Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

TL;DR

This paper develops a Carleman estimate for an anisotropic hyperbolic equation with memory, applies it to derive observability inequalities, and uses these results to solve an inverse source problem with stability in viscoelasticity models.

Contribution

It introduces a new Carleman estimate for anisotropic hyperbolic equations with memory and applies it to inverse problems and controllability in viscoelasticity.

Findings

Established a Carleman estimate with boundary data.

Derived an observability inequality for the initial state.

Proved Lipschitz stability for the inverse source problem.

Abstract

We consider an anisotropic hyperbolic equation with memory term: $$ \partial_t^2 u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_ju) + \int^t_0 \sum_{| \alpha| \le 2} b_{\alpha}(x,t,\eta)\partial_x^{\alpha}u(x,\eta) d\eta + F(x,t) $$ for $x \in \Omega$ and $t\in (0,T)$ or $\in (-T,T)$, which is a model equation for viscoelasticity. First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary $\Gamma \times (-T,T)$. Second we apply the Carleman estimate to establish a both-sided estimate of $| u(\cdot,0)|_{H^3(\Omega)}$ by $\partial_{\nu}u|_{\Gamma\times (0,T)}$ under the assumption that $\partial_tu(\cdot,0) = 0$ and $T>0$ is sufficiently large, $\Gamma \subset \partial\Omega$ satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in $F(x,t)$ and we establish a both-sided Lipschitz stability estimate.

Equations

$${\partial }_t^{2}u(x,t)=\sum _{i,j=1}^n{\partial }_i(a_{ij}(x){\partial }_{j}u)+{\int }_0^t\sum _{|\alpha |\le 2}{b}_{\alpha }(x,t,\eta ){\partial }_x^{\alpha }u(x,\eta )d\eta +F(x,t)$$

$${\partial }_t^{2}u(x,t)=\sum _{i,j=1}^n{\partial }_i(a_{ij}(x){\partial }_{j}u)+{\int }_0^t\sum _{|\alpha |\le 2}{b}_{\alpha }(x,t,\eta ){\partial }_x^{\alpha }u(x,\eta )d\eta +F(x,t),x\in \Omega ,$$

$$-T<t<T.$$

$$\left\{\begin{array}[]{rl}&a_{ij}=a_{ji}\in C^{1}(\overline{\Omega}),\quad 1\leq i,j\leq n,\\ &\mbox{there exists a constant $\mu_{0}>0$ such that}\\ &\sum_{i,j=1}^{n}a_{ij}(x)\xi_{i}\xi_{j}\geq\mu_{0}\sum_{i=1}^{n}\xi_{i}^{2},\quad x\in\overline{\Omega},\thinspace\xi_{1},...,\xi_{n}\in\mathbb{R}.\end{array}\right.$$

$$\left\{\begin{array}[]{rl}&\nabla_{x,t}\partial_{t}^{k}b_{\alpha}\in C(\overline{\Omega}\times[0,T]^{2}),\quad k=0,1,2\quad\mbox{if $|\alpha|=2$},\\ &\partial_{t}^{k}b_{\alpha}\in L^{\infty}(\Omega\times(0,T)^{2}),\quad k=0,1,2\quad\mbox{if $|\alpha|\leq 1$},\\ \end{array}\right.$$

$$\partial_{\nu}u=\sum_{i,j=1}^{n}a_{ij}(\partial_{j}u)\nu_{i}\quad\mbox{on $\partial\Omega$}.$$

$$Au:=\sum _{i,j=1}^n{\partial }_i(a_{ij}(x){\partial }_{j}u),B(t,\eta )u:=\sum _{|\alpha |\le 2}{b}_{\alpha }(x,t,\eta ){\partial }_x^{\alpha }u(x,\eta ).$$

$$a(x,\xi )=A_0(x)\xi \cdot \xi =\sum _{i,j=1}^n{a}_{ij}(x){\xi }_i{\xi }_j$$

$$\{p,q\}(x,\xi )=\sum _{j=1}^n\left(\frac{\partial p}{\partial {\xi }_j}\frac{\partial q}{\partial x_j}-\frac{\partial p}{\partial x_j}\frac{\partial q}{\partial {\xi }_j}\right)(x,\xi ).$$

$$d(x)=|x-x_0{|}^2,x\in {\mathbb{R}}^n$$

$$\{a,\{a,d\}\}(x,\xi )\ge {\mu }_1|A_0(x{)}^{-1}\xi {|}^2,x\in \overline{\Omega },\xi \in {\mathbb{R}}^n$$

$$\Gamma =\{x\in \partial \Omega ;(x-x_0)\cdot \nu (x)\ge 0\}.$$

$$Q=\Omega \times (-T,T),$$

$$\psi (x,t)=|x-x_0{|}^2-\beta t^2,(x,t)\in \overline{Q}$$

$$\phi (x,t)=e^{\gamma \psi (x,t)},\Phi =\Phi (\gamma )=\underset{(x,t)\in \overline{Q}}{\max }\phi (x,t),$$

$$\chi(t)=\left\{\begin{array}[]{rl}1,&|t|\leq T-2\varepsilon,\\ 0,&|t|\geq T-\varepsilon.\end{array}\right.$$

$$\delta =\underset{x\in \overline{\Omega },T-2\epsilon \le |t|\le T-\epsilon }{\max }\phi (x,t).$$

$$F\in L^{2}(-T,T;H^{2}(\Omega)),\quad F=\partial_{\nu}F=0\quad\mbox{on $\partial\Omega$}.$$

$$v(x,t)=\chi (t){\partial }_t^{2}u(x,t)-\chi (t)F(x,t),x\in \Omega ,-T<t<T.$$

$${\int }_Q(s\gamma \phi |{\nabla }_{x,t}v{|}^2+s^3{\gamma }^3{\phi }^3|v{|}^2)e^{2s\phi }dxdt$$

$$\le C{\int }_Q|AF{|}^2{e}^{2s\phi }dxdt+C\Vert u{\Vert }_{H^1(-T,T;H^2(\Omega ))}^2{s}^2{\Phi }^2{e}^{2s\delta }+C{e}^{Cs}\Vert {\partial }_{\nu }u{\Vert }_{H^2(-T,T;L^2(\Gamma ))}^2$$

