RETRACTED ARTICLE: Existence results for the general Schrödinger equations with a superlinear Neumann boundary value problem

Abstract

The main goal of this paper is to study the general Schrödinger equations with a superlinear Neumann boundary value problem in domains with conical points on the boundary of the bases. First the formulation and the complex form of the problem for the equations are given, and then the existence result of solutions for the above problem is proved by the complex analytic method and the fixed point index theory, where we absorb the advantages of the methods in recent works and give some improvement and development. Finally, we are also interested in the asymptotic behavior of solutions of the mentioned equation. These results generalize some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrödinger equations or of delay Schrödinger equations.

1 Introduction

This article deals with solutions of the general Schrödinger equation with a superlinear Neumann boundary value problem. To clarify our aim, we will introduce a class of Schrödinger equations (see [7, 22])

$$ \begin{aligned} &L_{\varepsilon}g= \operatorname{div} \bigl(\omega_{\varepsilon}(x) |\nabla g|^{\varrho(x)-2}\nabla g\bigr)=0 \quad \text{in } S, \\ &\frac{\partial g}{\partial n}=0 \quad \text{on } \partial{S}, \end{aligned} $$

(1)

where $\epsilon> 0$ is a small parameter, S is a compact metric space in $\mathbb{R}^{n} (n\ge2)$, $\omega_{\varepsilon}(x)$ is a positive weight. Assume that the domain is divided by the hyperplane $\varSigma=\{x:x_{n}=0\}$ into two parts $S^{(1)}=S\cap\{x:x_{n}>0\}$, $S^{(2)}=S\cap\{x:x_{n}<0\}$, and that

$$\begin{aligned}& \omega_{\varepsilon}(x)= \textstyle\begin{cases} \varepsilon,& \text{if }x\in S^{(1)}, \\ 1,&\text{if } x\in S^{(2)}, \end{cases}\displaystyle \quad \varepsilon \in(0,1], \\& \varrho(x)= \textstyle\begin{cases} q, &\text{if }x\in S^{(1)}, \\ \varrho, &\text{if }x\in S^{(2)}, \end{cases}\displaystyle \quad 1< q< \varrho. \end{aligned}$$

The general theory of PDEs like (1) with variable exponent has gained the interest of many mathematicians in recent years. We refer to the surveys [1, 8, 14, 15, 23, 27].

From a physical point of view, such Schrödinger equations with a superlinear Neumann boundary value problem have gained a lot of interest in recent years, in particular in the context of systems for the mean field dynamics of Bose–Einstein condensates [2, 5] and in applications to fields like nonlinear and fibers optics [25].

To define the solution of (1), we introduce a class of functions related to the exponent $\varrho(x)$ (see [30])

$$\biggl\{ g_{\varrho}:g_{\varrho}\in W^{1,1}_{\mathrm{loc}}(T), g_{\varrho}= \int _{T} g(x)\,d\varrho(x)\in L^{1}_{\mathrm{loc}}(T) \biggr\} . $$

This set is a Sobolev space of functions, locally summable on S together with their first order generalized derivatives. It follows that there exists a good approximation of $g_{\varrho}$ based on a set of independent and identically distributed random samples $\mathbf{w}=\{w_{i}\}_{i=1}^{m}=\{(s_{i}, t_{i})\}_{i=1}^{m} \in Z^{m}$ drawn according to the measure ϱ.

To the best of our knowledge, this notion of indirect observability was introduced for the first time in the context of coupled elliptic equations in [7], to obtain an exact indirect controllability result, in which one wants to drive back the fully coupled system to equilibrium by controlling only one component of the system. In 2017, Lai, Sun and Li (see [17]) used a two level energy method to estimate the solution of (1). In the case when $\omega_{\varepsilon}(x)$ and $\varrho(x)$ are fixed constants, there have been many results about the existence, uniqueness, blowing-up and so on; we refer to the bibliography (see [19, 29]). It follows that the hypothesis space is a Hilbert space $\mathfrak{H}_{E}$ induced by a Mercer kernel K which is a continuous, symmetric, and positive semi-definite function on $S\times S$ (see [24]). Space $\mathfrak {H}_{E}$ is the completion of the linear span of the set of functions $\{E_{s} :=E(s,\cdot) : s \in S \}$ with respect to the inner product

$$\Biggl\langle \sum_{i=1}^{n} \xi_{i} E_{s_{i}}, \sum_{l=1}^{m} \o_{j} E_{t_{j}} \Biggr\rangle _{E} := \sum _{i=1}^{n} \sum_{l=1}^{m} \xi_{i} \o_{j} E(s_{i} , t_{j}). $$

The reproducing property in $\mathfrak{H}_{E}$ is (see [3])

$$ g(s)=\langle g, E_{s}\rangle_{E}, $$

(2)

where $g\in \mathfrak{H}_{E}$ and $s\in S$.

Then by (2), we have (see [4])

$$ \|g\|_{\infty}\leq\kappa\|g\|_{E} $$

for any $g\in\mathfrak{H}_{E}$, where

$$\kappa:= \sup_{t, s\in S} \bigl\vert E(s,t) \bigr\vert < \infty. $$

It implies that $\mathfrak{H}_{E}\subseteq C(S)$.

We define the approximation $g_{\mathbf{w},\chi}$ of $g_{\varrho}$ by (see [16])

$$ \begin{aligned} &g_{\mathbf{w},\chi}(s)=g_{\mathbf{w},\zeta,\chi,s}(s)=g_{\mathbf{w},\zeta ,\chi,s}(u)|_{u=s}, \\ &g_{\mathbf{w},\zeta,\chi,s} :=\arg\min_{f\in\mathfrak{H}_{E}} \Biggl\{ \frac{1}{m}\sum_{i=1}^{m}\varPhi \biggl( \frac{s}{\zeta},\frac{s_{i}}{\zeta} \biggr) \bigl(t_{i}-g(s_{i}) \bigr)^{2}+\chi\|g\|_{E}^{2} \Biggr\} , \end{aligned} $$

(3)

where $\chi=\chi(m)>0$ is a regularization parameter, $\zeta =\varsigma(m)>0$ is a window width, and Φ is defined as follows (see [12]):

$$\begin{aligned} (1) &\quad \varPsi(s,t)\leq1, \quad \forall s,t \in\mathbb {R}^{n}, \\ (2) &\quad \varPsi(s,t)\geq c_{q}, \quad \forall|s-t|\leq1, \\ (3) &\quad \bigl\vert \varPsi(s,t_{1})-\varPsi(s,t_{2}) \bigr\vert \leq c_{\varPsi }|t_{1}-t_{2}|^{s}, \quad \forall s, t_{1}, t_{2} \in\mathbb{R}^{n}, \end{aligned}$$

where q, $c_{q}$ and $c_{\varPsi}>0$ are positive constants and $q>n+1$.

Scheme (3) shows that regularization not only ensures computational stability but also preserves localization property for the algorithm. In this paper, we further study the asymptotic behaviors of solutions of (1).

We adopt the coefficient-based regularization and the data-dependent hypothesis space (see [6, 20, 26])

$$ \begin{aligned} &g_{\mathbf{w},\varsigma}(s)=g_{\mathbf{w},\zeta,\varsigma,s}(s)=g_{\mathbf{w},\zeta,\varsigma,s}(u)|_{u=s}, \\ &g_{\mathbf{w},\zeta,\varsigma,s}=\arg\min_{f\in\mathfrak{H}_{E, \mathbf{w}} } \Biggl\{ \frac{1}{m}\sum_{i=1}^{m}\varPsi \biggl( \frac{s}{\zeta},\frac {s_{i}}{\zeta} \biggr) \bigl(g(s_{i})-t_{i} \bigr)^{2}+\varsigma\sum_{i=1}^{m}| \xi_{i}|^{q} \Biggr\} , \end{aligned} $$

(4)

where $1\leq q\leq2$, and

$$\begin{aligned}& \mathfrak{H}_{E, \mathbf{w}}= \Biggl\{ g(s)=\sum_{i=1}^{m} \xi_{i}E(s,s_{i}):\xi =(\xi_{1},\dots, \xi_{m})\in\mathbb{R}^{m},m\in \mathbb{N} \Biggr\} , \\& \varsigma=\varsigma(m)>0. \end{aligned}$$

Compared with scheme (3), the first advantage of (4) is the efficacy of computations without any optimization processes. Another advantage is that we can choose a suitable parameter q according to the research interest, e.g., smoothness and sparsity.

