\ &\quad + \int_{\mathbb{T}^1} \langle -\Pi(\gamma)(\gamma_x, \gamma_x), -\Pi(\gamma(x))(\varphi(x), \varphi(x)) \rangle dx, \\ &= \int_{\mathbb{T}^1} \langle \varphi^r \partial_r S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma^k + 2S_{jk}(\gamma) \partial_x \gamma^j \partial_x \varphi^k, \varphi \rangle dx \\ &\quad + \int_{\mathbb{T}^1} \langle \Pi(\gamma)(\gamma_x, \gamma_x), \Pi(\gamma)(\varphi, \varphi) \rangle dx. \end{aligned}$$

Meanwhile, we also notice from the definition of $I(s)$ that for every $s$,

$$I(s) = \int_{\mathbb{T}^1} 0 \, dx = 0.$$

Hence, the equality (A.2) holds. This finishes the proof of the identity (A.1).

Next, we show that $\mathcal{L}_\gamma$ is self-adjoint. Given two tangent fields $\varphi, \psi$ along the geodesic $\gamma$. Similarly, we construct a map

$$\begin{aligned} c(\cdot, \cdot, \cdot) : (-\varepsilon, \varepsilon) \times (-\varepsilon, \varepsilon) \times \mathbb{T}^1 &\to \mathcal{N}, \\ (s, t, x) &\mapsto \exp_{\gamma(x)} (s\varphi(x) + t\psi(x)). \end{aligned}$$

For any $s, t \in (-\varepsilon, \varepsilon)$, $c(s, t, \cdot)$ is a closed curve on $\mathcal{N}$ that is close to $\gamma$. We have

$$c(s, x) = \gamma(x) + s\varphi(x) + t\psi(x) - \frac{1}{2}\Pi(s\varphi(x) + t\psi(x), s\varphi(x) + t\psi(x)) + O(|(s, t)|^3).$$

Consider an auxiliary function

$$II(s,t) := \int_{\mathbb{T}^1} \left\langle -\Pi(c(s,t,x)) \left( \frac{\partial c}{\partial x}(s,t,x), \frac{\partial c}{\partial x}(s,t,x) \right), \frac{\partial c}{\partial t}(s,t,x) \right\rangle dx = 0.$$

Then, direct calculation yields

$$\begin{aligned} \frac{\partial}{\partial s} II(0, 0) &= \int_{\mathbb{T}^1} \langle \varphi^r \partial_r S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma^k + 2S_{jk}(\gamma) \partial_x \gamma^j \partial_x \varphi^k, \psi \rangle dx \\ &\quad + \int_{\mathbb{T}^1} \langle \Pi(\gamma)(\gamma_x, \gamma_x), \Pi(\gamma)(\varphi, \psi) \rangle dx. \end{aligned}$$
By the symmetry of the second fundamental form, we obtain

$$
\begin{align*}
& \int_{\mathbb{T}^1} \langle \varphi^r \partial_r S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma^k + 2S_{jk}(\gamma) \partial_x \gamma^j \partial_x \varphi^k, \psi \rangle \,dx \\
&= -\int_{\mathbb{T}^1} \langle \Pi(\gamma)(\gamma_x, \gamma_x), \Pi(\gamma)(\varphi, \psi) \rangle \,dx \\
&= \int_{\mathbb{T}^1} \langle \psi^r \partial_r S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma^k + 2S_{jk}(\gamma) \partial_x \gamma^j \partial_x \psi^k, \varphi \rangle \,dx,
\end{align*}
$$

thus $\mathcal{L}_\gamma$ is self-adjoint. This, together with the identity (A.1) as well as the symmetry

$$
\langle R(\gamma_x, \varphi)\gamma_x, \psi \rangle = \langle R(\gamma_x, \psi)\gamma_x, \varphi \rangle
$$

yields

$$
\langle \mathcal{L}_{\gamma} \varphi, \psi \rangle_{L^2(\mathbb{T}^1)} = - \langle \nabla_{\gamma_x} \varphi, \nabla_{\gamma_x} \psi \rangle_{L^2(\mathbb{T}^1)} + \int_{\mathbb{T}^1} \langle R(\gamma_x, \varphi)\gamma_x, \psi \rangle \,dx = \langle \varphi, \mathcal{L}_{\gamma} \psi \rangle_{L^2(\mathbb{T}^1)}.
$$

Finally, we show that $\mathcal{L}_{\gamma}\gamma_x = 0$. Using the above identity for $\varphi = \gamma_x$ we obtain

$$
\langle \mathcal{L}_{\gamma} \gamma_x, \psi \rangle_{L^2(\mathbb{T}^1)} = 0.
$$

Recall that $\gamma$ satisfies the geodesic equation (2.6). Then, we have

$$
\Delta\gamma(x+s) + S_{jk}(\gamma(x+s))\partial_x\gamma^j(x+s)\partial_x\gamma^k(x+s) = 0 \quad \forall x \in \mathbb{T}^1 \quad \forall s \in (-\varepsilon, \varepsilon).
$$

Differentiating the preceding equation with respect to the variable $s$ at $s=0$, we obtain

$$
0 = \Delta\gamma_x + \gamma_x^r \partial_r S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma^k + 2S_{jk}(\gamma) \partial_x \gamma^j \partial_x \gamma_x^k = \mathcal{L}_\gamma \gamma_x \quad \forall x \in \mathbb{T}^1.
$$

This finishes the proof of Lemma 2.8.

**A.3.** *Proof of Lemma 5.1.* To simplify notations, we identify $\mathbb{T}^1$ with the interval $[0, 1]$ with endpoints identified and denote $\gamma_x$ by $\dot{\gamma}$. The proof is composed of two steps.