$${\int }_Q\left(\sum _{|\alpha |\le 2}{s}^2{\phi }^2|\chi {\partial }_x^{\alpha }u{|}^2+|\chi {\partial }_x^{\alpha }{\partial }_{t}u{|}^2\right)e^{2s\phi }dxdt$$

$$\le C{\int }_Q|AF{|}^2{e}^{2s\phi }dxdt+C\Vert u{\Vert }_{H^1(-T,T;H^2(\Omega ))}^2{s}^2{\Phi }^2{e}^{2s\delta }+C{e}^{Cs}\Vert {\partial }_{\nu }u{\Vert }_{H^2(-T,T;L^2(\Gamma ))}^2$$

$$F,{\partial }_{t}F\in L^2(-T,T;H^2(\Omega )).$$

$${\int }_Q\left(\sum _{|\alpha |\le 2}{s}^2{\phi }^2|\chi {\partial }_x^{\alpha }u{|}^2+|\chi {\partial }_x^{\alpha }{\partial }_{t}u{|}^2+|\chi {\partial }_x^{\alpha }{\partial }_t^{2}u{|}^2\right)e^{2s\phi }dxdt$$

$$\le C{\int }_Q(|AF{|}^2+|A{\partial }_{t}F{|}^2)e^{2s\phi }dxdt+C\Vert u{\Vert }_{H^2(-T,T;H^2(\Omega ))}^2{s}^2{\Phi }^2{e}^{2s\delta }+C{e}^{Cs}\Vert {\partial }_{\nu }u{\Vert }_{H^3(-T,T;L^2(\Gamma ))}^2$$

$$a_{ij}(x)=\left\{\begin{array}[]{rl}p(x),&i=j,\\ 0,&i\neq j\end{array}\right.$$

$$\left\{\begin{array}[]{rl}&\partial_{t}^{2}y(x,t)=Ay+\int^{t}_{0}\sum_{|\alpha|\leq 2}b_{\alpha}(x,t,\eta)\partial_{x}^{\alpha}y(x,\eta)d\eta,\quad x\in\Omega,\thinspace 0<t<T,\\ &y|_{\partial\Omega}=0,\quad 0<t<T,\\ &y(x,0)=a(x),\quad\partial_{t}y(x,0)=0,\quad x\in\Omega.\\ \end{array}\right.$$

$$T>\frac{{\max }_{x\in \overline{\Omega }}|x-x_0|}{\sqrt{\beta }}.$$

$$C^{-1}\Vert {\partial }_{\nu }y{\Vert }_{H^2(0,T;L^2(\partial \Omega ))}\le \Vert y(\cdot ,0){\Vert }_{H^3(\Omega )}\le C\Vert {\partial }_{\nu }y{\Vert }_{H^2(0,T;L^2(\Gamma ))}.$$

Full text

[TABLE]

for $x\in\Omega$ and $t\in(0,T)$ or $\in(-T,T)$, which is a model equation for viscoelasticity. First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary $\Gamma\times(-T,T)$. Second we apply the Carleman estimate to establish a both-sided estimate of $\|u(\cdot,0)\|_{H^{3}(\Omega)}$ by $\partial_{\nu}u|_{\Gamma\times(0,T)}$ under the assumption that $\partial_{t}u(\cdot,0)=0$ and $T>0$ is sufficiently large, $\Gamma\subset\partial\Omega$ satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in $F(x,t)$ and we establish a both-sided Lipschitz stability estimate.

1 Introduction and main results

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary $\partial\Omega$. We consider an integro-hyperbolic equation

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[TABLE]

We assume that

[TABLE]

Here and henceforth let $\alpha=(\alpha_{1},...,\alpha_{n})\in(\mathbb{N}\cup\{0\})^{n}$ be a multi-index and we set $|\alpha|=\alpha_{1}+\cdots+\alpha_{n}$, $\partial_{i}=\frac{\partial}{\partial x_{i}}$, $1\leq i\leq n$, $\partial_{t}=\frac{\partial}{\partial t}$, $\partial_{x}^{\alpha}=\partial_{1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}}$.

Throughout this paper, we assume

[TABLE]

if not specified. Let $\nu=\nu(x)=(\nu_{1},...,\nu_{n})$ be the unit outward normal vector to $\partial\Omega$ at $x$ and let

[TABLE]

For concise description, we set

[TABLE]

The equation (1.1) is a model equation for the viscoelasticity.

We point out that for some materials, the effects of memory cannot be neglected without failing the analysis, as observed by Volterra [56]. He embraced Boltzmann model, according to which the stress has to depend linearly on strain history.

Our integro-differential equation (1.1) serves as a model for describing the viscoelastic properties of those materials whose properties are different along several directions. There is a huge number of papers treating viscoelastic models, as shown, e.g., in the book Renardy, Hrusa and Nohel [53]. With no pretension to be exhaustive, we cite the papers Dafermos [17] and Edelstein and Gurtin [19]. In particular, in a pioneering work [17], Dafermos studied an abstract formulation of our equation, giving as an application the case of an anisotropic viscoelastic equation.

In this paper, we first establish a Carleman estimate for (1.1). A Carleman estimate is a weighted $L^{2}$-estimate for solutions to a partial differential equation which holds uniformly in large parameter $s>0$, and was derived by Carleman [13] for proving the unique continuation property. Second we apply the Carleman estimate to prove an estimate of initial value by data on suitable lateral boundary data, which is called an observability inequality. Finally we discuss an inverse source problem. More precisely, the external force $F$ is assumed to cause the action, but in practice it is often that $F$ is not a priori known and so we have to identify by available data for the sake of accurate analysis of the system. We are concerned with the determination of a spatial component $f(x)$ of $F(x,t):=R(x,t)f(x)$ with given $R(x,t)$. The form $R(x,t)f(x)$ is special but in applications we model the external force in a more special form $F(x,t)=\lambda(t)f(x)$ where $\lambda(t)$ is the time changing ratio and $f$ is the spatial distribution of the external force.

