Masaru Ikehata
email: ikehataprobe@gmail.com

Yavar Kian
email: yavar.kian@univ-amu.fr

THE ENCLOSURE METHOD FOR THE DETECTION OF VARIABLE ORDER IN FRACTIONAL DIFFUSION EQUATIONS

Keywords: 35R30, 35L05 enclosure method, inverse problem, time-fractional diffusion equation, space-dependent variable order, anomalous diffusion u

This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of the time domain enclosure method enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.

Introduction

In the present article we consider a model of anomalous diffusion described by variable order time fractional diffusion equation on Ω a bounded domain of R 3 with C 2 -boundary. Namely, we fix α ∈ L ∞ (Ω) satisfying ess. inf x∈Ω α(x) > 0 and ess. sup x∈Ω α(x) < 1. Then, given g(x, t), (x, t) ∈ ∂Ω×] 0, ∞[, let u = u g (x, t), with (x, t) ∈ Ω× ] 0, ∞[, denote the solution of the following initial boundary value problem:

           (∂ α(x) t
-∆)u = 0, (x, t) ∈ Ω× ] 0, ∞[, u(x, 0) = 0, x ∈ Ω, u(x, t) = g(x, t), (x, t) ∈ ∂Ω×] 0, ∞[.

(1.1)
Here, the symbol ∂ α(x) t denotes the Caputo fractional derivative of order α(x) with respect to t, that is

∂ α(x) t
where Γ is the Gamma function. We consider solutions u = u g of the problem (1.1) lying in the space C 1 ([0, ∞[; L 2 (Ω)) ∩ C([0, ∞[; H 2 (Ω)). We denote by ν the outer unit normal vector field on ∂Ω.

In this article we consider the inverse problem of determining the region of variation and additional information about the amplitude of variation of the fractional order α(x) appearing in (1.1). More precisely, let D be a nonempty bounded open subset of Ω with C 2 -boundary such that D ⊂ Ω. Assume that the order α(x) in (1.1) takes the form

α(x) =    α 0 , x ∈ Ω \ D, α 0 + h(x), x ∈ D, (1.2)
where α 0 ∈ ]0, 1[ and the function h belongs to L ∞ (D) and satisfies -α 0 < ess.inf x∈D h(x) ≤ ess.sup x∈D h(x) < 1 -α 0 .

We impose the jump condition (A.I)/(A.II) of α from α 0 across ∂D as follows.

(A.I) ∃C > 0 ∃γ ≥ 0 h(x) ≥ C dist (x, ∂D) γ a.e. x ∈ D;

(A.II) ∃C > 0 ∃γ ≥ 0 -h(x) ≥ C dist (x, ∂D) γ a.e. x ∈ D.

To briefly describe the difference between the two conditions sometimes we write α >> α 0 if (A.I) is satisfied; α << α 0 if (A.II) is satisfied. Our inverse problem can be stated as follows.

Problem. Assume that α 0 is known and both D and h are unknown. Given g (to be specified later) we extract information about the location and shape of D and qualitative property of h from the Neumann data ∂ ν u g on ∂Ω over the time interval ]0, ∞[.

Recall that the initial boundary value problem (1.1) is frequently used as a model for anomalous diffusion in complex media with applications in different fields such as geophysics, environmental and biological problems. Such diffusion process are often described by problem (1.1) with a constant order α (see [START_REF] Adams | Field study of dispersion in a heterogeneous aquifer 2. Spatial moments analysis[END_REF][START_REF] Carcione | Theory and simulation of timefractional fluid diffusion in porous media[END_REF]). However, in some complex media the presence of heterogeneous regions displays space inhomogeneous variations and the constant order fractional dynamic models are not robust for long times (see [START_REF] Fedotov | Subdiffusive master equation with space-dependent anomalous exponent and structural instability[END_REF][START_REF] Fedotov | Asymptotic behavior of the solution of the space dependent variable order fractional diffusion equation: ultraslow anomalous aggregation[END_REF]). For such problems, the variable order time-fractional model is considered as more relevant for describing the space-dependent anomalous diffusion process (see e.g. [START_REF] Sun | Variable-order fractional differential operators in anomalous diffusion modeling[END_REF]). Indeed, several variable order diffusion models have been successfully applied in different problems of sciences and engineering, including Chemistry [START_REF] Chen | Variable-order time-fractional derivative model for chloride ions sub-diffusion in concrete structures[END_REF], Rheology [START_REF] Smit | Rheological models containing fractional derivatives[END_REF], Biology [START_REF] Glöckle | A Fractional Calculus Approach to Self-Similar Protein Dynamics[END_REF], Hydrogeology [START_REF] Atangana | Stability and convergence of a time-fractional variable order Hantush aquation for a deformable aquifer[END_REF] and Physics [START_REF] Stickler | Continuous time anomalous diffusion in a composite medium[END_REF][START_REF] Zhang | Fractional Feynman-Kac equation with space-dependent anomalous exponent[END_REF]. In this context, the goal of our inverse problem is to determine information about the variable order α which play a fundamental role in the anomalous mechanism leading to the model (1.1).