Step 1. *Constructing a frame via parallel transport.* Choose a base point $x_0 = 0 \in [0, 1]$. Select an orthonormal basis $\{\mathbf{h}_1, \mathbf{h}_2, \dots, \mathbf{h}_R\}$ of $T_{\gamma(0)}\mathcal{N}$ with the property that $e_1 = \dot{\gamma}(0)$. For each $x \in [0, 1]$, we denote the parallel transport along $\gamma$ (with respect to the Levi-Civita connection) as

$$
P_{0 \to x} : T_{\gamma(0)}\mathcal{N} \to T_{\gamma(x)}\mathcal{N}
$$

Define, for $p = 1, \dots, R$,

$$
\mathbf{f}_p(x) = P_{0 \to x}(\mathbf{h}_p).
$$

Because parallel transport is an isometry, for every $x$ the set $\{\mathbf{f}_1(x), \dots, \mathbf{f}_R(x)\}$ forms an orthonormal basis of $T_{\gamma(x)}\mathcal{N}$. Moreover, since $\gamma$ is a geodesic, its velocity is parallel along $\gamma$, $\nabla_{\dot{\gamma}}\dot{\gamma} = 0$. It follows that

$$
\mathbf{f}_1(x) = P_{0 \to x}(\mathbf{h}_1) = P_{0 \to x}(\dot{\gamma}(0)) = \dot{\gamma}(x) \quad \forall x \in [0, 1].
$$

Step 2. *Adjusting the frame via holonomy correction to achieve periodicity.* The construction above yields a smooth frame $\{\mathbf{f}_i(x)\}$ on $[0, 1]$. However, in general the holonomy

$$
P_{0 \to 1} : T_{\gamma(0)}\mathcal{N} \to T_{\gamma(1)}\mathcal{N} = T_{\gamma(0)}\mathcal{N}
$$

need not equal the identity. Since $P_{0 \to 1}$ is an orthogonal transformation (an element of $O(R)$), there exists a smooth path

$$
R : [0, 1] \to O(R)
$$
such that

$$R(0) = I \quad \text{and} \quad R(1) = P_{0 \to 1}^{-1}.$$

Moreover, because we want to preserve the fact that $f_1(x) = \dot{\gamma}(x)$, we may choose $R(x)$ such that

$$R(x)\mathbf{h}_1 = \mathbf{h}_1 \quad \forall x \in [0, 1].$$

Define the modified frame

$$\tilde{\mathbf{f}}_p(x) = R(x)\mathbf{f}_p(x) \quad \forall x \in [0, 1] \text{ and } p = 1, \dots, R.$$

Then $\tilde{\mathbf{f}}_i(x)$ is smooth on $[0, 1]$. At $x = 0$,

$$\tilde{\mathbf{f}}_p(0) = R(0)\mathbf{f}_p(0) = I \cdot \mathbf{h}_p = \mathbf{h}_p,$$

and at $x = 1$,

$$\tilde{\mathbf{f}}_p(1) = R(1)f_p(1) = P_{0 \to 1}^{-1}(P_{0 \to 1}(\mathbf{h}_p)) = \mathbf{h}_p.$$

Thus, $\tilde{\mathbf{f}}_p(0) = \tilde{\mathbf{f}}_p(1)$ for each $p$, and the modified frame is a smooth, periodic frame on $\mathbb{T}^1$.

Moreover, since $R(x)\mathbf{h}_1 = \mathbf{h}_1$ for all $x$, we have

$$\tilde{\mathbf{f}}_1(x) = R(x)\mathbf{f}_1(x) = R(x)P_{0 \to x}(\mathbf{h}_1) = P_{0 \to x}(\mathbf{h}_1) = \dot{\gamma}(x)$$

for all $x \in \mathbb{T}^1$. To simplify the notations we rename $\tilde{\mathbf{f}}_p$ as $\mathbf{f}_p$. This finishes the proof of this construction.

## APPENDIX B. MORE DETAILS ON THE PROPAGATION OF SMALLNESS

In this section we provide more details on the proofs of Propositions 3.1, 3.3.

**B.1. More details on the proof of Proposition 3.1.** This part is devoted to the proof of Lemma 3.5. First, we define the $i^{th}$-component of $\Pi_T(\phi)f$ as $g^i \forall i = 1, 2, \dots, N$. Recall that under the null coordinate equation (2.4) can be expressed as

$$-\phi_{uv}^i = S_{jk}^i(\phi)\phi_u^j\phi_v^k + \frac{1}{4}a\phi_t^i + g^i := F^i \quad \forall i = 1, \dots, N.$$

Therefore $\phi^i$ can be characterized as follows using direct integration

$$\phi^i(u, v) = 2 \int_0^t \int_{x-t+s}^{x+t-s} F^i(s, y)dyds + \int_{-v}^u \phi_u^i(u_0, -u_0)du_0 + \phi^i(-v, v),$$

where, $(u_0, v_0)$ is related to the $(u, v)$ coordinate, while $(s, y)$ is to be understood in the original $(t, x)$ coordinate.