For the statement of the Carleman estimate, we need to introduce notations. We set $A_{0}(x)=(a_{ij}(x))_{1\leq i,j\leq n}$ and

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for all $x\in\Omega$ and $\xi=(\xi_{1},...,\xi_{n})\in\mathbb{R}^{n}$. Given functions $p(x,\xi)$ and $q(x,\xi)$, we define the Poisson bracket by

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We set

[TABLE]

with fixed $x_{0}\in\mathbb{R}^{n}\setminus\overline{\Omega}$. In addition to (1.2), throughout this paper, we assume that there exists a constant $\mu_{1}>0$ such that

[TABLE]

(e.g., Bellassoued and Yamamoto [9], [10]). For proving a Carleman estimate, it is known that we need some condition like (1.5), which ic called the pseudo-convexity (e.g., Hörmande [25]). We refer to Yao [58] which discusses anisotropic materials without intregral terms and shows a counterexample to the observability inequality without such condition for the principal part.

Next as subboundary where we take boundary data of the solution $u$, we define

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Here and henceforth $(\cdot,\cdot)$ denotes the scalar product in $\mathbb{R}^{n}$.

Furthermore we set

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[TABLE]

and

[TABLE]

where $\beta>0$ is chosen sufficienly small for the constant $\mu_{1}>0$ in (1.5) and $\gamma>0$ is a second large paramater and chosen later.

The conditions (1.5) and (1.6) pose extra conditions for $a_{ij}$ and $\Gamma$ respectively and are a sufficient condition for the Carleman estimate below stated.

Now we introduce a cut-off function $\chi$ in $t$. For fixed sufficiently small $\varepsilon>0$, let $\chi\in C^{\infty}(\mathbb{R})$ satisfy $0\leq\chi\leq 1$ in $\mathbb{R}$ and

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We further set

[TABLE]

Then $\varphi(x,t)\leq\delta$ for $x\in\overline{\Omega}$ and $T-2\varepsilon\leq|t|\leq T-\varepsilon$.

Now we are ready to state our first main result.

**Theorem 1.1.

**Let

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We set

[TABLE]

(i) There exists a constant $\gamma_{0}>0$ such that for $\gamma>\gamma_{0}$, we can choose constants $s_{0}=s_{0}(\gamma)>0$ and $C>0$ such that

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

for all $s>s_{0}$ and $u\in L^{2}(Q)$ satisfying $\partial_{x}^{\alpha}u\in H^{2}(-T,T;L^{2}(\Omega))\cap L^{2}(-T,T;H^{2}(\Omega))$ for all $|\alpha|\leq 2$ and $u|_{\partial\Omega}=0$.

(ii) Moreover we assume

[TABLE]

Then

[TABLE]

[TABLE]

for all $s>s_{0}$ and $u\in H^{2}(Q)$ satisfying $\partial_{x}^{\alpha}u,\partial_{t}u\in H^{2}(Q)$ for all $|\alpha|\leq 2$ and $u|_{\partial\Omega}=0$.

In (ii) of the theorem, we can rewrite (1.13) in terms of $v$, but we omit.

Inequalities (1.12) and (1.13) hold for each solution $u$ to (1.1) and both are weighted with $e^{2s\varphi(x,t)}$ and uniform for sufficiently large $s>0$ in the sense that the constant $C>0$ is independent of all large $s>0$. Such an inequality is called a Carleman estimate. The Carleman estimate is effectively applied to the unique continuation for partial differential equations, the observability inequality and inverse problems. In this paper, by Theorem 1.1 we establish the observability inequality (Theorem 1.2) and the Lipschitz stability in an inverse source problem (Theorem 1.3) for (1.1).

As for general treatments on Carleman estimates for partial differential equations without integral terms, we refer to Hörmander [25], Isakov [38]. There are many works concerning Carleman estimates for partial differential equations without integral terms. Since inverse problems are often concerned with the determination of the principal coefficients $a_{ij}(x)$, we have to concretely realize the condition (1.5). As for such concrete Carleman estimates which give sufficient conditions for (1.5) and more directly applicable to inverse problems, see Amirov and Yamamoto [1]. We refer to Baudouin, de Buhan and Ervedaza [2], Imanuvilov [27], Khaĭdarov [42], Romanov [54] which establish Carleman estimates for hyperbolic equations. For Carleman estimates for parabolic equations, in addition to Isakov [38], [39], Isakov and Kim [40], see Fursikov and Imanuvilov [22], Imanuvilov [26], Imanuvilov, Puel and Yamamoto [30], Yamamoto [57]. For elliptic Carleman estimates where the right-hand side is estimated in $H^{-1}$-space, see Imanuvilov and Puel [29].

As for isotropic hyperbolic equations with integral terms, Cavaterra, Lorenzi and Yamamoto [14] established a Carleman estimate and applied it for proving a stability result for some inverse source problem. Here the isotropic hyperbolic equation means

[TABLE]

in (1.1). After [14], in the case where $A$ and $B$ are isotropic Lamé operators, the following works discuss Carleman estimates and inverse problems: de Buhan and Osses [18], Lorenzi, Messina and Romanov [51], Lorenzi and Romanov [52], Romanov and Yamamoto [55]. However, to the best knowledge of the authors, there are no publications on Carleman estimates for anisotropic hyperbolic equations with integral terms $\int^{t}_{0}\sum_{|\alpha|\leq 2}b_{\alpha}\partial_{x}^{\alpha}ud\eta$. For this anisotropic case, differently from the isotropic case, we need a lot of technicalities because $A$ and $B$ are not commutative modulo lower-order terms of derivatives.

Now we present two applications of the Carleman estimate (Theorem 1.1). First we derive a partial observability inequality of estimating one component of a pair of initial values.

Let $y\in L^{2}(\Omega\times(0,T))$ satisfy $\partial_{x}^{\alpha}y\in H^{2}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$ with $|\alpha|\leq 2$ and

[TABLE]

Let $\Gamma\subset\partial\Omega$ and $T>0$ be given.

**Partial observability inequality.

**Estimate $a(x)$ by $\partial_{\nu}y|_{\Gamma\times(0,T)}$.

Thanks to the integral term $\int^{t}_{0}\sum_{|\alpha|\leq 2}b_{\alpha}(x,t,\eta)\partial_{x}^{\alpha}y(x,\eta)d\eta$, our method requests that $y(\cdot,0)=0$ or $\partial_{t}y(\cdot,0)=0$ in $\Omega$, and here we discuss only the case of $\partial_{t}y(\cdot,0)=0$.