To study the approximation quality of $g_{\mathbf{w},\varsigma}$, we derive an upper bound of the error

$$\|g_{\mathbf{w},\varsigma}-g_{\varrho}\|_{\varrho_{S}} $$

with

$$\bigl\Vert g(\cdot) \bigr\Vert _{\varrho_{S}}:=\biggl( \int_{S} \bigl\vert g(\cdot) \bigr\vert ^{2}\, d{\varrho _{S}}\biggr)^{\frac{1}{2}} $$

and establish its convergence rate as $m \to\infty$ (see [10]).

The remainder of this paper is organized as follows. In Sect. 2, we will provide the main results. In Sect. 3, some basic, but important estimates and properties are summarized. The proofs of main results will be given in Sect. 4. Section 5 contains the conclusions of the paper.

2 Main results

We first formulate some basic notations and assumptions.

Let $\varrho_{S}$ be the marginal distribution of ϱ on S and $L_{\varrho_{S}} ^{2}(S)$ be the Hilbert space of functions from S to T, which are square-integrable with respect to $\varrho_{S}$ with the norm denoted by $\|\cdot\| _{\varrho_{S}}$. The integral operator $L_{E}:L_{\varrho_{S}} ^{2}(S)\rightarrow L_{\varrho_{S}} ^{2}(S)$ is defined by

$$(L_{E} g) (s) = \int_{S} E(s,t)g(t)\,d\varrho_{S}(t), $$

where $s\in S$.

Let $\{\mu_{i}\}$ be the eigenvalues of $L_{E}$ and $\{e_{i}\}$ be the corresponding eigenfunctions. Then for $g\in L_{\varrho_{S}} ^{2}(S)$,

$$L_{E}^{r}(g)=\sum_{i=1}^{\infty} \mu_{i}^{r}\langle g,e_{i}\rangle _{L_{\varrho_{S}} ^{2}}e_{i}; $$

see [9]. We assume that $g_{\varrho}$ satisfies the regularity condition $L_{E}^{-r}g_{\varrho}\in L^{2}_{\varrho_{S}}$, where $r>0$.

We show the following useful feature of the capacity of $\mathfrak {H}_{E, \mathbf{w}}$ when the $l^{2}$-empirical covering number is used (see [11]), namely

$$ \log\mathfrak {N}_{2}(B_{1},\epsilon)\leq c_{p}\epsilon^{-p}, $$

(5)

where $\epsilon>0$, $B_{1}= \{f\in\mathfrak{H}_{E, \mathbf{w}}: \| g\|_{E}\leq1 \}$, $0< p<2$ and $c_{p}>0$ (see [22]).

We use the projection operator to obtain a faster learning rate under the condition $|y|\leq M$ almost surely (see [18, 21]).

Definition 2.1

Let $A>0$. Then the projection operator $\gamma_{A}$ on the space of solutions $g:S\rightarrow\mathbb{R}$ is defined as

$$ \gamma_{A}(g) (s)= \textstyle\begin{cases} M, & \text{if } g(s)>M, \\ g(s), & \text{if } \vert g(s) \vert \leq M, \\ -M, & \text{if } g(s)< -M. \end{cases} $$

(6)

We assume all the constants are positive and independent of δ, m, χ, ς and ζ. Now we state our main results.

Theorem 1

Suppose $L_{E}^{-r}g_{\varrho}\in L^{2}_{\varrho_{S}}$ with $r>0$, and (5) holds with $0< p<2$ and $0<\delta<1$. Then we have

$$ \bigl\Vert \gamma_{A}(g_{\mathbf{w},\varsigma}) - g_{\varrho}\bigr\Vert _{\varrho_{S}}^{2} \leq \widetilde{D} \biggl(\frac{1}{m} \biggr)^{\tau(r)}\log \biggl(\frac {2}{\delta} \biggr), $$

(7)

where

$$\tau(r) = \textstyle\begin{cases}\min \{\frac{q}{[r(2p+2q+pq)+pq]}, 1 \} (\frac{2r}{1+\tau} ), & 0< r< \frac{1}{2}, \\ \frac{2q}{(2p+2q+3pq)(1+\tau)}, & r\geq1/2. \end{cases} $$

It follows that (see [13])

$$ \mathfrak{E}_{s}(g)= \int_{Z}\varPsi \biggl(\frac{s}{\zeta},\frac {u}{\zeta} \biggr) \bigl(g(u)-t\bigr)^{2}\,d\varrho(u,t), \quad \forall g: S \rightarrow\mathbb{R}, $$

is a solution of (1). In order to estimate $\|\gamma(g_{\mathbf{w},\varsigma})-g_{\varrho}\| ^{2}_{\varrho_{S}}$, we invoke the following proposition in [28].

Proposition 1

Let $f\in\mathfrak{H}_{E}\cup\{{g_{\varrho}}\}$ satisfy the Lipschitz condition on S, that is,

$$ \bigl\vert g(u)-g(v) \bigr\vert \leq c_{0}|u-v|, $$

(8)

where $u, v\in S$ and $c_{0}$ is a positive constant. Then

$$ \bigl\Vert \gamma(g_{\mathbf{w},\varsigma })-g_{\varrho} \bigr\Vert ^{2}_{\varrho_{S}}\leq\frac{\zeta^{-\tau }}{c_{q}c_{\tau}} \int_{S} \bigl\{ \mathfrak{E}_{s}\bigl( \gamma(g_{\mathbf{w},\zeta,\varsigma ,s})\bigr)-\mathfrak{E}_{s}(g_{\varrho}) \bigr\} \,d\varrho _{S}(s)+8c_{0}M\zeta. $$

(9)

Then we need an upper bound of the integral in (9). In order to get it, we only need to give its decomposition by using $g_{\mathbf{w},\chi}$ which provides a crucial connection between $g_{\mathbf{w},\varsigma}$ and the regularization function $g_{\chi}$, while different regularization parameters χ and ς are adopted.

Here $g_{\chi}$ is given by

$$ g_{\chi}:=\arg\min_{f\in\mathfrak{H}_{E}} \bigl\{ \| g-g_{\varrho} \|_{\varrho_{S}}^{2}+\chi\|g\|_{E}^{2} \bigr\} . $$

Define

$$\begin{aligned}& \begin{aligned} \mathfrak{S}(\mathbf{w},\chi,\varsigma)&= \int_{S} \bigl\{ \mathfrak {E}_{s}\bigl( \gamma_{A}(g_{\mathbf{w}, \zeta,\varsigma,s})\bigr)-\mathfrak {E}_{\mathbf{w},s}\bigl( \gamma_{A}(g_{\mathbf{w}, \zeta,\varsigma,s})\bigr) \\ &\quad {}+\mathfrak{E}_{\mathbf{w},s}(g_{\chi })-\mathfrak{E}_{s}(g_{\chi}) \bigr\} \,d\varrho_{S}(s), \end{aligned} \\& \begin{aligned} \mathfrak{H}(\mathbf{w},\chi,\varsigma)&= \int_{S} \bigl\{ \bigl(\mathfrak {E}_{\mathbf{w},s}\bigl( \gamma_{A}(g_{\mathbf{w}, \zeta,\varsigma ,s})\bigr)+\varsigma\varOmega_{\mathbf{w}}(g_{\mathbf{w}, \zeta,\varsigma ,s}) \bigr) \\ &\quad {}-\bigl(\mathfrak{E}_{\mathbf{w},s}(g_{\chi})+\chi \|g_{\chi}\|_{E}^{2}\bigr) \bigr\} \,d\varrho _{S}(s), \end{aligned} \\& \mathfrak{D}(\chi)=\|g_{\chi}-g_{\varrho}\|_{\varrho_{S}}^{2}+ \chi \|g_{\chi}\|_{E}^{2}. \end{aligned}$$

Remark

$\mathfrak{S}(\mathbf{w},\chi,\varsigma)$, $\mathfrak{H}(\mathbf{w},\chi ,\varsigma)$ and $\mathfrak{D}(\chi)$ are solutions of (1).