The inverse problems of determining fractional orders, which is one of the most important inverse problems for fractional diffusion equations, have been extensively studied these last decades. We refer to [START_REF] Li | Inverse problems of determining parameters of the fractional partial differential equations[END_REF] for a survey about this topic (see also [START_REF] Jin | A tutorial on inverse problems for anomalous diffusion processes[END_REF] for an overview of inverse problems for fractional diffusion equations). Without being exhaustive we can mention the works of [START_REF] Alimov | Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation[END_REF][START_REF] Cheng | Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation[END_REF][START_REF] Hatano | Determination of order in fractional diffusion equation[END_REF][START_REF] Janno | Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation[END_REF][START_REF] Janno | Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements[END_REF][START_REF] Kian | Simultaneous determination of coefficients and internal source of a diffusion equation from a single measurement[END_REF][START_REF] Liao | Identifying a fractional order and a space source term in a time-fractional diffusion wave equation simultaneously[END_REF][START_REF] Li | Uniqueness in inverse boundary value problems for fractional diffusion equations[END_REF][START_REF] Li | Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation[END_REF][START_REF] Li | Unique determination of fractional order and source term in a fractional diffusion equation from sparse boundary data[END_REF][START_REF] Yamamoto | Uniqueness in determining fractional orders of derivatives and initial values[END_REF] devoted to the determination of single or multiple constant fractional orders, sometimes together with other parameters (coefficients or internal sources), from several class of observational data. We mention also the recent works [START_REF] Jin | Recovering multiple fractional orders in time-fractional diffusion in an unknown medium[END_REF][START_REF] Jin | Recovery of the order of derivation for fractional diffusion equations in an unknown medium[END_REF] where the determination of constant fractional order have been studied in the context of an unknown medium (unknown source, coefficients, domain...). All the above mentioned results have been devoted to the determination of constant fractional order. The only result that we are aware of dealing with the determination of variable fractional order depending on the space variable can be found in [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF]. Here the authors proved the determination of general order α ∈ L ∞ (Ω) from the knowledge of Neumann data ∂ ν u g on ∂Ω× ]0, T [ with an arbitrary fixed T > 0 for infinitely many input g having the form g(x, t) = t κ g(x) with a constant κ ∈]2, ∞[. The aim of the present article is to prove the detection of the region of variation and the amplitude of variation of α by using the enclosure method initiated in [START_REF] Ikehata | Reconstruction of the support function for inclusion from boundary measurements[END_REF] where the infinite boundary measurements under consideration in [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF] are replaced by a single boundary measurement for some class of suitable input g. We apply here the most recent version of the time domain enclosure method developed in [START_REF] Ikehata | The enclosure method for inverse obstacle scattering over a finite time interval: IV. Extraction from a single point on the graph of the response operator[END_REF][START_REF] Ikehata | The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data[END_REF].