The observability inequality is regarded as a dual problem to the exact controllability, and for a hyperbolic type of equations without integral terms, there have been enormous works. Here we refer only to Komornik [48], Lions [50], and Yao [58] which discusses anisotropic hyperbolic equations without integral terms. For proving observability inequalities, the multiplier method is commonly applied, but also a Carleman estimate is applicable for wider classes of partial differential equations (e.g., Kazemi and Klibanov [41], Klibanov and Malinsky [45]). As for the first application of Theorem 1.1, we show an observability inequality for (1.14).

**Theorem 1.2.

**We assume that $\Gamma$ satisfies (1.6), and

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Then there exists a constant $C>0$ such that

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for each solution $y$ to (1.14) with $y(\cdot,0)\in H^{3}(\Omega)\cap H^{1}_{0}(\Omega)$.

In this theorem, we can replace (1.3) by weaker condition

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By the finiteness of the propagation speed, we need to assume (1.15), and also a geometric condition (1.6) on the observation subboundary $\Gamma$ is assumed. This is the same for the inverse source problem stated below.

Finally we discuss an inverse source problem. That is, we consider

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Here we assume

[TABLE]

Let $\Gamma\subset\partial\Omega$ be given and $T>0$ be fixed. Then we discuss

**Inverse source problem.

**Determine $f(x)$, $x\in\Omega$ from $\partial_{\nu}u|_{\Gamma\times(0,T)}$.

As the stability for the inverse problem, we prove

**Theorem 1.3.

**We assume (1.6) and (1.15). Then there exists a constant $C>0$ such that

[TABLE]

for each $f\in H^{2}_{0}(\Omega)$.

The second inequality in (1.18) asserts the Lipschitz stability for our inverse problem. The first inequality means that our estimate is the best possible estimate for the inverse source problem.

Our argument for the inverse problem is based on Bukhgeim and Klibanov [12], which relies on a Carleman estimate. Klibanov [44] corresponds to the full version of [12]. Since Bukhgeim and Klibanov [12], their methodology has been developed for various equations and we can refer to many papers on inverse problems of determining spatially varying coefficients and components of source terms. As a partial list of references on inverse problems for hyperbolic and parabolic equations by Carleman estimates, we refer to Baudouin and Yamamoto [3], Bellassoued [5], [6], Bellassoued and Yamamoto [8], Benabdallah, Cristofol, Gaitan and Yamamoto [11], Cristofol, Gaitan and Ramoul [16], Imanuvilov and Yamamoto [31] - [34], Klibanov [43], [44], Klibanov and Yamamoto [47], Yamamoto [57], Yuan and Yamamoto [60].

As for similar inverse problems for the Navier-Stokes equations, see Bellassoued, Imanuvilov and Yamamoto [7], Choulli, Imanuvilov, Puel and Yamamoto [15], Fan, Di Cristo, Jiang and Nakamura [20], Fan, Jiang and Nakamura [21]. Gaitan and Ouzzane [23], and Gölgeleyen and Yamamoto [24] discuss inverse problems for transport equations by Carleman estimate, and Imanuvilov, Isakov and Yamamoto [28], Imanuvilov and Yamamoto [35] - [37] discuss Carleman estimates and inverse problems for non-stationary isotropic Lamé systems, which are related to our equation for the viscoelasticity. See Yuan and Yamamoto [59] about a Carleman estimate and inverse problems for a plate equation. As related books on Carleman estimates and inverse problems, see Beilina and Klibanov [4], Klibanov and Timonov [46], Lavrent’ev, Romanov and Shishat$\cdot$skiĭ[49].

This paper is composed of five sections. In section 2, we prove Theorem 1.1 and section 3 is devoted to providing fundamental energy estimates. In sectios 4 and 5, we prove Theorems 1.2 and 1.3 respectively.

2 Proof of Theorem 1.1

The proof is combination of hyperbolic and elliptic Carleman estimates (Lemmata 2.1 and 2.2) with another key lemma (Lemma 2.3) which can incorporate the integral term in (1.1). We divide the proof into five steps.

**First Step.

**Henceforth we set

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Under the assumption (1.7), a Carleman estimate for hyperbolic equations is known (e.g., Bellassoued and Yamamoto [9], [10]).

**Lemma 2.1.

**There exists a constant $\gamma_{0}>0$ such that for $\gamma>\gamma_{0}$, we can choose constants $s_{0}=s_{0}(\gamma)>0$ and $C=C(\gamma)>0$ such that

[TABLE]

[TABLE]

for all $s>s_{0}$ and $u\in H^{2}(Q)$ satisfying $u|_{\partial\Omega}=0$.

Moreover we have a Carleman estimate for the elliptic operator $A$ without the extra conditions on $a_{ij}$.

**Lemma 2.2.

**Let $p\in\mathbb{R}$ be given. There exists a constant $\gamma_{0}>0$ such that for $\gamma>\gamma_{0}$, we can choose constants $s_{0}>0$ and $C>0$ such that

[TABLE]

[TABLE]

for all $s>s_{0}$ and $y\in L^{2}(-T,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$.

In the case of $p=-1$, Lemma 2.2 is classical and we refer to Lemma 7.1 in Bellassoued and Yamamoto [10] for example. For completeness, we give the proof of Lemma 2.2 for arbitrary $p\in\mathbb{R}$ on the basis of the case of $p=-1$ in Appendix.

Second Step.

For gaining compact supports in time for functions under consideration, we use the cut-off function. That is, we recall that we choose a cut-off function $\chi\in C^{\infty}(\mathbb{R})$ such that $0\leq\chi\leq 1$ and

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In the succeeding arguments, we notice that all the terms with derivatives of $\chi$ can be regarded as of minor orders with respect to the large parameter $s$.

For treating an integral, it is essential to introduce a new function

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Then

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We set

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Then we have

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and so

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[TABLE]

Then $S(x,t)$ satisfies

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By (2.3), we note that $|S(x,t)|\neq 0$ only if $T-2\varepsilon\leq|t|\leq T-\varepsilon$. Moreover, since

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we obtain

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By $F|_{\partial\Omega}=0$ and (2.5), we see that $v|_{\partial\Omega}=0$. Thus

[TABLE]

where

[TABLE]

and $c_{\alpha,k}\in W^{1,\infty}(-T,T;L^{2}(\Omega))$, $\widetilde{c}_{\alpha}\in W^{1,\infty}(\Omega\times(-T,T)^{2})$.