Theorem 2

Let $g_{\mathbf{w},\zeta,\varsigma,s}$ be defined as in (4) and let

$$ \mathfrak{E}_{\mathbf{w},s}(g)=\frac{1}{m}\sum _{i=1}^{m}\varPsi \biggl(\frac{s}{\zeta}, \frac{s_{i}}{\zeta} \biggr) \bigl(g(s_{i})-t_{i} \bigr)^{2} $$

(10)

be a solution of (1). Then we have

$$ \int_{S} \bigl\{ \mathfrak{E}_{s}\bigl( \gamma_{A}(g_{\mathbf{w},\zeta,\varsigma ,s})\bigr)-\mathfrak{E}_{s}(g_{\varrho}) \bigr\} \,d\varrho_{S}(s)\leq \mathfrak{S}(\mathbf{w},\chi,\varsigma)+ \mathfrak{H}(\mathbf{w},\chi ,\varsigma)+\mathfrak{D}(\chi). $$

(11)

3 Lemmas

Some basic, but important estimates and properties of solutions $\gamma_{A}(g)$ are summarized in the following lemma.

Lemma 1

Under the assumptions of Theorem 1, we have

$$ \int_{S} \gamma_{A}(g_{\varrho};t) \,dt \geq \frac{\tau}{2 (1 + \delta\tau )} \bigl( \gamma_{A}(g_{\varrho};0) - \widetilde{ \gamma}_{A}(g_{\chi};0) \bigr). $$

(12)

Proof

We will split the proof into four steps.

Step 1. Obtaining estimates of the terms:

$$\int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt,\qquad \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt,\qquad \widetilde{E}_{g}(g_{\chi}; \tau) + \widetilde{\gamma}_{A}(g_{\chi};0). $$

We take the sum of the inner products with $g_{\chi}(t)$ and $-g_{\varrho}(t)$, respectively, and obtain

$$\begin{aligned} &\bigl\langle g_{\varrho}''(t) - \partial_{g}^{2} g_{\varrho}(t) + \delta g_{\chi}(t), g_{\chi}(t)\bigr\rangle _{\mathbb{R}^{N},g} \\ &\quad {} - \bigl\langle g_{\chi}''(t) - \partial_{g}^{2} g_{\chi}(t) + \delta g_{\varrho}(t), g_{\varrho}(t)\bigr\rangle _{\mathbb{R}^{N},g} = 0 \end{aligned}$$

in $(\mathbb{R}^{N}, \Vert \cdot \Vert _{\mathbb{R}^{N}, g})$.

Hence, integrating the latter equation over $t \in(0,\tau)$, we have

$$\int_{S} \bigl( \bigl\langle g_{\varrho}''(t) , g_{\chi}(t) \bigr\rangle _{\mathbb {R}^{N},g} - \bigl\langle g_{\chi}''(t) , g_{\varrho}(t) \bigr\rangle _{\mathbb {R}^{N},g} + \delta \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} - \delta \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr) \,dt= 0, $$

and

$$\begin{aligned}& \int_{S} \bigl\langle g_{\varrho}''(t) , g_{\chi}(t) \bigr\rangle _{\mathbb {R}^{N},g} \,dt = \bigl[ \bigl\langle g_{\varrho}'(t) , g_{\chi}(t) \bigr\rangle _{\mathbb {R}^{N},g}\bigr]_{0}^{\tau}- \int_{S} \bigl\langle g_{\varrho}'(t) , g_{\chi}'(t) \bigr\rangle _{\mathbb {R}^{N},g} \,dt, \\& \int_{S} \bigl\langle g_{\chi}''(t) , g_{\varrho}(t) \bigr\rangle _{\mathbb {R}^{N},g} \,dt = \bigl[ \bigl\langle g_{\chi}'(t) , g_{\varrho}(t) \bigr\rangle _{\mathbb{R}^{N},g} \bigr]_{0}^{\tau}- \int_{S} \bigl\langle g_{\chi}'(t) , g_{\varrho}'(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt, \end{aligned}$$

which yields

$$ \delta \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt = \bigl[ X_{g}(t) \bigr]_{0}^{\tau}+ \delta \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt, $$

(13)

where

$$X_{g}(t) := \bigl\langle g_{\chi}'(t) , g_{\varrho}(t) \bigr\rangle _{\mathbb{R}^{N},g} - \bigl\langle g_{\varrho}'(t) , g_{\chi}(t) \bigr\rangle _{\mathbb{R}^{N},g}. $$

On the other hand,

$$\begin{aligned}& \begin{aligned} \bigl\vert \bigl\langle g_{\chi}'(t) , g_{\varrho}(t) \bigr\rangle _{\mathbb{R}^{N},g} \bigr\vert &= \bigl\vert \bigl\langle \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) , \bigl(-\partial_{g}^{2} \bigr)^{1/2}g_{\varrho}(t) \bigr\rangle _{\mathbb{R}^{N},g} \bigr\vert \\ &\leq \frac{\varepsilon_{1} \Vert (-\partial_{g}^{2} )^{-1/2} g_{\chi }'(t) \Vert _{\mathbb{R}^{N},g}^{2}}{2} + \frac{ \Vert (-\partial_{g}^{2} )^{1/2} g_{\varrho}(t) \Vert _{\mathbb{R}^{N},g}^{2}}{2 \varepsilon_{1}}, \end{aligned} \\& \bigl\vert \bigl\langle g_{\varrho}'(t) , g_{\chi}(t) \bigr\rangle _{\mathbb {R}^{N},g} \bigr\vert \leq \frac{\|g_{\varrho}'(t)\|_{\mathbb{R}^{N},g}^{2}}{2 \varepsilon_{1}} + \frac{\varepsilon_{1} \|g_{\chi}(t)\|_{\mathbb{R}^{N},g}^{2}}{2} \end{aligned}$$

for all $\varepsilon_{1} > 0$.

In view of the latter two inequalities, we have

$$ \bigl\vert \bigl[ X_{g}(t) \bigr]_{0}^{\tau}\bigr\vert \leq\frac{1}{\varepsilon_{1}} \bigl( \gamma_{A}(g_{\varrho}; \tau) + \gamma_{A}(g_{\varrho};0) \bigr) + \varepsilon_{1} \bigl( \widetilde{\gamma}_{A}(g_{\chi};\tau) + \widetilde{ \gamma}_{A}(g_{\chi};0) \bigr). $$

(14)

Using (13) and (14), we have

$$\begin{aligned} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \leq& \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{1}{ \varepsilon_{1} \delta} \bigl( \gamma_{A}(g_{\varrho};\tau) + \gamma_{A}(g_{\varrho};0) \bigr) \\ &{} + \frac{\varepsilon_{1}}{\delta} \bigl( \widetilde{ E}_{g}(g_{\chi }; \tau) + \widetilde{\gamma}_{A}(g_{\chi};0) \bigr) \end{aligned}$$

(15)

for each $\varepsilon_{1} >0$.