1.1. Statement of the main result. Now let us describe the main result of this paper. For this purpose, we start by introducing the class of input under consideration in (1.1). Let η > 0 and 0 < R 1 < R 2 . Let B η be the open ball with radius η such that B η ∩ Ω = ∅. Let B R 1 and B R 2 be two concentric balls with radius R 1 and R 2 , respectively such that Ω ⊂ B R 1 (see Figure 1). Let m = 0, 1, • • • . For a complex number τ with Re τ > 1, we choose the solution w 0

,m (x) = w 0 ,m (x, τ ) ∈ H 2 (R 3 ) with = ext, int of the equation (∆ -τ α 0 )w 0 ,m + τ α 0 -1 Ψ ,m (x) = 0, x ∈ R 3 , where    Ψ ext,m (x) = (η 2 -|x -p| 2 ) m χ Bη (x), Ψ int,m (x) = (R 2 2 -|x -p| 2 ) m (|x -p| 2 -R 2 1 ) m χ B R 2 \B R 1 (x)
, the point p denotes the center of B η when = ext; the common center of B R 1 and B R 2 when = int. Note that both Ψ ext,m (x) and Ψ int,m (x) are non-negative for all x ∈ R 3 . Note that the restriction of w 0 ,m onto Ω satisfies (∆ -

τ α 0 )w 0 ,m = 0, x ∈ Ω. (1.3)
In the present article we consider the following class of input g

g ,m (x, t) = e t 2π ∞ -∞ e its (1 + is) -5 w 0 ,m (x, 1 + is)ds, x ∈ ∂Ω, t ∈ [0, ∞[, (1.4)
where = ext, int.

In order to state our main result we need to consider first the forward problem. Namely, we consider solutions of (1.1) lying in the space

C 1 ([0, ∞[; L 2 (Ω))∩C([0, ∞[; H 2 (Ω))
. This means that we consider solutions of (1.1) in a strong sense as stated in [START_REF] Kian | Well-posedness for weak and strong solutions of non-homogeneous initial boundary value problems for fractional diffusion equations[END_REF]Definition 2.2]. In addition to this property, we will show in the next Proposition 1.2 that e -t g ,m ∈ L ∞ (R + ; H that, for all τ ∈ C satisfying Reτ > 1, the Laplace transform

û( • , τ ) = ∞ 0 e -τ t u(t, • ) dt of u is lying in H 2 (Ω) and it solves the boundary value problem    (∆ -τ α(x) )û(x, τ ) = 0, x ∈ Ω, û(x, τ ) = g ,m (x, τ ), x ∈ ∂Ω.
(1.5)

All these properties are stated in the following proposition.

Proposition 1.1. Let g ,m be given by (1.4). Then, we have e -t g ,m ∈ L ∞ (R + ; H 3 2 (∂Ω)) and, for all complex τ satisfying Reτ > 1, we have

g ,m (x, τ ) = τ -5 w 0 ,m (x, τ ), x ∈ ∂Ω. (1.6)
Moreover, for g = g ,m , the problem (1.1) admits a unique strong solution

u ,m ∈ C 1 ([0, ∞[; L 2 (Ω)) ∩ C([0, ∞[; H 2 (Ω)) satisfying e -t u ∈ W 1,∞ (R + ; H 2 (Ω)).
Finally, for all complex τ satisfying Reτ > 1, the Laplace transform in time u ,m of u ,m solves (1.5), we have e

-τ t ∂ ν u ,m ∈ L 1 (R + ; H 3 2 (∂Ω)) and ∂ ν u ,m (x, τ ) = ∂ ν u ,m (x, τ ), x ∈ ∂Ω. (1.7)
We mention that the only other works that we are aware of dealing with the existence of solutions of (1.1) with variable order can be found in [START_REF] Kian | Equivalence of definitions of solutions for some class of fractional diffusion equations[END_REF][START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF] and only [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF] considered this problem with non-homogenous boundary condition. In Proposition 1.1, we extend the analysis of [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF] to more general class of Dirichlet boundary conditions of the form (1.4) and, in contrast to [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF] who considered solutions defined in terms of Laplace transform in time, we prove the unique existence of strong solutions of (1.1).

Applying Proposition 1.1, we introduce the following indicator function which is one of the key ingredient of the enclosure method. Definition 1.2. Define

I ,m (τ ) = ∂Ω ∂ ν u ,m (x, τ ) -τ -5 ∂ ν w 0 ,m (x, τ ) τ 5 w 0 ,m (x, τ ) dS(x), τ > 1. (1.8)
This indicator function can be computed from the data ∂ ν u g on ∂Ω over time interval ]0, ∞[ which is the Neumann data of the solution of (1.1) with g = g ,m . Applying Proposition 1.1, we can transform the indicator function (1.8) in the following way.