Applying Lemma 2.1 to (2.8), we have

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[TABLE]

for $s>s_{0}$.

**Third Step.

**For estimating the integral term in (1.1) with the weight $e^{2s\varphi}$, we need to prove

**Lemma 2.3.

**Let $q\geq 0$. Then

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This type of inequality is essential for applications of Carleman estimates to inverse problems (Bukhgeim and Klibanov [12], Klibanov [44]) and the inequality not involving the cut-off function $\chi$, is proved in [44], [46].

**Proof.

**It suffices to prove for $t\geq 0$, because the proof for $t\leq 0$ is similar. By the Cauchy-Schwarz inequality, we have

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[TABLE]

Noting that

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by integration by parts, we obtain

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[TABLE]

Here we used

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by (2.3).

Therefore we can shift the first term on the right-hand side into the left-hand side, we have

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Choosing $\gamma>0$ and $s>0$ sufficiently large and noting that $\varphi=e^{\gamma\psi}$ and $\psi\geq 0$ in $Q$, we can obtain $1-\frac{|q-1|}{2}\frac{1}{s\varphi}\geq\frac{1}{2}$. Therefore

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Substituting this into (2.12), by (2.11) we can complete the proof of Lemma 2.3.

**Fourth Step.

**Henceforth $\mu(t)$ generically denotes functions in $L^{\infty}(\mathbb{R})$ such that

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Henceforth we denote $\chi^{\prime}(t)=\frac{d\chi}{dt}(t)$, $\chi^{\prime\prime}(t)=\frac{d^{2}\chi}{dt^{2}}(t)$. We note that $|\partial_{t}(\chi^{2})|$, $|\chi^{\prime}(t)|^{2}$, $|\chi^{\prime\prime}(t)|^{2}$ can be replaced by $\mu(t)$ in the following estimation.

Applying Lemma 2.3 to the third term on the right-hand side of (2.10) and noting that $\chi\partial_{x}^{\alpha}\partial_{t}u=\partial_{x}^{\alpha}\partial_{t}(\chi u)-\chi^{\prime}\partial_{x}^{\alpha}u$, we have

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We set

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Then

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Therefore

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[TABLE]

for $s>s_{0}$.

We will estimate the second and the third terms on the right-hand side of (2.13). By (2.4), we have

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in $Q$, and so we apply (2.2) with $p=2$, we obtain

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Next we apply Lemma 2.3 with $q=2$ to the second term on the right-hand side and, similarly to (2.13), we obtain

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[TABLE]

Therefore

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Choosing $\gamma>0$ sufficiently large, we can absorb the second term on the right-hand side into the left-hand side. Moreover we choose $s_{0}(\gamma)>0$ sufficiently larger such that $s_{0}(\gamma)\varphi(x,t)\geq s_{0}(\gamma)e^{-\gamma\max_{\overline{Q}}|\psi(x,t)|}\geq 1$, and by $s\varphi\geq 1$ we obtain

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[TABLE]

for $s>s_{0}$.

Now we estimate the third term on the right-hand side of (2.13). By (2.4) we have

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Apply (2.2) with $p=0$, and we obtain

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By an argument similar to (2.14) for the third term on the right-hand side, we have

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Applying (2.15), we obtain

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[TABLE]

for $s>s_{0}$.

Applying (2.15) and (2.16) to (2.13), we obtain

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Choosing $\gamma>0$ sufficiently large, we can absorb the second term on the right-hand side into the left-hand side. Therefore

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[TABLE]

for $s>s_{0}$. By the definition of $U_{1}$, the estimate (2.17) proves (1.11).

Next we will prove (1.12). The addition of (2.15) and (2.16) yields

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[TABLE]

Applying (2.17) in (2.18), we obtain

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By (2.4) and $\partial_{\nu}F|_{\partial\Omega}=0$, we have $\partial_{\nu}v=\chi\partial_{\nu}\partial_{t}^{2}u-\chi\partial_{\nu}F=\chi\partial_{\nu}\partial_{t}^{2}u$ on $\partial\Omega$. Consequently the estimate (1.12), and the proof of Theorem 1.1 (i) is completed.

**Fifth Step.

**We will prove Theorem 1.1 (ii). By (2.4), we have

[TABLE]

Henceforth $U_{2}$ generically denotes a function satisfying

[TABLE]

for $(x,t)\in Q$. Applying Lemma 2.2 to (2.19) with $p=0$, we obtain

[TABLE]

We estimate the second term on the right-hand side as follows. By $\partial_{t}u(\cdot,0)=0$, we have

[TABLE]

Lemma 2.3 with $q=1$ yields

[TABLE]

Consequently

[TABLE]

Thus we can absorb the second term on the right-hand side into the left-hand side, so that

[TABLE]

Fixing $\gamma>0$ sufficiently large and applying Theorem 1.1 (i) to the second term on the right-hand side, we obtain

[TABLE]

[TABLE]

Next, setting $v_{1}=\partial_{t}v$, we differentiate the first equation in (2.8) in $t$, and we have

[TABLE]

Since

[TABLE]

and

[TABLE]

in terms of (2.22) and (2.23), we apply Lemma 2.1 to (2.21) to obtain

[TABLE]

Applying (2.20) and Theorem 1.1 (i) to the third and the second terms on the right-hand side respectively, we see

[TABLE]

By (2.4), we have $v=\chi\partial_{t}^{2}u-\chi F$ and

[TABLE]

on $\partial\Omega$. Choosing $s>0$ and $\gamma>0$ sufficiently large, we can absorb the fourth terms on the right-hand side into the left-hand side, we have

[TABLE]

[TABLE]

for $s>s_{0}$.

Substituting (2.24) into (2.20), we obtain

[TABLE]

Thus the proof of Theoerem 1.1 (ii) is completed.

We close this section with the following lemma which is nothing but (2.24) where we fix $\gamma>0$ sufficiently large. The lemma plays an essential role for the proof of Theorem 1.3.

**Lemma 2.4.

**Let $u\in H^{2}(-T,T;H^{2}(\Omega))$ satify (1.16) and let (1.17) hold. Under the same assumptions in Theorem 1.1 (ii), we have

[TABLE]

for $s>s_{0}$. Here we set

[TABLE]

3 Energy estimates

For proving Theorems 1.2 and 1.3, we show energy estimates for hyperbolic equations with integral terms. Such an energy estimate is classical for hyperbolic equations without integral terms (e.g., Komornik [48], Lions [50]), but the presence of the integral terms makes extra estimation demanded.