*On the characterization of $\phi_t$.*

By taking the time derivative the function $\phi_t$ is governed by

$$-\phi_{t,uv}^i = (\nabla S_{jk}^i(\phi) \cdot \phi_t) \phi_u^j \phi_v^k + S_{jk}^i(\phi) (\phi_u^j \phi_{t,v}^k + \phi_v^j \phi_{t,u}^k) + \frac{1}{4}a\phi_{tt}^i + g_t^i := G^i,$$

for every $i = 1, \dots, N$. Taking advantage of the preceding formula we can express the value of $\phi_t$, basically due to the finite speed of propagation. This is linked to the proof of Proposition 2.2 in [35], more precisely, to the “propagation of the smallness” part as Section 2.4.2 therein. To characterize of propagation of the wave, recall the setting in [35, equations (2.30)–(2.31)], we work with the vertical trapezoidal regions for $\alpha + 2\pi < \beta$ and $l \le \pi$

$$P_{\alpha, \beta}^l(y) := \{(t, x) : x \in [y, y+l], t \in [\alpha + x - y, \beta - x + y]\},$$
and the $L_x^\infty L_t^2$-norm on it as

$$ \|\psi\|_{L_x^\infty L_t^2(P_{\alpha,\beta}^l(y))} := \sup_{x \in [y, y+l]} \|\psi(t, x)\|_{L_t^2(\alpha+x-y, \beta-x+y)}. $$

Thus, recalling the calculation in [35, Section 2.4.2], for every point $(u, v) \in P_{\alpha,\beta}^l(x_0)$

$$ \begin{aligned} -\phi_t^i(u, v) &= -\phi_t^i(2x_0 + v, v) - \int_{2x_0+v}^u \phi_{t,u}^i(u_0, u_0 - 2x_0) du_0 + \int_{2x_0+v}^u \int_{u_0-2x_0}^v G^i(u_0, v_0) dv_0 du_0, \\ &= -\frac{1}{2} (\phi_t^i(u - x_0, x_0) + \phi_t^i(v + x_0, x_0) + \phi_x^i(u - x_0, x_0) - \phi_x^i(v + x_0, x_0)) \\ &\quad + \int_{2x_0+v}^u \int_{u_0-2x_0}^v \left( (\nabla S_{jk}^i(\phi) \cdot \phi_t) \phi_u^j \phi_v^k + \frac{1}{4} a \phi_{tt}^i + g_t^i \right) (u_0, v_0) dv_0 du_0 \\ &\quad + \int_{2x_0+v}^u \int_{u_0-2x_0}^v S_{jk}^i(\phi) (\phi_u^j \phi_{t,v}^k + \phi_v^j \phi_{t,u}^k) (u_0, v_0) dv_0 du_0. \end{aligned} $$

Thanks to the integration by parts

$$ \begin{aligned} & \int_{2x_0+v}^u \int_{u_0-2x_0}^v (S_{jk}^i(\phi) \phi_u^j \phi_{t,v}^k) (u_0, v_0) \, dv_0 du_0 \\ &= - \int_{2x_0+v}^u \int_{u_0-2x_0}^v (\nabla S_{jk}^i(\phi) \cdot \phi_v) \phi_u^j \phi_t^k (u_0, v_0) \, dv_0 du_0 \\ & \quad - \int_{2x_0+v}^u \int_{u_0-2x_0}^v S_{jk}^i(\phi) \phi_{uv}^j \phi_t^k (u_0, v_0) \, dv_0 du_0 \\ & \quad + \int_{2x_0+v}^u S_{jk}^i(\phi) \phi_u^j \phi_t^k (u_0, v) - S_{jk}^i(\phi) \phi_u^j \phi_t^k (u_0, u_0 - 2x_0) \, du_0 \\ &= - \int_{2x_0+v}^u \int_{u_0-2x_0}^v (\nabla S_{jk}^i(\phi) \cdot \phi_v) \phi_u^j \phi_t^k (u_0, v_0) \, dv_0 du_0 \\ & \quad + \int_{2x_0+v}^u \int_{u_0-2x_0}^v S_{jk}^i(\phi) \phi_t^k \left( S_{lm}^j(\phi) \phi_u^l \phi_v^m + \frac{1}{4} a \phi_t^j + g^j \right) (u_0, v_0) \, dv_0 du_0 \\ & \quad + \int_{2x_0+v}^u S_{jk}^i(\phi) \phi_u^j \phi_t^k (u_0, v) - S_{jk}^i(\phi) \phi_u^j \phi_t^k (u_0, u_0 - 2x_0) \, du_0 \end{aligned} $$

and

$$ \begin{aligned} & \int_{2x_0+v}^u \int_{u_0-2x_0}^v (S_{jk}^i(\phi) \phi_v^j \phi_{t,u}^k) (u_0, v_0) \, dv_0 du_0 \\ &= - \int_v^{u-2x_0} \int_{v_0+2x_0}^u (S_{jk}^i(\phi) \phi_v^j \phi_{t,u}^k) (u_0, v_0) \, du_0 dv_0 \\ &= \int_v^{u-2x_0} \int_{v_0+2x_0}^u (\nabla S_{jk}^i(\phi) \cdot \phi_u) \phi_v^j \phi_t^k (u_0, v_0) \, du_0 dv_0 \\ & \quad + \int_v^{u-2x_0} \int_{v_0+2x_0}^u S_{jk}^i(\phi) \phi_{uv}^j \phi_t^k (u_0, v_0) \, du_0 dv_0 \\ & \quad + \int_v^{u-2x_0} S_{jk}^i(\phi) \phi_v^j \phi_t^k (v_0 + 2x_0, v_0) - S_{jk}^i(\phi) \phi_u^j \phi_t^k (u_0, v_0) \, dv_0 \\ &= \int_v^{u-2x_0} \int_{v_0+2x_0}^u (\nabla S_{jk}^i(\phi) \cdot \phi_u) \phi_v^j \phi_t^k (u_0, v_0) \, du_0 dv_0 \end{aligned} $$
$$
\begin{align*}
& - \int_v^{u-2x_0} \int_{v_0+2x_0}^u S_{jk}^i(\phi) \phi_t^k \left( S_{lm}^j(\phi) \phi_u^l \phi_v^m + \frac{1}{4} a \phi_t^j + g^j \right) (u_0, v_0) \, du_0 \, dv_0 \\
& \qquad + \int_v^{u-2x_0} S_{jk}^i(\phi) \phi_v^j \phi_t^k (v_0 + 2x_0, v_0) - S_{jk}^i(\phi) \phi_u^j \phi_t^k (u_0, v_0) \, dv_0 \\
= & - \int_{2x_0+v}^u \int_{u_0-2x_0}^v (\nabla S_{jk}^i(\phi) \cdot \phi_u) \phi_v^j \phi_t^k (u_0, v_0) \, dv_0 \, du_0 \\
& + \int_{2x_0+v}^u \int_{u_0-2x_0}^v S_{jk}^i(\phi) \phi_t^k \left( S_{lm}^j(\phi) \phi_u^l \phi_v^m + \frac{1}{4} a \phi_t^j + g^j \right) (u_0, v_0) \, dv_0 \, du_0 \\
& \qquad + \int_v^{u-2x_0} S_{jk}^i(\phi) \phi_v^j \phi_t^k (v_0 + 2x_0, v_0) - S_{jk}^i(\phi) \phi_u^j \phi_t^k (u_0, v_0) \, dv_0
\end{align*}
$$