Let the function w ,m belongs to H 2 (Ω) and satisfies

   (∆ -τ α(x) )w ,m = 0, x ∈ Ω, w ,m (x, τ ) = w 0 ,m (x, τ ), x ∈ ∂Ω.
(1.9)

In view of (1.6), we have

w 0 ,m (x, τ ) = τ 5 g ,m (x, τ ), τ > 1, x ∈ ∂Ω
and (1.5) together with (1.9) yields

w ,m (x, τ ) = τ 5 u ,m (x, τ ), τ > 1, x ∈ Ω.
Applying this together with (1.7) to (1.8), we obtain the more familiar expression of the indicator function

I ,m (τ ) = ∂Ω (∂ ν w ,m (x, τ ) -∂ ν w 0 ,m (x, τ ))w 0 ,m (x, τ ) dS(x), τ > 1.
(1.10) Now we state the main result of this paper.

Fixing K = supp Ψ ,m , we get K =    B η , if = ext, B R 2 \ B R 1 , if = int. Besides we have dist (K , D) =    dist (p, D) -η, if = ext, R 1 -R D (p), if = int,
Theorem 1.1 Let T be an arbitrary positive number. We have

lim τ →∞ e τ α 0 2 T I ,m (τ ) =            0 if T < 2 dist (K , D), ∞ if α >> α 0 and T > 2dist (K , D), -∞ if α << α 0 and T > 2dist (K , D).
(1.11)

If (A.I) or (A.II) are satisfied, then there exists a positive number τ 0 such that, for all τ ≥ τ 0 |I ,m (τ )| > 0 and we have the one line formula

lim τ -→∞ τ -α 0 2 log |I ,m (τ )| = -2dist (K , D). (1.12)
Let us observe that Theorem 1.1 give several important information about the domain of variation D and the amplitude of variation h of the variable order α. Namely, formula (1.11) gives a target distinction and range estimate at the same time, that means one can distinguish whether α >> α 0 or α << α 0 together with T > 2 dist (K , D) or T < 2 dist (K , D) (see figure 1 for more detail) by using the asymptotic behavior of the indicator function as τ → ∞. Formula (1.12) gives us a direct way of extracting information about the geometry of D from the indicator function.

To the best of our knowledge, in Theorem 1.1 we obtain the first result of extraction of information about the variable order α from a single boundary measurement of the solution of (1.1). Indeed, the only other work treating this type of problem can be found in [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF] where the authors considered the problem of recovering the full knowledge of α itself from infinite boundary measurements. We give in this article an application of the enclosure method, considered so far mainly for inverse source or inverse obstacle problem [START_REF] Ikehata | Reconstruction of a source domain drom the Cauhy data[END_REF][START_REF] Ikehata | Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data[END_REF], to a new class of inverse obstacle problem, that is the problem of extracting information about the region of the jump of space dependent variable order of fractional time derivative in the governing equation from the background one.

This article is organized as follows. Section 2 is devoted to the proof of the result about the froward problem stated in Proposition 1.1 where we show the unique existence of strong solutions of (1.1) having some specific properties. In Section 3, we complete the proof of our main result stated in Theorem 1.1 by assuming Lemma 3.2 whose proof is postponed to Section 4. Finally, in Section 5 we give some additional remarks about our results with possible extension of our analysis.

Proof of Proposition 1.1

In all this proof C > 0 will be a constant independent of τ that may change from line to line.

2.1. Proof of (1.6). Let us first observe that for all complex τ ∈ C + := {z ∈ C : Re z > 0} the Fourier transform Fw 0 ,m in x of w 0 ,m is given by

Fw 0 ,m (ξ, τ ) = τ α 0 -1 FΨ ,m (ξ) |ξ| 2 + τ α 0 . We fix r = |1 + τ | > 1 and θ ∈ ] -π 2 , π 2 [ such that 1 + τ = re iθ . Using the fact that Ψ ,m ∈ L 2 (R 3 ), we get w 0 ,m ( • , 1 + τ ) 2 H 2 (R 3 ) = |1 + τ | 2(α 0 -1) R 3 |ξ| 2 + 1 ||ξ| 2 + (1 + τ ) α 0 | 2 |FΨ ,m (ξ)| 2 dξ ≤ |1 + τ | 2(α 0 -1) R 3 |ξ| 2 + 1 |ξ| 2 + r α 0 cos(α 0 θ) 2 |FΨ ,m (ξ)| 2 dξ ≤ |1 + τ | 2(α 0 -1) R 3 |ξ| 2 + 1 |ξ| 2 + cos( α 0 π 2 ) 2 |FΨ ,m (ξ)| 2 dξ ≤ C|1 + τ | 2(α 0 -1) R 3 |FΨ ,m (ξ)| 2 dξ ≤ C|1 + τ | 2(α 0 -1) Ψ ,m 2
L 2 (R 3
) . Here we have used the fact that cos( α 0 π