**Lemma 3.1.

**We assume that

[TABLE]

and

[TABLE]

If $u\in H^{2}(0,T;L^{2}(\Omega))\cap H^{1}(0,T;H^{1}_{0}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$ satisfies

[TABLE]

and

[TABLE]

then there exists a constant $C>0$, which is independent of choice of $u$, such that

[TABLE]

Here and henceforth we set

[TABLE]

Proof of Lemma 3.1. We multiply (3.3) with $\partial_{t}u(x,t)$ and integrate over $\Omega$: By (3.1), (3.2) and (3.4) and integration by parts, we obtain

[TABLE]

and

[TABLE]

Next we calculate

[TABLE]

For $|\alpha|=2$, we set $\partial_{x}^{\alpha}u=\partial_{i}\partial_{j}u$ with $1\leq i,j\leq n$. Noting

[TABLE]

we write

[TABLE]

Here

[TABLE]

Therefore

[TABLE]

where

[TABLE]

[TABLE]

For $|\alpha|=1$, we have

[TABLE]

[TABLE]

and for $|\alpha|=0$, we use the Poincaré inequality to obtain

[TABLE]

[TABLE]

Hence

[TABLE]

Thus we obtain

[TABLE]

We choose $r\in(0,T)$ arbitrarily. Integrating both sides with respect to $t$ in $(0,r)$, we reach

[TABLE]

Here we have

[TABLE]

Moreover, for any $\varepsilon>0$, we can choose a constant $C_{\varepsilon}>0$ such that

[TABLE]

Moreover (3.9) and (3.10) yield

[TABLE]

and

[TABLE]

Thus we obtain

[TABLE]

Choosing $\varepsilon>0$ sufficiently small such that $C_{2}\varepsilon<\frac{1}{2}$, we have

[TABLE]

The Gronwall inequality implies

[TABLE]

Thus the proof of Lemma 3.1 is completed.

Next, on the basis of Lemma 3.1, we prove an energy estimate in Sobolev spaces of higher orders for the solution to (1.16). We assume that $u\in H^{2}(0,T;H^{2}(\Omega))$ satisfies (1.16), $\partial_{\nu}u\in H^{3}(0,T;L^{2}(\Gamma))$ and that (1.17) holds. Then we prove

**Lemma 3.2.

**There exists a constant $C>0$ such that

[TABLE]

for each $f\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$.

Proof of Lemma 3.2. We set

[TABLE]

and

[TABLE]

We set

[TABLE]

Then

[TABLE]

[TABLE]

and

[TABLE]

Indeed we can verify (3.14) as follows. We differentiate (1.16) in $t$ to have

[TABLE]

Noting that $u(x,t)=\int^{t}_{0}u_{1}(x,\xi)d\xi$ by $u(\cdot,0)=0$ and changing the orders of integration, we obtain

[TABLE]

and

[TABLE]

[TABLE]

Therefore (3.14) is verified. The systems (3.15) and (3.16) can be verified similarly by noting that

[TABLE]

Applying Lemma 3.1 to (3.14) - (3.16), we have

[TABLE]

Next we have to estimate $\|\partial_{t}^{j}u(\cdot,t)\|_{H^{2}(\Omega)}$, $k=0,1,2$. Since $\partial_{t}u(\cdot,t)=\int^{t}_{0}\partial_{t}^{2}u(\cdot,\eta)d\eta$ and $u(\cdot,t)=\int^{t}_{0}(t-\eta)\partial_{t}^{2}u(\cdot,\eta)d\eta$ by $u(\cdot,0)=\partial_{t}u(\cdot,0)=0$ in $\Omega$, it suffices to estimate $\|\partial_{t}^{2}u(\cdot,t)\|_{H^{2}(\Omega)}$. By (3.17) we have

[TABLE]

Therefore (3.15) implies

[TABLE]

and so

[TABLE]

Since $\partial_{t}^{2}u(\cdot,t)|_{\partial\Omega}=0$, we apply the a priori estimate for the elliptic boundary value problem, by (3.18) we obtain

[TABLE]

The Gronwall inequality yields

[TABLE]

Thus the proof of Lemma 3.2 is completed.

In this section we further show the following lemma for the hyperbolic equation without integral terms.

**Lemma 3.3.

**We assume (3.1). Let $w\in H^{2}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))\cap C([0,T];H^{1}_{0}(\Omega))\cap C^{1}([0,T];L^{2}(\Omega))$ satisfy

[TABLE]

(i) There exists a constant $C>0$ such that

[TABLE]

(ii) There exists a constant $C>0$ such that

[TABLE]

Proof of Lemma 3.3. The estimate (3.20) is proved by the multiplier method (e.g., Komornik [48]). That is, we choose $h:\Omega\longrightarrow{\mathbb{R}}^{n}$ such that $h\in C^{1}(\overline{\Omega})$ and $h|_{\partial\Omega}=\nu$ which is the unit outward normal vector to $\partial\Omega$. Then multiplyng (3.19) with $h\cdot\nabla w$ and integrating over $\Omega\times(0,T)$, we see (3.20). We omit the details and see e.g., [48] for the complete proof.

A usual energry estimate yields

[TABLE]

Since (3.19) is time-reversing, we can consider (3.19) by regarding $t$ as initial time and, applying (3.22), we obtain (3.21). Thus the proof of Lemma 3.3 is completed.

4 Proof of Theorem 1.2

Our proof is a modification of Kazemi and Klibanov [41] and Klibanov and Malinsky [45] which discuss hyperbolic equations without integral terms.