So

$$\int_{S} \bigl\langle g_{\chi}''(t) - \partial_{g}^{2} g_{\chi}(t) + \delta g_{\varrho}(t), \bigl(-\partial_{g}^{2} \bigr)^{-1}g_{\chi}(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt = 0, $$

which yields

$$\begin{aligned} & \int_{S} \bigl\langle \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}''(t), \bigl(- \partial_{g}^{2}\bigr)^{-1/2}g_{\chi}(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt \\ &\quad {}+ \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \delta \int_{S} \bigl\langle g_{\varrho}(t), \bigl(- \partial_{g}^{2}\bigr)^{-1}g_{\chi}(t)\bigr\rangle _{\mathbb{R}^{N},g} \,dt = 0. \end{aligned}$$

Integrating by parts, we have

$$\begin{aligned} & \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &\quad = \bigl[ Y_{g}(t) \bigr]_{0}^{\tau}+ \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \delta \int_{S} \bigl\langle g_{\varrho}(t), \bigl(- \partial_{g}^{2} \bigr)^{-1}g_{\chi}(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt, \end{aligned}$$

(16)

where

$$Y_{g}(t)= \bigl\langle \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t), \bigl(- \partial_{g}^{2} \bigr)^{-1/2}g_{\chi}(t) \bigr\rangle _{\mathbb{R}^{N},g}. $$

However, for this term we have

$$\begin{aligned} \bigl\vert \bigl[ Y_{g}(t) \bigr]_{0}^{\tau}\bigr\vert \leq& \bigl\vert \bigl\langle \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(\tau), \bigl(- \partial_{g}^{2} \bigr)^{-1/2}g_{\chi}(\tau) \bigr\rangle _{\mathbb{R}^{N},g} \bigr\vert \\ &{} + \bigl\vert \bigl\langle \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(0), \bigl(- \partial_{g}^{2} \bigr)^{-1/2}g_{\chi}(0) \bigr\rangle _{\mathbb{R}^{N},g} \bigr\vert \\ \leq&\frac{1}{2 \sqrt{\delta_{0}}} \bigl[ \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(\tau) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(0) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \bigr] \\ &{} +\frac{\sqrt{\delta_{0}}}{2} \bigl[ \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}(\tau) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert \bigl(-\partial_{g}^{2}\bigr)^{-1/2}g_{\chi}(0) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \bigr]. \end{aligned}$$

(17)

Moreover,

$$\bigl\Vert \bigl(-\partial_{g}^{2}\bigr)^{-1/2}g_{\chi}( \tau) \bigr\Vert _{\mathbb {R}^{N},g}^{2} + \bigl\Vert \bigl(- \partial_{g}^{2}\bigr)^{-1/2}g_{\chi}(0) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \leq\frac{1}{\delta_{0}} \bigl( \bigl\Vert g_{\chi}(\tau ) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{\chi}(0) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \bigr). $$

Inserting the latter inequality into (17), we have

$$ \bigl\vert \bigl[ Y_{g}(t)\bigr]_{0}^{\tau}\bigr\vert \leq\frac{1}{\sqrt{\delta_{0}}} \bigl( \widetilde{ E}_{g}(g_{\chi}; \tau) + \widetilde{\gamma}_{A}(g_{\chi};0) \bigr). $$

(18)

On the other hand,

$$\begin{aligned} & \biggl\vert \delta \int_{S} \bigl\langle g_{\varrho}(t), \bigl(- \partial_{g}^{2} \bigr)^{-1}g_{\chi}(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt \biggr\vert \\ &\quad \leq\frac{\delta}{2} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{\delta}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1}g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt . \end{aligned}$$

So

$$\begin{aligned} & \biggl\vert \delta \int_{S} \bigl\langle g_{\varrho}(t), \bigl(- \partial_{g}^{2} \bigr)^{-1}g_{\chi}(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt \biggr\vert \\ &\quad \leq\frac{\delta}{2} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{\delta}{2 \delta_{0}^{2}} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt. \end{aligned}$$

(19)

Using (16), (18), (19) and (15), we have

$$\begin{aligned} & \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &\quad \leq\frac{1}{\varepsilon_{1} \delta} \bigl( \gamma_{A}(g_{\varrho }; \tau) + \gamma_{A}(g_{\varrho};0) \bigr) + \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &\qquad {} + \biggl( \frac{1}{\sqrt{\delta_{0}}} + \frac{\varepsilon _{1}}{\delta} \biggr) \bigl( \widetilde{ E}_{g}(g_{\chi};\tau) + \widetilde{ \gamma}_{A}(g_{\chi };0) \bigr). \end{aligned}$$

(20)

Next, we estimate $\widetilde{ E}_{g}(g_{\chi};\tau) + \widetilde{\gamma}_{A}(g_{\chi};0)$. For this purpose, we take the inner product with $(-\partial_{g}^{2} )^{-1}g_{\chi}'(t)$ in the space $(\mathbb{R}^{N}, \Vert \cdot \Vert _{\mathbb{R}^{N}, g})$ to obtain

$$\frac{d}{dt}\widetilde{ E}_{g}(g_{\chi};t) = - \delta \bigl\langle \bigl(-\partial_{g}^{2}\bigr)^{-1/2}g_{\varrho}(t) , \bigl(-\partial_{g}^{2}\bigr)^{-1/2}g_{\chi }'(t) \bigr\rangle _{\mathbb{R}^{N},g}. $$

It follows that

$$\begin{aligned} &\widetilde{E}_{g}(g_{\chi};\tau) + \widetilde{ \gamma}_{A}(g_{\chi };0) \\ &\quad = 2 \widetilde{\gamma}_{A}(g_{\chi};0) - \delta \int_{S} \bigl\langle \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\varrho}(t) , \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt. \end{aligned}$$

We now estimate the second term of the right-hand side of the above equation as

$$\begin{aligned} & \biggl\vert \delta \int_{S} \bigl\langle \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\varrho }(t) , \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt \biggr\vert \\ &\quad \leq \frac{\delta}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\varrho}(t) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \,dt + \frac{\delta}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &\quad \leq \frac{\delta}{2 \delta_{0}} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{\delta}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \,dt. \end{aligned}$$

(21)

Moreover, by (20) and having in mind (21), we can write

$$\begin{aligned} & \biggl[ 1 - \frac{\delta}{2\sqrt{\delta_{0}}} - \frac{ \varepsilon _{1} }{2} \biggr] \bigl( \widetilde{ E}_{g}(g_{\chi};\tau) + \widetilde{ E}_{g}(g_{\chi };0) \bigr) \\ &\quad \leq2 \widetilde{ E}_{g}(g_{\chi};0) + \frac{(\delta_{0} +1) \delta}{2 \delta_{0}} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{1}{2\varepsilon_{1} } \bigl( \gamma_{A}(g_{\varrho};\tau) + \gamma_{A}(g_{\varrho};0) \bigr). \end{aligned}$$

So

$$\begin{aligned} &\biggl( 1 - \frac{\delta}{\sqrt{\delta_{0}}} \biggr) \bigl( \widetilde{ E}_{g}(g_{\chi }; \tau) + \widetilde{ E}_{g}(g_{\chi};0)\bigr) \\ &\quad \leq\frac{(\delta_{0} +1 ) \delta}{\delta_{0}} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + 4 \widetilde{ E}_{g}(g_{\chi};0) + \bigl( \gamma_{A}(g_{\varrho};\tau) + \gamma_{A}(g_{\varrho};0) \bigr), \end{aligned}$$

which implies that

$$\begin{aligned} &\widetilde{E}_{g}(g_{\chi};\tau) + \widetilde{ E}_{g}(g_{\chi};0) \\ &\quad \leq\frac{ \delta}{\sqrt{\delta_{0}} - \delta} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{4 \sqrt{\delta_{0}}}{\sqrt{\delta_{0}} - \delta} \widetilde{ E}_{g}(g_{\chi};0) \\ &\qquad {} + \frac{1}{\sqrt{\delta_{0}} - \delta} \bigl( \gamma _{A}(g_{\varrho}; \tau) + \gamma_{A}(g_{\varrho};0) \bigr). \end{aligned}$$

(22)

Step 2. Improving estimates (15) and (20).