2 ) > 0 since α 0 ∈ ]0, 1[. Thus, we find

(1 + τ ) -5 w 0 ,m ( • , 1 + τ )| ∂Ω H 3 2 (∂Ω) ≤ C|1 + τ | -5 w 0 ,m ( • , 1 + τ ) H 2 (R 3 ) ≤ C|1 + τ | α 0 -6 , τ ∈ C + . (2.1)
This together with (1.4) yields e -t g ,m ∈ L ∞ (R + ; H 3 2 (∂Ω)). Moreover, using similar arguments as above, one can check that the map τ → w 0 ,m ( • , 1 + τ ) is holomorphic with respect to τ ∈ C + as a map taking values in H 2 (R 3 ). And it follows that the map τ → (1 + τ ) -5 w 0 ,m ( • , 1 + τ )| ∂Ω is holomorphic with respect to τ ∈ C + as a map taking values in H 3 2 (∂Ω). Therefore, applying [START_REF] Rudin | Real and complex analysis[END_REF] Theorem 19.2 and the note, we deduce that

g ,m (x, 1 + τ ) = e -t g ,m (x, τ ) = (1 + τ ) -5 w 0 ,m (x, 1 + τ ), x ∈ ∂Ω, τ ∈ C + .
This identity clearly implies (1.6).

2.2.

Proof of the unique existence of solutions of (1.1) with g = g ,m and (1.7).

For τ ∈ C + let v( • , τ ) be the solution of    ∆v(x, τ ) -τ α(x) v(x, τ ) = 0, x ∈ Ω, v(x, τ ) = g ,m (x, τ ), x ∈ ∂Ω. (2.2) One can split v(x, 1 + τ ) into two terms v(x, 1 + τ ) = (1 + τ ) -5 w 0 ,m (x, 1 + τ ) + z(x, 1 + τ ), x ∈ Ω, τ ∈ C + where z( • , 1 + τ ) solves    ∆z(x, 1 + τ ) -(1 + τ ) α(x) z(x, 1 + τ ) = -(1 + τ ) -5 G(x, 1 + τ ), x ∈ Ω, z(x, 1 + τ ) = 0, x ∈ ∂Ω . and G(x, 1 + τ ) = ∆w 0 ,m (x, 1 + τ ) -(1 + τ ) α(x) w 0 ,m (x, 1 + τ ),
x ∈ Ω. Therefore, fixing ∆ 0 the Laplacian with Dirichlet boundary condition acting L 2 (Ω), for all τ ∈ C + , we obtain [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF], we have

z( • , 1 + τ ) = (1 + τ ) -5 (-∆ 0 + (1 + τ ) α(x) ) -1 [∆w 0 ,m ( • , 1 + τ ) -(1 + τ ) α(x) w 0 ,m (1 + τ, • )]. Following Proposition 2.1 in
(-∆ 0 + (1 + τ ) α(x) ) -1 B(L 2 (Ω)) ≤ C|1 + τ | α 0 U -2α 0 L , where 0 < α 0 L = ess.inf x∈Ω α(x) ≤ ess.sup x∈Ω α(x) = α 0 U < 1. It follows that, for all v ∈ L 2 (Ω), we have (-∆ 0 + (1 + τ ) α(x) ) -1 v H 2 (Ω) ≤ C -∆ 0 (-∆ 0 + (1 + τ ) α(x) ) -1 v L 2 (Ω) ≤ C v -(1 + τ ) α(x) (-∆ 0 + (1 + τ ) α(x) ) -1 v L 2 (Ω) ≤ C 1 + |1 + τ | α 0 U (-∆ 0 + (1 + τ ) α(x) ) -1 B(L 2 (Ω)) v L 2 (Ω) ≤ C|1 + τ | 2α 0 U -2α 0 L v L 2 (Ω)
. From this estimate and (2.1), we obtain