These equations further deduce that for any $(t, x) \in P_{\alpha,\beta}^l(x_0)$ with $x = x_0 + d$,

$$
\begin{align*}
-\phi_t^i(t, x) &= -2 \underbrace{\int_{x_0}^x \int_{t-x+y}^{t+x-y} W^i(s, y) \, ds \, dy}_{=:H_1^i(t,x)} \\
&\quad -\underbrace{\frac{1}{2} \left( \phi_t^i(u-x_0, x_0) + \phi_t^i(v+x_0, x_0) \right)}_{=:H_2^i(t,x)} \underbrace{-\frac{1}{2} \int_{t-d}^{t+d} \phi_{tx}^i(s, x_0) \, ds}_{=:H_3^i(t,x)} \\
&\quad -\underbrace{\frac{1}{2} \int_{x_0}^x a(y) \phi_t^i(t+x-y, y) + g^i(y) \, dy + \frac{1}{2} \int_{x_0}^x a(y) \phi_t^i(t-x+y, y) + g^i(y) \, dy}_{=:H_4^i(t,x)} \\
&\quad + \underbrace{\int_{2x_0+v}^u S_{jk}^i(\phi) \phi_u^j \phi_t^k(u_0, v) \, du_0 - \int_{2x_0+v}^u S_{jk}^i(\phi) \phi_u^j \phi_t^k(u_0, u_0-2x_0) \, du_0}_{=:H_5^i(t,x)} \\
&\quad + \underbrace{\int_v^{u-2x_0} S_{jk}^i(\phi) \phi_v^j \phi_t^k(v_0+2x_0, v_0) \, dv_0 - \int_v^{u-2x_0} S_{jk}^i(\phi) \phi_u^j \phi_t^k(u, v_0) \, dv_0}_{=:H_6^i(t,x)} \\
&=: \sum_{j=1}^6 H_j^i(t, x),
\end{align*}
$$

where $W^i$ is given by

$$
W^i := (\nabla S_{jk}^i(\phi) \cdot \phi_t) \phi_u^j \phi_v^k - (\nabla S_{jk}^i(\phi) \cdot \phi_v) \phi_u^j \phi_t^k - (\nabla S_{jk}^i(\phi) \cdot \phi_u) \phi_v^j \phi_t^k \\
\qquad + S_{jk}^i(\phi) \phi_t^k \left( 2S_{lm}^j(\phi) \phi_u^l \phi_v^m + \frac{1}{2} a \phi_t^j + 2g^j \right).
$$

On the estimates of $\|\phi_t\|_{L^\infty_x L^2_t(\alpha+d,\beta-d)}$.

To be compared with [35, Step 1 of the proof of Lemma 2.5], we notice that the estimates on $H_2^i, H_3^i, H_5^i, H_6^i$ remain the same: for $d \in (0, S)$, there are

$$
\begin{align*}
\| H_2^i(t, x) \|_{L_t^2(\alpha+d, \beta-d)} &\lesssim \| \phi_t(t, x_0) \|_{L_t^2(\alpha, \beta)}, \\
\| H_3^i(t, x) \|_{L_t^2(\alpha+d, \beta-d)} &\lesssim \| \langle \partial_t \rangle^{-1} (\eta_\alpha^\beta[\tau](t) \phi_{tx}(t, x_0)) \|_{L_t^2(\mathbb{R})}, \\
\| H_5^i(t, x) \|_{L_t^2(\alpha+d, \beta-d)} + \| H_6(t, x) \|_{L_t^2(\alpha+d, \beta-d)} &\lesssim \sqrt{d} \sqrt{E(0)} \| \phi_t \|_{L^\infty_x L_t^2(P)}.
\end{align*}
$$
As for $H_2^i$ and $H_4^i$, the contribution of the extra term $g^j$ is bounded by its $L_{t,x}^2$-norm which is further controlled by $\delta_2 E(\phi[0])$, otherwise the other estimates remain the same. More precisely,

$$ 
\begin{aligned}
& \int_{\alpha+d}^{\beta-d} \left( \int_{x_0}^{x} a(y) \phi_t^i(t+x-y, y) + g^i(y) dy \right)^2 dt \\
& \lesssim d \int_{\alpha+d}^{\beta-d} \int_{x_0}^{x} \left( (\phi_t^i)^2(t+x-y, y) + (g^i(y))^2 \right) dy dt \\
& \lesssim d^2 \|\phi_t\|_{L_x^\infty L_t^2(P)}^2 + d \|g^i\|_{L_{t,x}^2(P)}^2
\end{aligned}
$$

thus

$$ \|H_4(t, x)\|_{L_t^2(\alpha+d, \beta-d)} \lesssim d \|\phi_t\|_{L_x^\infty L_t^2(P)} + (d\delta_2 E(\phi[0]))^{1/2}. $$