We make the even extension of $y$ to $(-T,0)$:

[TABLE]

Then, by $\partial_{t}y(\cdot,0)=0$, we can verify that

[TABLE]

[TABLE]

Hence

[TABLE]

Next we will estimate $\|y\|_{H^{1}(-T,T;H^{2}(\Omega))}$. By (4.1) and (4.2), it is sufficient to estimate for $0<t<T$, that is, $\|y\|_{H^{1}(0,T;H^{2}(\Omega))}$. Similarly to the proof of Lemma 3.2, we set

[TABLE]

[TABLE]

We recall that

[TABLE]

Noting that $y(x,t)=\int^{t}_{0}y_{1}(x,\xi)d\xi+a(x)$ and $y_{1}(x)=\int^{t}_{0}y_{2}(x,\xi)d\xi$ and $y_{1}(x,0)=\partial_{t}y(x,0)=0$, in a way similar to (3.13) and (3.14), we can

[TABLE]

and

[TABLE]

Applying Lemma 3.1 to (4.4) and (4.5), we obtain

[TABLE]

By the first equation in (4.4), we have

[TABLE]

and

[TABLE]

Applying the a priori estimate for the elliptic boundary value problem, we reach

[TABLE]

¿From (4.6) and the Gronwall inequality it follows that

[TABLE]

Since $y(\cdot,t)=\int^{t}_{0}\partial_{t}y(\cdot,\xi)d\xi+a$, it follows from (4.7) that

[TABLE]

Thus

[TABLE]

Now we choose $\varepsilon>0$ and $\delta>0$ in (1.10). The assumption (1.15) implies

[TABLE]

and

[TABLE]

by $x_{0}\not\in\overline{\Omega}$. Therefore there exist $\varepsilon_{0}>0$ and $\varepsilon\in\left(0,\frac{T}{4}\right)$ such that

[TABLE]

By (4.10) we note that $\varphi(x,\pm T)<1$ for $x\in\overline{\Omega}$, and so $0<\delta<1$ if we choose $\varepsilon>0$ sufficiently small in (1.10). Then we can set $\delta=1-\varepsilon_{0}$ with $\varepsilon_{0}>0$. Then

[TABLE]

We set

[TABLE]

Noting that $F=0$ in $Q$, we apply (1.11) in Theorem 1.1 to (1.14) where we fix $\gamma>0$ large and applying (4.8) and so

[TABLE]

[TABLE]

for all large $s>0$. On the other hand, similarly to (2.8), we see

[TABLE]

where

[TABLE]

for $(x,t)\in Q$. Here, in terms of $\chi(0)=1$, $\chi^{\prime}(0)=0$ and the first equation in (1.14), we calculated $z(x,0)=\partial_{t}^{2}y(x,0)$ and $\partial_{t}z(x,0)=\partial_{t}^{3}y(x,0)$.

Applying (3.21) in Lemma 3.3, we fix sufficiently small $\delta_{0}>0$ such that $0<\delta_{0}<2\varepsilon$ and

[TABLE]

Here we recall that $E(z)(t)=E(t)=\int_{\Omega}\left(|\partial_{t}z(x,t)|^{2}+\sum_{i,j=1}^{n}a_{ij}(x)\partial_{i}z(x,t)\partial_{j}z(x,t)\right)dx$. By (4.14) and (4.8), we obtain

[TABLE]

[TABLE]

At the last inequality, in view of the Poincaré inequality, $z(\cdot,0)=Aa$ and the a priori estimate for the boundary value problem for $A$, we used

[TABLE]

[TABLE]

Hence, by (4.15), we have

[TABLE]

We further choose small $\delta_{0}>0$ such that $1-C\delta_{0}>0$ and fix. Then $E(0)\leq C_{1}E(t)$ for $0\leq t\leq\delta_{0}$. Therefore

[TABLE]

for all large $s>0$. Substituting this into the left-hand side of (4.12) and using (4.17), we obtain

[TABLE]

for all large $s>0$. Hence

[TABLE]

for all large $s>0$. Choosing $s>0$ sufficiently large such that $C_{1}\delta_{0}-Cs^{2}e^{-4s\varepsilon_{0}}>0$, we complete the proof of the second inequality in the conclusion of Theorem 1.2.

Next we prove the first inequality of the conclusion. In place of $z=\chi\partial_{t}^{2}y$, we set $y_{2}=\partial_{t}^{2}y$. Then, similarly to (2.8), we have

[TABLE]

where $\widetilde{S_{1}}$ satisfies (4.14). Similarly to (4.16) and (4.17), we can verify

[TABLE]

Applying (3.20) in Lemma 3.3, we have

[TABLE]

[TABLE]

By $\partial_{\nu}\partial_{t}y=0$ on $\partial\Omega$, we see

[TABLE]

and

[TABLE]

Therefore

[TABLE]

and

[TABLE]

With (4.19), we complete the proof of the first inequality. Thus the proof of Theorem 1.2 is completed.

5 Proof of Theorem 1.3

Once that the Carleman estimate Theorem 1.1 is established, we can prove Theorem 1.3 by an argument similar to Imanuvilov and Yamamoto [33]. See also Bellassoued and Yamamoto [10].

**First Step.

**We set $F(x,t)=R(x,t)f(x)$ and

[TABLE]

Similarly to (2.8) in $\Omega\times(0,T)$, using $f=\partial_{\nu}f=0$ on $\partial\Omega$, by (2.4) we can verify

[TABLE]

and

[TABLE]

We make the even extension of $v$ to $(-T,0)$:

[TABLE]

Accordingly we make the even extensions of $\chi AF+\chi J+S$. Then, by $v(x,0)=\partial_{t}v(x,0)=0$ for $x\in\Omega$ we can prove that

[TABLE]

[TABLE]

and

[TABLE]

We recall that $Q=\Omega\times(-T,T)$. Hence, setting $v_{1}=\partial_{t}v$, we have

[TABLE]

Here we used that $\partial_{t}v_{1}(x,0)=\chi AF(x,0)$ by $\partial_{t}v_{1}=\partial_{t}^{2}v$ and (2.7), (2.9).

We set

[TABLE]

We write (5.5) in terms of $z$. First we have

[TABLE]

and

[TABLE]

Moreover

[TABLE]

and so

[TABLE]

Hence

[TABLE]

Using $a_{ij}=a_{ji}$ in the second term on the right-hand side, we obtain

[TABLE]

Thus (5.5) yields

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

We rewrite Lemma 2.4 in terms of $z=v_{1}e^{s\varphi}$. First $s^{3}|z|^{2}=s^{3}|v_{1}|^{2}e^{2s\varphi}$ and then $\partial_{t}z=(\partial_{t}v_{1})e^{s\varphi}+s(\partial_{t}\varphi)z$, so that

[TABLE]

and we have similar estimates for $s|\nabla z|^{2}$. Hence

[TABLE]

and so

[TABLE]

for $s>s_{0}$. Here we include $\Phi^{2}$ into $C>0$. Henceforth we set

[TABLE]

We estimate $\|u\|^{2}_{H^{2}(-T,T;H^{2}(\Omega))}$ by Lemma 3.2, so that we obtain

[TABLE]

[TABLE]

for $s>s_{0}$.