z( • , 1 + τ ) H 2 (Ω) ≤ C|1 + τ | 2α 0 U -2α 0 L -5 w 0 ,m ( • , 1 + τ ) H 2 (R 3 ) +|1 + τ | α 0 U w 0 ,m ( • , 1 + τ ) L 2 (R 3 )
≤ C|1 + τ | 3α 0 U +α 0 -6 . This estimate implies that the solution v( • , 1 + τ ) of (2.2) satisfies the following estimate

v( • , 1 + τ ) H 2 (Ω) ≤ C|1 + τ | 3α 0 U +α 0 -6 , (2.3)
where we recall that 3α 0 U +α 0 -6 < -2. Besides, in view of Proposition 2.1 in [START_REF] Kian | On time-fractional diffusion equations with spacedependent variable order[END_REF], we know that the map τ → (-∆ 0 + (1 + τ ) α(x) ) -1 is holomorphic with respect to τ ∈ C + as a map taking values in B(L 2 (Ω)) and from the above properties we deduce that τ → v( • , 1 + τ ) is holomorphic with respect to τ ∈ C + as a map taking values in H 2 (Ω).

Thus, applying again Theorem 19.2 and note in [START_REF] Rudin | Real and complex analysis[END_REF] as well as estimate (2.3), we deduce that for

u ,m (x, t) = e t 2π ∞ -∞ e its v(x, 1 + is) ds, x ∈ Ω, t ∈ ]0, ∞[, we have e -t u ,m ∈ W 1,∞ (R + ; H 2 (Ω)) ∩ C 1 ([0, ∞[; L 2 (Ω)) ∩ C([0, ∞[; H 2 (Ω)) and u ,m (x, 1 + τ ) = e -t u ,m (x, τ ) = v(x, 1 + τ ), τ ∈ C + , x ∈ Ω.
This proves that for all complex τ satisfying Re τ > 1, u ,m (•, τ ) solves the boundary value problem (1.5). Therefore, in order to complete the proof of the proposition we need to prove that u ,m is the unique solution of (1.1) for g = g ,m satisfying e -t u ,m ∈ W 1,∞ (R + ; H 2 (Ω)). Let us show first that u ,m solves (1.1) for g = g ,m . Note that, in view of estimate (1.7), we have

u ,m (x, 0) = 1 2π ∞ -∞ v(x, 1 + is) ds, x ∈ Ω.
On the other hand, for all R > 0, fixing the contour C R = {1 + Re iθ : θ ∈ [-π/2, π/2]} and applying the residue theorem we deduce that

1 2π R -R v(x, 1 + is) ds = 1 2iπ C R v(x, 1 + τ ) dτ x ∈ Ω.
Now sending R → ∞ and applying estimate (2.3), we get

u ,m (•, 0) L 2 (Ω) = 1 2π ∞ -∞ v(•, 1 + is) ds L 2 (Ω) ≤ lim sup R→∞ 1 2iπ C R v(•, 1 + τ ) dτ L 2 (Ω) ≤ lim sup R→∞ 1 2π π/2 -π/2 R v(•, 1 + Re iθ ) L 2 (Ω) dθ = 0.
It follows that u ,m (•, 0) = 0. Using the fact that e -t u ,m ∈ W 1,∞ (R + ; H 2 (Ω)) and applying the properties of fractional derivative (see e.g. [36, pp. 80]), for all complex τ satisfying Re τ > 1 we deduce that

τ α u ,m (•, τ ) -τ α-1 u ,m (•, 0) = ∂ α t u ,m (•, τ ).
Combining this with the fact that u ,m (•, 0) = 0, we deduce that for y = ∂ α t u ,m -∆u ,m and for all complex τ satisfying Re τ > 1, we have