We also know that the contribution of $g$ in $H_1^i$ is bounded by

$$ \int_{x_0}^{x} \int_{t-x+y}^{t+x-y} S_{jk}^{i}(\phi) \phi_{t}^{k} g^{j}(s, y) ds dy \lesssim \|\phi_{t}^{k}\|_{L_{t,x}^{2}} \|g\|_{L_{t,x}^{2}} \lesssim (\delta_{2})^{1/2} E(\phi[0]) $$

thus

$$ |H_1|(t, x) \lesssim (\sqrt{d}E(\phi[0]) + d\sqrt{E(\phi[0])}) \|\phi_t\|_{L_x^\infty L_t^2(P)} + (\delta_2)^{1/2} E(\phi[0]). $$

In conclusion, for $x = x_0 + d$ and for any $\tau \in (0, 1)$, we have

$$ \|\phi_t(t, x)\|_{L_t^2(\alpha+d, \beta-d)} \lesssim \|\phi_t(t, x_0)\|_{L_t^2(\alpha, \beta)} + \left\| \langle \partial_t \rangle^{-1} \left( \eta_{\alpha}^{\beta}[\tau](t) \phi_{tx}(t, x_0) \right) \right\|_{L_t^2(\mathbb{R})} \\ + \sqrt{d} \|\phi_t\|_{L_x^\infty L_t^2(P)} + (\delta_2 E(\phi[0]))^{1/2} $$

Hence, by choosing $d$ small enough, we obtain

$$ 
\begin{aligned}
& \|\phi_t(t, x)\|_{L_x^\infty L_t^2(P^{d, \beta}(x_0))} \\
& \lesssim \|\phi_t(t, x_0)\|_{L_t^2(\alpha, \beta)} + \left\| \langle \partial_t \rangle^{-1} \left( \eta_{\alpha}^{\beta}[\tau](t) \phi_{tx}(t, x_0) \right) \right\|_{L_t^2(\mathbb{R})} + (\delta_2 E(\phi[0]))^{1/2}.
\end{aligned}
$$

This yields the first estimate in Lemma 3.5.

*On the choice of $z$ such that $\phi_{tx}(t, z)$ is small.*

This part is the same as [35, Step 2 of the proof of Lemma 2.5]. We can find some point $\bar{z} \in [x_0 + \frac{5S_0}{8}, x_0 + \frac{7S_0}{8}]$ such that

$$ 
\begin{aligned}
& \int_{\mathbb{R}} \left( \langle \partial_t \rangle^{-1} \left( \eta_{\alpha+S_0+\tau_0}^{\beta-S_0-\tau_0}[\tau_0](t) \phi_{tx}(t, \bar{z}) \right) \right)^2 dt \\
& \lesssim \sqrt{E(\phi[0])} \|\phi_t(t, x)\|_{L_x^\infty(x_0+S_0/2, x_0+S_0; L_t^2(\alpha+S_0, \beta-S_0))}.
\end{aligned}
$$

Therefore, we finish the proof of Lemma 3.5.

**B.2. More details on the proof of Proposition 3.3.** Similar to [35, Proposition 2.2], it is a direct consequence of the following two auxilary properties: Lemma B.1 and Lemma B.2. Since these two lemmas are analog to [35, Lemma 2.4, Lemma 2.5] concerning $\phi_t$, we omit the detailed proofs. Actually, the analysis here is even simpler: in [35] the authors have studied a nonlinear equation on $\phi_t$:

$$ \square \phi_t = (|\phi_t|^2 - |\phi_x|^2)\phi_t + 2(\phi_t \cdot \phi_{tt} - \phi_x \cdot \phi_{tx})\phi + a\phi_{tt}, $$
while, in the current setting, we deal with a linear equation (3.2) concerning the function $\varphi$:

$$ \square \varphi + \mathbf{A} \partial_x \varphi + \mathbf{C} \partial_t \varphi + \mathbf{B} \varphi = 0. $$

Recall the truncated function $\mu$, the cutoff function $\eta_{\alpha}^{\beta}[\tau]$, the domain $P_{\alpha,\beta}^{l}(y)$, and the norm $\| \cdot \|_{L_x^{\infty} L_t^2 (P_{\alpha,\beta}^{l}(y))}$ defined in (3.3)-(3.5).

**Lemma B.1.** There exists some effectively computable $C_0 > 0$ such that, for any $\varepsilon \in (0, 1)$, for any $\tau \in (0, 1)$, if there is some solution $\varphi$ of (3.2)

$$ \int_{-16\pi}^{16\pi} \int_{\mathbb{T}^1} \chi_{\omega} |\varphi|^2(t, x) dxdt \le \varepsilon \|\varphi[0]\|_{L^2 \times H^{-1}}^2, $$

then, there exists some $x_0 \in [0, 2\pi)$ such that

$$ \|\varphi(t, x_0)\|_{L_t^2(-16\pi, 16\pi)}^2 + \|\langle\partial_t\rangle^{-1} (\eta_{-15\pi}[\tau]\varphi_x)(t, x_0)\|_{L_t^2(\mathbb{R})}^2 \le C_0 \frac{\sqrt{\varepsilon}}{\tau^2} \|\varphi[0]\|_{L^2 \times H^{-1}}^2. $$

This lemma is similar to [35, Lemma 2.4], where the authors obtained estimates on $\|\phi_t(t, x_0)\|_{L_t^2(-16\pi, 16\pi)}^2$ and $\|\langle\partial_t\rangle^{-1} (\eta_{-15\pi}[\tau]\phi_{tx})(t, x_0)\|_{L_t^2(\mathbb{R})}^2$.