Taking $\varepsilon_{1} = 1$ in (15) yields

$$\begin{aligned} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt & \leq \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{1}{\delta} \bigl( \gamma_{A}(g_{\varrho};\tau) + \gamma _{A}(g_{\varrho};0) \bigr) \\ &\quad {} + \frac{1}{\delta} \bigl( \widetilde{ E}_{g}(g_{\chi}; \tau) + \widetilde{\gamma}_{A}(g_{\chi};0) \bigr). \end{aligned}$$

Inserting (22) into the latter inequality, we have

$$\begin{aligned} & \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &\quad \leq\frac{C_{7}}{\delta(\sqrt{\delta_{0}} - \delta)} \bigl( \gamma_{A}(g_{\varrho}; \tau) + \gamma_{A}(g_{\varrho};0) \bigr) \\ &\qquad {} +\frac{1}{\delta(\sqrt{\delta_{0}} - \delta)} \widetilde{ E}_{g}(g_{\chi};0) + \frac{1}{\sqrt{\delta_{0}} - \delta} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt. \end{aligned}$$

(23)

On the other hand, equation (20) implies that

$$\begin{aligned} & \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &\quad \leq \biggl( \frac{1}{\sqrt{\delta_{0}}} + \frac{1}{\delta} \biggr) \bigl( \widetilde{ E}_{g}(g_{\chi};\tau) + \widetilde{ \gamma}_{A}(g_{\chi };0) \bigr) \\ &\qquad {} + \frac{C_{2}}{\delta} \bigl( \gamma_{A}(g_{\varrho}; \tau) + \gamma_{A}(g_{\varrho};0) \bigr) + \int_{S} \bigl\Vert g_{\varrho }(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \end{aligned}$$

and we have

$$\begin{aligned} & \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &\quad \leq\frac{1}{\delta(\sqrt{\delta_{0}}-\delta)}\bigl( \gamma _{A}(g_{\varrho}; \tau) + \gamma_{A}(g_{\varrho};0) \bigr) \\ &\qquad {} + \frac{1}{\delta(\sqrt{\delta_{0}}-\delta)} \widetilde{ E}_{g}(g_{\chi};0) + \frac{1}{\sqrt{\delta_{0}}-\delta} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \end{aligned}$$

(24)

from (22).

Step 3. Estimating $\gamma_{A}(g_{\varrho};\tau ) + \gamma_{A}(g_{\varrho};0)$.

We have

$$ \frac{d}{dt} \gamma_{A}(g_{\varrho};t)= - \delta\bigl\langle g_{\chi }(t) , g_{\varrho}'(t) \bigr\rangle _{\mathbb{R}^{N},g} $$

(25)

from (22), (23) and (24), which gives

$$\gamma_{A}(g_{\varrho};\tau) - \gamma_{A}(g_{\varrho};0) = - \delta \int_{S} \bigl\langle g_{\chi}(t) , g_{\varrho}'(t) \bigr\rangle _{\mathbb{R}^{N},g} \,dt. $$

It follows that

$$\begin{aligned} &\gamma_{A}(g_{\varrho};\tau) + \gamma_{A}(g_{\varrho};0) \\ &\quad \leq2\gamma_{A}(g_{\varrho};0) + \frac{\delta}{2 \varepsilon_{2}} \int_{S} \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{\delta\varepsilon_{2}}{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \end{aligned}$$

for each $\varepsilon_{2} > 0$, and we have

$$\begin{aligned} & \biggl[ 1 - \frac{ \varepsilon_{2} }{2 (\sqrt{\delta_{0}}-\delta)} \biggr] \bigl(\gamma_{A}(g_{\varrho}; \tau) + \gamma_{A}(g_{\varrho};0)\bigr) \\ &\quad \leq2\gamma_{A}(g_{\varrho};0) + \frac{\delta}{2 \varepsilon_{2}} \int_{S} \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{\varepsilon_{2} }{2 (\sqrt{\delta_{0}}-\delta)} \widetilde{ E}_{g}(g_{\chi};0) \\ &\qquad {} + \frac{\delta\varepsilon_{2}}{2 (\sqrt{\delta_{0}}-\delta)} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \end{aligned}$$

in view of (23).

Next we have

$$\begin{aligned} \gamma_{A}(g_{\varrho};\tau) + \gamma_{A}(g_{\varrho};0) \leq& \gamma_{A}(g_{\varrho};0) + \widetilde{ E}_{g}(g_{\chi};0) \\ &{} + \frac{\delta}{\sqrt{\delta_{0}}-\delta} \int_{S} \bigl( \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt. \end{aligned}$$

(26)

Inserting the latter inequality into equations (22)–(24), we obtain

$$\begin{aligned}& \begin{aligned}[b] \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt & \leq\frac{1}{\delta( \sqrt{\delta_{0}}-\delta)} \bigl( \gamma_{A}(g_{\varrho};0) + \widetilde{ E}_{g}(g_{\chi};0) \bigr) \\ &\quad {} + \frac{1}{( \sqrt{\delta_{0}}-\delta)^{2}} \int_{S} \bigl( \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt, \end{aligned} \end{aligned}$$

(27)

$$\begin{aligned}& \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{-1/2}g_{\chi}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\& \quad \leq\frac{1}{\delta( \sqrt{\delta_{0}}-\delta)} \bigl( \gamma _{A}(g_{\varrho};0) + \widetilde{ E}_{g}(g_{\chi};0) \bigr) \\& \qquad {} + \frac{1}{( \sqrt{\delta_{0}}-\delta)^{2}} \int_{S} \bigl( \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt, \end{aligned}$$

(28)

$$\begin{aligned}& \widetilde{ E}_{g}(g_{\chi};\tau) + \widetilde{ \gamma}_{A}(g_{\chi };0) \\& \quad \leq\frac{1}{\sqrt{\delta_{0}}-\delta} \bigl( \gamma_{A}(g_{\varrho};0) + \widetilde{ E}_{g}(g_{\chi};0) \bigr) \\& \qquad {} + \frac{\delta}{(\sqrt{\delta_{0}}-\delta)^{2}} \int_{S} \bigl( \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \bigr)\,dt. \end{aligned}$$

(29)

Step 4. Estimating $\int_{S} \gamma _{A}(g_{\varrho};t) \,dt$.

From (25), we have

$$\gamma_{A}(g_{\varrho};t) = \gamma_{A}(g_{\varrho};0) - \delta \int_{0}^{t} \bigl\langle g_{\chi}(s) , g_{\varrho}'(s) \bigr\rangle _{\mathbb{R}^{N},g} \,ds. $$

It follows that

$$ \gamma_{A}(g_{\varrho};t) \geq \gamma_{A}(g_{\varrho};0) - \frac {\delta}{ 2 \varepsilon_{3}} \int_{S} \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \,dt - \frac{\delta\varepsilon_{3}}{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt $$

(30)

for all $\varepsilon_{3}>0$.