**Second Step.

**We will carry out the energy estimate. We multiply (5.6) with $\partial_{t}z$ and integrate by parts over $\Omega\times(-T,0)$. Then

[TABLE]

Here by $z|_{\partial\Omega}=0$ and $a_{ij}=a_{ji}$, integrating by parts, we see

[TABLE]

Using $z(\cdot,0)=z(\cdot,-T)=0$ in $\Omega$ and noting that $a_{ij}$ is independent of $t$, we obtain

[TABLE]

Next, by the Cauchy-Schwarz inequality, we have

[TABLE]

[TABLE]

Here we extended the integral domain $\Omega\times(-T,0)$ to $\Omega\times(-T,T)=:Q$. Moreover we note

[TABLE]

and

[TABLE]

By (2.22) and (1.13), we have

[TABLE]

and (2.23) yields

[TABLE]

Consequently substituting these inequalities and applying Lemma 2.4 and (5.9) to estimate

[TABLE]

from (5.11) we reach

[TABLE]

[TABLE]

for $s>s_{0}$.

By (5.7), (5.10) and (5.12), using $\chi(0)=1$, we see

[TABLE]

for $s>s_{0}$.

**Third Step.

**We complete the proof by an elliptic Carleman estimate. Since

[TABLE]

we estimate

[TABLE]

and

[TABLE]

and so

[TABLE]

By $\varphi(x,t)\leq\varphi(x,0)$, we replace the second term on the right-hand side by the second term on the left-hand side, and we apply $|R(\cdot,0)|>0$ on $\overline{\Omega}$ by (1.17). Therefore

[TABLE]

[TABLE]

Since

[TABLE]

and $e^{-\gamma\beta t^{2}}-1<0$ for $t\neq 0$, the Lebesgue theorem yields

[TABLE]

as $s\to\infty$.

Therefore by choosing $s>0$ sufficiently large, the estimate (5.13) implies

[TABLE]

for $s>s_{0}$. By $\nabla\varphi(x,0)=2\gamma(x-x_{0})\varphi\neq 0$ for $x\in\overline{\Omega}$, we apply the Carleman estimate for the elliptic operator $A$ of the second order which is similar to Lemma 2.2 (here we fix $\gamma$), we have

[TABLE]

for $s>s_{0}$. Again choosing $s>0$ sufficienly large, we can absorb the first term on the right-hand side into the left-hand side, multiplying with $s$ and replacing $se^{Cs}$ by $e^{Cs}$, we have

[TABLE]

for $s>s_{1}$.

Hence

[TABLE]

that is,

[TABLE]

for all $s>s_{0}$. We choose $s>0$ sufficiently large so that $1-Cs^{3}e^{-4\varepsilon_{0}s}>0$. Then $\|f\|^{2}_{H^{2}(\Omega)}\leq C_{1}e^{Cs}D^{2}$. Noting that

[TABLE]

we complete the proof of the second inequality of (1.18).

Finally we have to prove the first inequality in (1.18). We set $w=\partial_{t}^{3}u$. Similarly to (2.8), we can obtain

[TABLE]

where $\widetilde{B}(t)=B(t,t)$ and

[TABLE]

Therefore by (1.4) we see

[TABLE]

for $x\in\Omega$ and $0<t<T$. We set

[TABLE]

Then we readily verify that $E(0)\leq C\|f\|^{2}_{H^{2}(\Omega)}$. Applying (3.20) to (5.15) and noting (5.16), we obtain

[TABLE]

The second term on the right-hand side is estimated by Lemma 3.2, so that

[TABLE]

Finally, since

[TABLE]

by $f\in H^{2}_{0}(\Omega)$ and $u(x,0)=\partial_{t}u(x,0)=0$, we have

[TABLE]

Consequently $\|\partial_{\nu}u\|_{H^{3}(0,T;L^{2}(\partial\Omega))}\leq C\|f\|_{H^{2}(\Omega)}$. Thus the proof of the first inequality, and so Theorem 1.3 is completed.

Acknowledgements. This work was completed when the third author was a guest professor at Sapienza Università di Roma in May - June 2016. The author thanks the university for that opportunity yielding the current joint work.

Appendix. Proof of Lemma 2.2 for general $p$.

It is known that

[TABLE]

[TABLE]

for all $s>s_{0}$ and $y\in L^{2}(-T,T;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$. See e.g., Bellassoued and Yamamoto [10] or we can prove (1) similarly to Yamamoto [57] where the parabolic Carleman estimate is proved. However here we omit the proof of (1). Now let $p\neq-1$, $\in\mathbb{R}$. We set

[TABLE]

and

[TABLE]

Then we directly verify

[TABLE]

and

[TABLE]

and

[TABLE]

[TABLE]

Therefore we have

[TABLE]

that is,

[TABLE]

By $a_{ij}\in C^{1}(\overline{\Omega})$, we can write

[TABLE]

By (3), applying (1) to (6), we obtain

[TABLE]

Choosing $s>0$ sufficiently large, we can absorb the second term on the right-hand side into the left-hand side and using

[TABLE]

by $y|_{\partial\Omega}=0$, we have

[TABLE]

[TABLE]

Similarly to (4) and (5), we see

[TABLE]

and

[TABLE]

Therefore, since

[TABLE]

we obtain

[TABLE]

[TABLE]

Here we used (8) for the final term. Moreover by (8) and (9), we have

[TABLE]

[TABLE]

Substituting (8) - (10) in (7), we obtain

[TABLE]

[TABLE]

for all $s>s_{0}$. Here in choosing $s_{0}(\gamma)$, we further assume that $s_{0}(\gamma)=s_{0}>\gamma$ to have

[TABLE]

with sufficiently large constant $C_{1}>0$. Multiplying (11) with $s^{p+1}$, we reach the conclusion and thus the proof of Lemma 2.2 with $p\in\mathbb{R}$ is completed.