**Lemma B.2.** There exist some effectively computable values $S_0$ and $C_{S_0}$ such that, for any $\tau \in (0, 1)$, for any $\alpha \in [-15\pi, 0)$, $\beta \in (2\pi, 15\pi]$, for any $z \in [0, 4\pi]$, one has

$$ \|\varphi\|_{L_x^\infty L_t^2(P_{\alpha,\beta}^{S_0}(z))} \le 2\|\varphi(t,z)\|_{L_t^2(\alpha,\beta)} + 6 \left\| \langle\partial_t\rangle^{-1} \left(\eta_{\alpha}^{\beta}[\tau](t)\varphi_x(t,z)\right) \right\|_{L_t^2(\mathbb{R})}, $$

and, moreover, by denoting $\tau_0 := S_0/16$, there exists some $\bar{x} \in (z + S_0/2, z + S_0)$ such that

$$ \left\| \langle \partial_t \rangle^{-1} \left( \eta_{\alpha+S_0+\tau_0}^{\beta-S_0-\tau_0}[\tau_0](t) \varphi_x(t, \bar{x}) \right) \right\|_{L_t^2(\mathbb{R})} $$

$$ \le C_{S_0} \left( \|\varphi[0]\|_{L^2 \times H^{-1}} \right)^{\frac{1}{2}} \left( \|\varphi(t, z)\|_{L_t^2(\alpha, \beta)} + \left\| \langle \partial_t \rangle^{-1} \left( \eta_{\alpha}^{\beta}[\tau_0](t) \varphi_x(t, z) \right) \right\|_{L_t^2(\mathbb{R})} \right)^{\frac{1}{2}}. $$

This lemma is similar to [35, Lemma 2.5], where the authors obtained estimates on $\|\phi_t\|_{L_x^\infty L_t^2(P_{\alpha,\beta}^{S_0}(z))}$ and $\|\langle\partial_t\rangle^{-1} (\eta_{\alpha+S_0+\tau_0}^{\beta-S_0-\tau_0}[\tau_0](t)\phi_{tx}(t, \bar{x}))\|_{L_t^2(\mathbb{R})}^2$.

By combining the above two properties, we obtain Proposition 3.3.

## APPENDIX C. MORE DETAILS ON THE WELL-POSEDNESS

This section is devoted to the proofs of some lemmas related to well-posedness issues, including Lemma 5.5, and Lemma 5.9.

**C.1. The proof of Lemma 5.5.** First we prove the basic properties of the nonlinear terms $K(x; \mathbf{w})$ and $K_1(x; \mathbf{w})\alpha$, namely inequalities (5.22)-(5.25),

$$ \begin{aligned} \| \sum_{j,k} A_{jk}^p(x; \mathbf{w}) \partial_\beta w^j \partial^\beta w^k \|_{L_{x,t}^2(D_T)} & \lesssim \sum_{j,k} \| \sum_\beta A_{jk}^i(x; \mathbf{w}) \partial_\beta w^j \partial^\beta w^k \|_{L_{u,v}^2(D_T)} \\ & \lesssim \sum_{j,k} \| A_{jk}^p(x; \mathbf{w}) (w_u^j w_v^k + w_v^j w_u^k) \|_{L_{u,v}^2(D_T)} \\ & \lesssim \sum_{j,k} \| A_{jk}^p(x; \mathbf{w}) \|_{L^\infty(D_T)} \| w_u^j \|_{L_v^\infty L_u^2(D_T)} \| w_v^k \|_{L_v^2 L_u^\infty(D_T)} \\ & \quad + \sum_{j,k} \| A_{jk}^p(x; \mathbf{w}) \|_{L^\infty(D_T)} \| w_v^j \|_{L_u^\infty L_v^2(D_T)} \| w_u^k \|_{L_u^2 L_v^\infty(D_T)} \end{aligned} $$
$$ \lesssim \|\mathbf{w}\|_{W_T}^2, $$

where we have used the identity (2.10), and then successively obtain

$$ \| \sum_j B_{r,j}^p(x; \mathbf{w}) \partial_x w^j \|_{L_{x,t}^2(D_T)} \lesssim \sum_j \| \mathbf{w} \|_{L^\infty(D_T)} \| \partial_x w^j \|_{L_{x,t}^2(D_T)} \lesssim \| \mathbf{w} \|_{W_T}^2, $$

$$ \| \sum_j C_{r,j}^p(x; \mathbf{w}) w^j \|_{L_{x,t}^2(D_T)} \lesssim \sum_j \| \mathbf{w} \|_{L^\infty(D_T)} \| w^j \|_{L_{x,t}^2(D_T)} \lesssim \| \mathbf{w} \|_{W_T}^2, $$

$$ \| \sum_j D_j^p(x; \mathbf{w}) \alpha^j(t, x) \|_{L_{x,t}^2(D_T)} \lesssim \sum_j \| \mathbf{w} \|_{L^\infty(D_T)} \| \alpha^j \|_{L_{x,t}^2(D_T)} \lesssim \| \mathbf{w} \|_{L^\infty(D_T)} \| \alpha \|_{L_{x,t}^2(D_T)} $$

for every $p, j, k = 1, 2, \dots, R$. Similarly,

$$ \| \sum_{\beta} A_{jk}^{p}(x; \mathbf{w}_1) \partial_{\beta} w_1^j \partial^{\beta} w_1^k - \sum_{\beta} A_{jk}^{p}(x; \mathbf{w}_2) \partial_{\beta} w_2^j \partial^{\beta} w_2^k \|_{L_{x,t}^2(D_T)} $$

$$ \lesssim \| A_{jk}^p(x; \mathbf{w}_1) w_{1u}^j w_{1v}^k + A_{jk}^p(x; \mathbf{w}_1) w_{1v}^j w_{1u}^k - A_{jk}^p(x; \mathbf{w}_2) w_{2u}^j w_{2v}^k - A_{jk}^p(x; \mathbf{w}_2) w_{2v}^j w_{2u}^k \|_{L_{u,v}^2(D_T)} $$