Integrating the latter inequality between 0 and τ, we obtain

$$\begin{aligned} \int_{S} \gamma_{A}(g_{\varrho};t) \,dt \geq& \tau \gamma_{A}(g_{\varrho};0) - \frac{\delta\tau}{2 \varepsilon_{3}} \int_{S} \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &{} - \frac{\delta\varepsilon_{3} \tau}{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt, \end{aligned}$$

and having in mind equation (27), we can improve the last estimate as follows:

$$\begin{aligned} \int_{S} \gamma_{A}(g_{\varrho};t) \,dt \geq& \tau \biggl[ 1 - \frac{\varepsilon_{3}}{2 ( \sqrt{\delta _{0}}-\delta)} \biggr] \gamma_{A}(g_{\varrho};0) - \frac{\varepsilon_{3} \tau}{2 ( \sqrt {\delta_{0}}-\delta)} \\ &{} \times\widetilde{ E}_{g}(g_{\chi};0) - \frac{ \delta\varepsilon_{3} \tau}{( \sqrt{\delta_{0}}-\delta)^{2}} \int_{S} \bigl\Vert g_{\varrho}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ &{} - \frac{\delta\tau}{2} \biggl[ \frac{1}{\varepsilon_{3}} + \frac{ \varepsilon_{3}}{(\sqrt {\delta_{0}} -\delta)^{2}} \biggr] \int_{S} \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt. \end{aligned}$$

So

$$\begin{aligned} \int_{S} \gamma_{A}(g_{\varrho};t) \,dt \geq& \tau \biggl[ 1 - \frac{\varepsilon_{3} }{2( \sqrt{\delta _{0}}-\delta)} \biggr] \gamma_{A}(g_{\varrho};0) - \frac{\varepsilon_{3} \tau}{2 (\sqrt {\delta_{0}}-\delta)} \\ &{} \times\widetilde{ E}_{g}(g_{\chi};0) - \frac{ \delta \varepsilon_{3} \tau}{\delta_{0} (\sqrt{\delta _{0}}-\delta)^{2}} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{1/2}g_{\varrho}(t) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \\ &{} -\frac{\delta\tau}{2} \biggl[ \frac{1}{\varepsilon_{3}} + \frac{\varepsilon_{3}}{(\sqrt {\delta_{0}} -\delta)^{2}} \biggr] \int_{S} \bigl\Vert g_{\varrho}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt, \end{aligned}$$

which yields

$$\int_{S} \gamma_{A}(g_{\varrho};t) \,dt \geq \frac{\tau}{2} \bigl( \gamma_{A}(g_{\varrho};0) - \widetilde{ E}_{g}(g_{\chi};0) \bigr) - \frac{\delta\tau}{\sqrt{\delta_{0}}-\delta} \int_{S} \gamma_{A}(g_{\varrho};t) \,dt. $$

In other words,

$$\biggl[ 1 + \frac{\delta \tau}{\sqrt{\delta_{0}}-\delta} \biggr] \int_{S} \gamma_{A}(g_{\varrho};t) \,dt \geq \frac{\tau}{2} \bigl( \gamma_{A}(g_{\varrho};0) - \widetilde{ E}_{g}(g_{\chi};0) \bigr). $$

Since $\delta\leq\sqrt{\delta_{0}}/2$, it follows that (12) holds. This completes the proof. □

The following result provides a uniform observability inequality.

Lemma 2

$$ L \int_{S} \biggl\vert \frac{y_{N}(t)}{g} \biggr\vert ^{2} \,dt \leq C(\tau) \delta^{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt, $$

(31)

where $h>0$ and $\tau>0$.

Proof

We first have the discrete identity

$$ \frac{L}{2} \int_{S} \biggl\vert \frac{y_{N}(t)}{g} \biggr\vert ^{2} \,dt = A + \bigl[X_{g}(t)\bigr]_{0}^{\tau}- B, $$

(32)

by Lemma 1, where

$$\begin{aligned}& A= \frac{g}{2} \sum_{l=0}^{N} \int_{S} \biggl[ \biggl\vert \frac{y_{l+1}(t) - y_{l}(t)}{g} \biggr\vert ^{2} + y'_{l}(t) y'_{l+1}(t) \biggr] \,dt, \\& X_{g}(t) = h \sum_{l=1}^{N} j \biggl( \frac{y_{l+1}(t)-y_{j-1}(t)}{2} \biggr) y_{l}'(t), \\& B=\delta h \sum_{l=1}^{N} \int_{S} j \biggl( \frac{y_{l+1}(t)-y_{j-1}(t)}{2} \biggr) v_{l}(t) \,dt. \end{aligned}$$

We now estimate separately A, $X_{g}$ and B.

Estimate for A. We have

$$\begin{aligned} A =& \frac{1}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{1/2} \vec {y} {g}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{1}{2} \int_{S} \bigl\Vert \vec{y} {g}'(t) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \,dt \\ &{} - \frac{g}{2} \sum_{l=0}^{N} \int_{S} \bigl(y_{l}'y_{l+1}'- \bigl\vert y_{l}' \bigr\vert ^{2} \bigr) \,dt \\ =& \frac{1}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{1/2} \vec {y} {g}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{1}{2} \int_{S} \bigl\Vert \vec{y} {g}'(t) \bigr\Vert _{\mathbb {R}^{N},g}^{2} \,dt \\ &{} - \frac{g}{2} \sum_{l=0}^{N} \int_{S} \bigl\vert y_{l+1}' - y_{l}' \bigr\vert ^{2} \,dt \\ \leq& \frac{1}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{1/2} \vec {y} {g}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{1}{2} \int_{S} \bigl\Vert \vec{y} {g}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt \\ =& \int_{S} \gamma_{A}(\vec{y} {g};t) \,dt . \end{aligned}$$

(33)

Estimate for $X_{g}$. Notice that

$$\begin{aligned} X_{g}(t) &= h \sum_{l=1}^{N} j \biggl( \frac{y_{l+1}-y_{j}}{2} \biggr) y_{l}' + h \sum _{l=1}^{N} j \biggl( \frac{y_{j}-y_{j-1}}{2} \biggr) y_{l}' \\ &= h \sum_{l=0}^{N} (jh) \biggl( \frac{y_{l+1}-y_{j}}{2g} \biggr) y_{l}' + h \sum _{l=0}^{N} \bigl((j+1)h \bigr) \biggl( \frac{y_{l+1}-y_{j}}{2g} \biggr) y_{l+1}'. \end{aligned}$$

So

$$\begin{aligned} \bigl\vert X_{g}(t) \bigr\vert \leq& \frac{L}{2} h \sum _{l=0}^{N} \biggl\vert \frac{y_{l+1}-y_{j}}{g} \biggr\vert \bigl\vert y_{l}' \bigr\vert + \frac{L}{2} h \sum_{l=0}^{N} \biggl\vert \frac{y_{l+1}-y_{j}}{g} \biggr\vert \bigl\vert y_{l+1}' \bigr\vert \\ \leq& \frac{L}{4} h \sum_{l=0}^{N} \biggl\vert \frac{y_{l+1}-y_{j}}{g} \biggr\vert ^{2} + \frac{L}{4} h \sum_{l=0}^{N} \bigl\vert y_{l}' \bigr\vert ^{2} \\ &{} + \frac{L}{4} h \sum_{l=0}^{N} \biggl\vert \frac{y_{l+1}-y_{j}}{g} \biggr\vert ^{2} + \frac{L}{4} h \sum_{l=0}^{N} \bigl\vert y_{l+1}' \bigr\vert ^{2} \\ =& \frac{L}{2} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{1/2} \vec{y} {g}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} + \frac{L}{2} \bigl\Vert \vec{y} {g}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2}. \end{aligned}$$

(34)