$$ \lesssim \| A_{jk}^p(x; \mathbf{w}_1) w_{1u}^j w_{1v}^k - A_{jk}^p(x; \mathbf{w}_2) w_{2u}^j w_{2v}^k \|_{L_{u,v}^2(D_T)} $$

$$ + \| A_{jk}^p(x; \mathbf{w}_1) w_{1v}^j w_{1u}^k - A_{jk}^p(x; \mathbf{w}_2) w_{2v}^j w_{2u}^k \|_{L_{u,v}^2(D_T)} $$

$$ \lesssim \| \mathbf{w}_1 - \mathbf{w}_2 \|_{L^\infty(D_T)} \| \mathbf{w}_1 \|_{W_T}^2 + \| \mathbf{w}_1 - \mathbf{w}_2 \|_{W_T} ( \| \mathbf{w}_1 \|_{W_T} + \| \mathbf{w}_2 \|_{W_T} ) $$

$$ \lesssim \| \mathbf{w}_1 - \mathbf{w}_2 \|_{W_T} ( \| \mathbf{w}_1 \|_{W_T} + \| \mathbf{w}_2 \|_{W_T} ). $$

as well as

$$ \| B_{r,j}^p(x; \mathbf{w}_1) \partial_x w_1^j - B_{r,j}^p(x; \mathbf{w}_2) \partial_x w_2^j \|_{L_{x,t}^2(D_T)} $$

$$ \lesssim \| B_{r,j}^p(x; \mathbf{w}_1) \partial_x w_1^j - B_{r,j}^p(x; \mathbf{w}_1) \partial_x w_2^j \|_{L_{x,t}^2(D_T)} $$

$$ + \| B_{r,j}^p(x; \mathbf{w}_1) \partial_x w_2^j - B_{r,j}^p(x; \mathbf{w}_2) \partial_x w_2^j \|_{L_{x,t}^2(D_T)} $$

$$ \lesssim \| \mathbf{w}_1 - \mathbf{w}_2 \|_{W_T} ( \| \mathbf{w}_1 \|_{W_T} + \| \mathbf{w}_2 \|_{W_T} ), $$

$$ \| C_{r,j}^p(x; \mathbf{w}_1) w_1^j - C_{r,j}^p(x; \mathbf{w}_2) w_2^j \|_{L_{x,t}^2(D_T)} $$

$$ \lesssim \| \mathbf{w}_1 - \mathbf{w}_2 \|_{W_T} ( \| \mathbf{w}_1 \|_{W_T} + \| \mathbf{w}_2 \|_{W_T} ), $$

and finally,

$$ \| \mathcal{K}_1(x; \mathbf{w}_1) \alpha_1 - \mathcal{K}_1(x; \mathbf{w}_2) \alpha_2 \|_{L_{t,x}^2(D_T)} $$

$$ \leq \| \mathcal{K}_1(x; \mathbf{w}_1) \alpha_1 - \mathcal{K}_1(x; \mathbf{w}_2) \alpha_1 \|_{L_{t,x}^2(D_T)} + \| \mathcal{K}_1(x; \mathbf{w}_2) \alpha_1 - \mathcal{K}_1(x; \mathbf{w}_2) \alpha_2 \|_{L_{t,x}^2(D_T)} $$

$$ \lesssim \| \mathbf{w}_1 - \mathbf{w}_2 \|_{L^\infty(D_T)} \| \alpha_1 \|_{L_{t,x}^2(D_T)} + \| \mathbf{w}_2 \|_{L^\infty(D_T)} \| \alpha_1 - \alpha_2 \|_{L_{t,x}^2(D_T)} $$

Now we come back to the proof of Lemma 5.5. Suppose the unique solution exists, then by (5.20)–(5.21),

$$ \| \mathbf{w} \|_{W_T} \leq C_T \left( \| (\mathbf{w}_0, \mathbf{w}_{0t}) \|_{\mathcal{H}} + \| \mathcal{K}(x; \mathbf{w}) + \alpha + \mathcal{K}_1(x; \mathbf{w}) \alpha + e \|_{L_{t,x}^2(D_T)} \right). $$

Next, we a priori assume that the unique solution satisfies $\|\mathbf{w}\|_{W_T} \le 1$ and obtain

$$ \| \mathbf{w} \|_{W_T} \le C_T \left( \| (\mathbf{w}_0, \mathbf{w}_{0t}) \|_{\mathcal{H}} + \| \alpha + e \|_{L_{t,x}^2(D_T)} \right) $$

$$ + C_T \left( \| \mathcal{K}(x; \mathbf{w}) \|_{L_{t,x}^2(D_T)} + \| \mathcal{K}_1(x; \mathbf{w}) \alpha \|_{L_{t,x}^2(D_T)} \right) $$

$$ \le C_T \left( \| (\mathbf{w}_0, \mathbf{w}_{0t}) \|_{\mathcal{H}} + \| \alpha \|_{L_{t,x}^2(D_T)} + \| e \|_{L_{t,x}^2(D_T)} \right) $$
$$ + C C_T \left( \|\mathbf{w}\|_{W_T}^2 + \|\mathbf{w}\|_{W_T} \|\alpha\|_{L_{t,x}^2(D_T)} \right). $$

Next, we turn to the second part of this lemma. Indeed, the difference of the two solutions, $\mathbf{w} := \mathbf{w}_1 - \mathbf{w}_2$, satisfies the following linear equation:

$$ \left\{ \begin{array}{l} \mathcal{L}_{\text{in}} \mathbf{w} = \mathcal{K}(x; \mathbf{w}_1) - \mathcal{K}(x; \mathbf{w}_2) + \chi_{\omega}(\alpha_1 - \alpha_2 + \mathcal{K}_1(x; \mathbf{w}_1)\alpha_1 - \mathcal{K}_1(x; \mathbf{w}_2)\alpha_2) + e_1 - e_2, \\ \mathbf{w}[0] = \mathbf{u}_1[0] - \mathbf{u}_2[0]. \end{array} \right. $$