Estimate for B. We have

$$\begin{aligned} B =& \delta h \sum_{l=1}^{N} \int_{S} j \biggl( \frac{y_{l+1}-y_{j}}{2} \biggr) v_{l} \,dt + \delta h \sum_{l=1}^{N} \int_{S} j \biggl( \frac{y_{j}-y_{j-1}}{2} \biggr) v_{l} \,dt \\ =& \delta h \sum_{l=1}^{N} \int_{S} j \biggl( \frac{y_{l+1}-y_{j}}{2} \biggr) v_{l} \,dt + \delta h \sum_{l=0}^{N} \int_{S} j \biggl( \frac{y_{l+1}-y_{j}}{2} \biggr) v_{l+1} \,dt \\ \leq& \frac{L }{4} h \sum_{l=0}^{N} \int_{S} \biggl\vert \frac {y_{l+1}-y_{j}}{g} \biggr\vert ^{2} \,dt + \frac{L\delta^{2}}{4} h \sum_{l=0}^{N} \int_{S} | v_{l} |^{2} \,dt \\ &{} + \frac{L}{4} h \sum_{l=0}^{N} \int_{S} \biggl\vert \frac {y_{l+1}-y_{j}}{g} \biggr\vert ^{2} \,dt + \frac{L\delta^{2}}{4} h \sum_{l=0}^{N} \int_{S} | v_{l+1} |^{2} \,dt \\ =& \frac{L}{2} \int_{S} \bigl\Vert \bigl(-\partial_{g}^{2} \bigr)^{1/2} \vec {y} {g}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt + \frac{L\delta^{2}}{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt. \end{aligned}$$

(35)

Next we obtain

$$\begin{aligned} \frac{L}{2} \int_{S} \biggl\vert \frac{y_{N}(t)}{g} \biggr\vert ^{2} \,dt \leq&(1+L) \int_{S} \gamma_{A}(\vec{y} {g};t) \,dt + L \bigl( \gamma _{A}(\vec{y} {g};\tau) + \gamma_{A}(\vec{y} {g};0) \bigr) \\ &{} + \frac{L \delta^{2}}{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt, \end{aligned}$$

(36)

due to (32) and (33)–(35).

Moreover,

$$ \gamma_{A}(\vec{y} {g};t) \leq\tau \delta^{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt . $$

(37)

In other words,

$$\begin{aligned} \frac{g}{2} \sum_{l=0}^{N} \biggl\vert \frac{y_{l+1} - y_{j}}{g} \biggr\vert ^{2} =& \frac{g}{2} \sum_{l=0}^{N} \Biggl\vert \sum _{k=1}^{N} \frac{\widehat{A}_{k}}{g}( \varphi_{k, j+1} - \varphi_{k, j}) \Biggr\vert ^{2} \\ =& \frac{g}{2} \sum_{l=0}^{N} \sum _{k=1}^{N} \widehat{A}_{k}^{2} \biggl\vert \frac{\varphi_{k, j+1} - \varphi_{k, j}}{g} \biggr\vert ^{2} \\ &{} + \frac{g}{2} \sum_{l=0}^{N} \sum_{\substack{k,k' =1 \\ k \neq k'}}^{N} \frac{\widehat{A}_{k} \widehat{A}_{k'} }{g}( \varphi_{k, j+1} - \varphi_{k, j}) ( \varphi_{k', j+1} - \varphi_{k', j}), \end{aligned}$$

where

$$\widehat{A}_{k} = \widehat{A}_{k}(t) = \frac{\delta}{\sqrt{\lambda_{k}(h)}} \int_{0}^{t} \sin\bigl( (t-s)\sqrt{ \lambda_{k}(h)} \bigr) \widehat{v}_{k}(s) \,ds. $$

So

$$\begin{aligned} \frac{g}{2} \sum_{l=0}^{N} \biggl\vert \frac{y_{l+1} - y_{j}}{g} \biggr\vert ^{2} =& \frac{g}{2} \sum_{k=1}^{N} \lambda_{k}(h) | \widehat{A}_{k} |^{2} \sum_{l=1}^{N} |\varphi_{k, j}|^{2} \\ =&\frac{h \delta^{2}}{2} \sum_{k=1}^{N} \biggl\vert \int_{0}^{t} \sin\bigl( (t-s)\sqrt{ \lambda_{k}(h)} \bigr) \widehat{v}_{k}(s) \,ds \biggr\vert ^{2} \sum_{l=1}^{N} | \varphi_{k, j}|^{2} \\ \leq& \frac{\tau \delta^{2}}{2} \int_{S} \sum_{k=1}^{N} \bigl\vert \widehat {v}_{k}(t) \bigr\vert ^{2} \,dt h \sum_{l=1}^{N} |\varphi_{k, j}|^{2} \\ =& \frac{\tau \delta^{2}}{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt. \end{aligned}$$

(38)

It follows that

$$ \frac{1}{2} \bigl\Vert \vec{y} {g}'(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} = \frac{g}{2} \sum _{k=1}^{N} \lambda_{k}(h) \bigl\vert \widehat{A}_{k}'(t) \bigr\vert ^{2} \sum _{l=1}^{N} |\varphi_{k, j}|^{2} \leq\frac{\tau \delta^{2}}{2} \int_{S} \bigl\Vert g_{\chi}(t) \bigr\Vert _{\mathbb{R}^{N},g}^{2} \,dt. $$

(39)

From (38)–(39) we deduce (37). Next, using (36) together with (37), we obtain the desired estimate (31). □

4 Proofs of main results

Now we derive the learning rates.

Proof of Theorem 1

Combining the three bounds of Step 1 in Lemma 1, we have

$$\begin{aligned} & \int_{S} \bigl\{ \mathfrak{E}_{s}\bigl( \gamma(g_{\mathbf{w},\zeta,\varsigma ,s})\bigr)-\mathfrak{E}_{s}(g_{\varrho}) \bigr\} \,d\varrho_{S}(s) \\ &\quad \leq D_{1}\log \biggl(\frac{2}{\delta} \biggr) \bigl\{ \chi^{\min\{ 2r,1\}}+ m^{-1}\chi^{\min\{2r-1,0\}} \\ &\qquad {} +m^{1-q}\varsigma\chi^{-q}+m^{\frac {-2q-2p+2pq}{(2+p)q}} \varsigma^{-\frac{2p}{q(2+p)}} \bigr\} . \end{aligned}$$

(40)

By substituting (40) into (9), we have

$$\begin{aligned} \bigl\Vert \gamma(g_{\mathbf{w},\varsigma})-g_{\varrho} \bigr\Vert ^{2}_{\varrho_{S}} \leq& D_{2}\log \biggl( \frac{2}{\delta} \biggr) \bigl\{ \zeta^{-\tau} \bigl\{ \chi^{\min\{2r,1\}}+ m^{-1}\chi^{\min \{2r-1,0\}} \\ &{} +m^{1-q}\varsigma\chi^{-q}+m^{\frac {-2q-2p+2pq}{(2+p)q}} \varsigma^{-\frac{2p}{q(2+p)}} \bigr\} +\zeta \bigr\} . \end{aligned}$$

When $0< r<1/2$,

$$\begin{aligned} \bigl\Vert \gamma(g_{\mathbf{w},\varsigma})-g_{\varrho} \bigr\Vert ^{2}_{\varrho_{S}} \leq& D_{2}\log \biggl( \frac{2}{\delta} \biggr) \bigl\{ \zeta^{-\tau} \bigl\{ \chi^{2r}+ m^{-1}\chi ^{2r-1}+m^{1-q}\varsigma\chi^{-q} \\ &{} +m^{\frac{-2q-2p+2pq}{(2+p)q}}\varsigma^{-\frac {2p}{q(2+p)}} \bigr\} +\zeta \bigr\} . \end{aligned}$$

Let $\chi=m^{-\tau_{1}}$, $\varsigma=m^{-\tau_{2}}$ and $\zeta =m^{-\tau_{3}}$. Then

$$ \bigl\Vert \gamma(g_{\mathbf{w},\varsigma})-g_{\varrho} \bigr\Vert ^{2}_{\varrho_{S}}\leq D_{3}\log \biggl( \frac{2}{\delta} \biggr) m^{-\tau}, $$

where

$$\begin{aligned} \tau =&\min \biggl\{ -\tau\tau_{3}+2r\tau_{1}, -\tau \tau_{3}+1+(2r-1)\tau_{1}, \\ &{} -\tau\tau_{3}+q-1+\tau_{2}-q\tau_{1}, \\ &{} -\tau\tau_{3}+\frac{2q+2p-2pq}{(2+p)q}-\frac {2p}{q(2+p)} \tau_{2}, \tau_{3} \biggr\} . \end{aligned}$$

To maximize the learning rate, we take

$$\begin{aligned} \tau_{\max} =&\max_{\tau_{1}, \tau_{3}}\min \biggl\{ \max _{\tau_{2}}\min \biggl\{ -\tau\tau_{3}+q-1+ \tau_{2}-q\tau _{1}, \\ &{} -\tau\tau_{3}+\frac{2q+2p-2pq}{(2+p)q}-\frac {2p}{q(2+p)} \tau_{2} \biggr\} , \\ &{} -\tau\tau_{3}+2r\tau_{1}, -\tau\tau _{3}+1+(2r-1)\tau_{1}, \tau_{3} \biggr\} . \end{aligned}$$

Let

$$ -\tau\tau_{3}+q-1+\tau_{2}-q\tau_{1}=-\tau \tau_{3}+\frac {2q+2p-2pq}{(2+p)q}-\frac{2p}{q(2+p)}\tau_{2}. $$

Then

$$\begin{aligned} \tau_{\max} =&\max_{\tau_{1}, \tau_{3}}\min \biggl\{ -\tau \tau_{3}+q-1-q\tau_{1} +\frac {-pq+4q+2p-2q^{2}-pq^{2}}{2p+2q+pq} \\ &{}+\frac{(2+p)q^{2}}{2p+2q+pq}\tau_{1},-\tau \tau_{3}+2r \tau_{1}, \\ &{}-\tau\tau_{3}+1+(2r-1)\tau _{1}, \tau_{3} \biggr\} \\ =&\max_{\tau_{3}}\min \biggl\{ \max_{\tau_{1}}\min \biggl\{ -\tau \tau_{3}+q-1-q\tau_{1} \\ &{}+\frac {-pq+4q+2p-2q^{2}-pq^{2}}{2p+2q+pq} \\ &{}+\frac{(2+p)q^{2}}{2p+2q+pq}\tau_{1},-\tau \tau_{3}+2r \tau_{1} \biggr\} , \\ &\max_{\tau_{1}}\min \bigl\{ -\tau\tau_{3}+1+(2r-1) \tau_{1},-\tau\tau_{3}+2r\tau_{1} \bigr\} , \tau_{3} \biggr\} . \end{aligned}$$

Let

$$\begin{aligned}& -\tau\tau_{3}+q-1-q\tau_{1}+\frac {-pq+4q+2p-2q^{2}-pq^{2}}{2p+2q+pq} \\& \quad {}+\frac{(2+p)q^{2}}{2p+2q+pq}\tau_{1}=-\tau\tau_{3}+2r\tau _{1}, \\& -\tau\tau_{3}+1+(2r-1)\tau_{1}=-\tau\tau_{3}+2r \tau _{1}. \end{aligned}$$

Then we have

$$\begin{aligned} \tau_{\max} =&\max_{\tau_{3}}\min \biggl\{ -\tau\tau _{3}+\frac{4qr}{2r(2p+2q+pq)+2pq},-\tau\tau_{3}+2r, \tau_{3} \biggr\} \\ =&\min \biggl\{ \max_{\tau_{3}}\min \biggl\{ -\tau \tau_{3}+\frac {4qr}{2r(2p+2q+pq)+2pq},\tau_{3} \biggr\} , \\ &\max_{\tau_{3}}\min \{-\tau\tau _{3}+2r, \tau_{3} \} \biggr\} \\ =&2r\min \biggl\{ -\frac{q\tau}{(1+\tau)[r(2p+2q+pq)+pq]} \\ &{}+\frac{q}{r(2p+2q+pq)+pq}, \frac {-\tau}{1+\tau}+1 \biggr\} . \end{aligned}$$

When $r\geq1/2$,

$$\begin{aligned} \bigl\Vert \gamma(g_{\mathbf{w},\varsigma})-g_{\varrho} \bigr\Vert ^{2}_{\varrho_{S}} \leq& D_{2}\log \biggl( \frac{2}{\delta} \biggr) \bigl\{ \zeta^{-\tau} \bigl\{ \chi+ m^{-1}+m^{1-q}\varsigma\chi ^{-q} \\ &{} +m^{\frac{-2q-2p+2pq}{(2+p)q}}\varsigma^{-\frac {2p}{q(2+p)}} \bigr\} +\zeta \bigr\} . \end{aligned}$$

Similarly, we choose

$$ \tau_{\max}=\frac{2q}{(1+\tau)(2p+2q+3pq)} $$

to maximize the convergence rate.

We complete the proof of Theorem 1. □

Proof of Theorem 2

$$\begin{aligned}& \int_{S} \bigl\{ \mathfrak{E}_{s}\bigl( \gamma_{A}(g_{\mathbf{w},\zeta ,\varsigma,s})\bigr)-\mathfrak{E}_{s}(g_{\varrho}) \bigr\} \,d\varrho _{S}(s) \\& \quad \leq \int_{S} \bigl\{ \mathfrak{E}_{s}\bigl( \gamma_{A}(g_{\mathbf{w},\zeta ,\varsigma,s})\bigr)-\mathfrak{E}_{s}(g_{\varrho})+ \varsigma\varOmega _{\mathbf{w}}(g_{\mathbf{w}, \zeta,\varsigma,s}) \bigr\} \,d\varrho _{S}(s) \\& \quad =\mathfrak{S}(\mathbf{w},\chi,\varsigma)+\mathfrak{H}(\mathbf{w},\chi , \varsigma)+ \int_{S} \bigl\{ \mathfrak{E}_{s}(g_{\chi})- \mathfrak {E}_{s}(g_{\varrho})+\chi\|g_{\chi} \|_{E}^{2} \bigr\} \,d\varrho_{S}(s), \end{aligned}$$

(41)

which yields

$$\begin{aligned} \mathfrak{E}_{s}(g_{\chi})-\mathfrak{E}_{s}(g_{\varrho})&= \int _{S}\varPsi \biggl(\frac{s}{\zeta},\frac{u}{\zeta} \biggr) \bigl(g_{\chi}(u)-g_{\varrho}(u)\bigr)^{2}\,d \varrho_{S}(u) \\ &\leq\|g_{\chi}-g_{\varrho}\|_{\varrho_{S}}^{2}. \end{aligned}$$

This completes the proof of Theorem 2. □

5 Conclusions

In this paper, we studied a class of Schrödinger equations with Neumann boundary condition $L_{\varepsilon}g= \operatorname{div}(\omega_{\varepsilon}(x) |\nabla g|^{\varrho(x)-2}\nabla g)=0$ on a compact metric space $S\subset\mathbb{R}^{n}$, $n\ge2$, with a positive weight $\omega_{\varepsilon}(x)$. We were interested in the asymptotic behavior of solutions of the mentioned equation. More precisely, we formulated conditions on a function g, which guarantee that the graph of at least one solution for the above-mentioned equation stays in the prescribed domain. These results generalized some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrödinger equations or of delay Schrödinger equations.