Thanks to the first part of this lemma, we know that $\|\mathbf{w}_1\|_{W_T}$ and $\|\mathbf{w}_2\|_{W_T}$ are small. Thus, due to the well-posedness of the linear equation (5.20)-(5.21), one has

$$ \begin{aligned} \|\mathbf{w}\|_{W_T} &\le C_T \left( \|\mathbf{u}_1[0] - \mathbf{u}_2[0]\|_{H^1 \times L^2(\mathbb{T}^1)} + \|\alpha_1 - \alpha_2 + e_1 - e_2\|_{L_{t,x}^2(D_T)} \right) \\ &\quad + C_T (\mathcal{N}(x; \mathbf{w}_1) - \mathcal{N}(x; \mathbf{w}_2) + \mathcal{N}_1(x; \mathbf{w}_1)\alpha_1 - \mathcal{N}_1(x; \mathbf{w}_2)\alpha_2) \\ &\le C_T \left( \|\mathbf{u}_1[0] - \mathbf{u}_2[0]\|_{H^1 \times L^2(\mathbb{T}^1)} + \|\alpha_1 - \alpha_2 + e_1 - e_2\|_{L_{t,x}^2(D_T)} \right) \\ &\quad + C C_T \|\mathbf{w}\|_{W_T} (\|\mathbf{w}_1\|_{W_T} + \|\mathbf{w}_2\|_{W_T}) \\ &\quad + C C_T \left( \|\mathbf{w}\|_{L^\infty(D_T)} \|\alpha_1\|_{L_{t,x}^2(D_T)} + \|\mathbf{w}_2\|_{L^\infty(D_T)} \|\alpha_1 - \alpha_2\|_{L_{t,x}^2(D_T)} \right). \end{aligned} $$

This concludes the proof of this lemma.

**C.2. The proof of Lemma 5.9.** This proof is directly inspired by [67, Lemma 3.4], where the author considered a single variable wave equation with potential terms.

Define a non-negative cutoff function $g \in C_c^\infty(-3\pi, 3\pi)$ such that $g(x) = 1$ in $[-2\pi, 2\pi]$. Using integration by parts we get

$$ \begin{aligned} & \int_{-3\pi}^{3\pi} g \partial_t^2 \varphi \cdot ((-\Delta + 1)^{-1} \varphi) \, dxdt \\ &= - \int_{-3\pi}^{3\pi} g \partial_t \varphi \cdot ((-\Delta + 1)^{-1} \partial_t \varphi) \, dxdt - \int_{-3\pi}^{3\pi} \dot{g} \partial_t \varphi \cdot ((-\Delta + 1)^{-1} \varphi) \, dxdt \\ &= - \int_{-3\pi}^{3\pi} g \|\varphi_t\|_{H^{-1}(\mathbb{T}^1)}^2 \, dt - \frac{1}{2} \int_{-3\pi}^{3\pi} \dot{g} \partial_t \left( ((-\Delta + 1)^{-\frac{1}{2}} \varphi)^2 \right) \, dxdt \\ &= - \int_{-3\pi}^{3\pi} g \|\varphi_t\|_{H^{-1}(\mathbb{T}^1)}^2 \, dt + \frac{1}{2} \int_{-3\pi}^{3\pi} \ddot{g} \left( ((-\Delta + 1)^{-\frac{1}{2}} \varphi)^2 \right) \, dxdt. \end{aligned} $$

By plugging equation (5.39) into the above identity we obtain

$$ \begin{aligned} & \int_{-2\pi}^{2\pi} \|\varphi_t\|_{H^{-1}(\mathbb{T}^1)}^2 \, dt \\ &\le \int_{-3\pi}^{3\pi} g \|\varphi_t\|_{H^{-1}(\mathbb{T}^1)}^2 \, dt \\ &= - \int_{-3\pi}^{3\pi} g \partial_t^2 \varphi \cdot ((-\Delta + 1)^{-1} \varphi) \, dxdt + \frac{1}{2} \int_{-3\pi}^{3\pi} \ddot{g} ((-\Delta + 1)^{-\frac{1}{2}} \varphi)^2 \, dxdt \\ &= - \int_{-3\pi}^{3\pi} g(\Delta \varphi + \mathbf{b}\partial_x\varphi + \mathbf{c}\varphi) \cdot ((-\Delta + 1)^{-1} \varphi) \, dxdt + \frac{1}{2} \int_{-3\pi}^{3\pi} \ddot{g} ((-\Delta + 1)^{-\frac{1}{2}} \varphi)^2 \, dxdt \\ &= - \int_{-3\pi}^{3\pi} g((\Delta - 1)\varphi + \mathbf{b}\partial_x\varphi + (\mathbf{c} + 1)\varphi) \cdot ((-\Delta + 1)^{-1} \varphi) \, dxdt + \frac{1}{2} \int_{-3\pi}^{3\pi} \ddot{g} ((-\Delta + 1)^{-\frac{1}{2}} \varphi)^2 \, dxdt \end{aligned} $$
$$ \lesssim \int_{-3\pi}^{3\pi} \int_{\mathbb{T}^1} \varphi^2(t, x) dxdt. $$

Therefore, we finish the proof of Lemma 5.9.

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SORBONNE UNIVERSITÉ, UNIVERSITÉ PARIS-DIDEROT SPC, CNRS, INRIA, LABORATOIRE JACQUES-LOUIS LIONS, LJLL, ÉQUIPE CAGE, F-75005 PARIS, FRANCE

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