ŷ(x, τ ) = τ α u ,m (x, τ ) -∆u ,m (x, τ ) = 0, x ∈ Ω.
Combining this with the uniqueness of the Laplace transform in time, we deduce that ∂ α t u ,m -∆u ,m = 0 in Ω × R + . Finally, (1.5) implies that for all complex τ satisfying Re τ > 1, we have u ,m (x, τ ) = g ,m (x, τ ), x ∈ ∂Ω and applying again the uniqueness of Laplace transform in time we deduce that u ,m = g ,m on ∂Ω × R + . Thus, u ,m solves (1.1) for g = g ,m . The uniqueness of solutions u of (1.1), with g = g ,m , satisfying e -t u ,m ∈ W 1,∞ (R + ; H 2 (Ω)) is the consequence of the uniqueness of the Laplace transform and the unique solvability of problem (1.5) for any complex τ satisfying Re τ > 1. Indeed, fix u a solution of (1.1) with g = 0 and satisfying e -t u ∈ W 1,∞ (R + ; H 2 (Ω)). Then, applying the Laplace transform in time to (1.1), we deduce that for any complex τ satisfying Re τ > 1, û(•, τ ) is well defined and it solves (1.5) with g ,m = 0. Then the uniqueness of solutions of (1.5) implies that for all complex τ satisfying Re τ > 1, û(•, τ ) = 0 and the uniqueness of the Laplace transform in time implies that u = 0.

Finally, since e -t u ,m ∈ L ∞ (R + ; H 2 (Ω)), for all τ > 1, we have

∞ 0 e -τ t ∂ ν u ,m H 1 2 (∂Ω) dt ≤ C ∞ 0 e -(τ -1)t e -t u ,m H 2 (Ω) dt ≤ C e -t u ,m L ∞ (R + ;H 2 (Ω)) < ∞.
Therefore, for a.e. x ∈ ∂Ω and all τ > 1, we have

∂ ν u ,m (x, τ ) = ∞ 0 e -τ t ∂ ν u ,m (x, t) dt = ∂ ν ∞ 0 e -τ t u ,m (x, t) dt = ∂ ν u ,m (x, τ ).
From this identity, we deduce (1.7).

Proof of Theorem 1.1

First we describe a basic system of inequalities which is a consequence of the expression (1.10) and the governing equations of (1.3) and (1.9) for w 0 ,m and w ,m , respectively. Lemma 3.1. We have, for all τ > 1

I ,m (τ ) ≥ Ω τ α 0 τ α(x) (τ α(x) -τ α 0 )(w 0 ,m ) 2 dx (3.1)
and

I ,m (τ ) ≤ Ω (τ α(x) -τ α 0 )(w 0 ,m ) 2 dx. (3.2)
We omit to describe the proof since the idea of the derivation is well known in the framework of the enclosure method. See [START_REF] Ikehata | Size estimation of inclusion[END_REF] and Proposition 4.1 in [START_REF] Ikehata | On finding an obstacle embedded in the rough background medium via the enclosure method in the time domain[END_REF].

It follows from (3.1) and (3.2) that (0) We have, for all τ > 1

|I ,m (τ )| ≤ τ α 0 (τ h L ∞ (D) + 1) D (w 0 ,m (x)) 2 dx. (3.3)
Note that the precise values of the power of τ is not important. (i) if α >> α 0 , then

I ,m (τ ) ≥ τ α 0 D (τ C dist (x,∂D) γ -1)(w 0 ,m (x)) 2 dx. (3.4) (ii) if α << α 0 , then I ,m (τ ) ≤ - τ α 0 τ C sup x∈D dist(x,∂D) γ D (τ C dist (x,∂D) γ -1)(w 0 ,m (x)) 2 dx. (3.5)
It is clear that w 0 ,m has the expression

w 0 ,m (x, τ ) = τ α 0 -1 v ,m (x; β)| β=α 0 , (3.6)
where the function v ,m (x; β) of x ∈ R 3 with β > 0 takes the form Lemma 3.2. Let β > 0, γ ≥ 0, C > 0 and m ≥ 0 be an integer. Then, there exist positive numbers τ 0 > 1, C 1 , C 2 and λ ∈ R such that, for all τ ≥ τ 0 we have

v ,m (x; β) =                1 4π Bη (η 2 -|y -p| 2 ) m e -τ β 2 |x-y| |x -y| dy, if = ext, 1 4π B R 2 \B R 1 (R 2 2 -|y -p| 2 ) m (|y -p| 2 -R 2
τ λ e 2τ β 2 dist (K ,D) D (τ Cdist (x,∂D) γ -1) v ,m (x; β) 2 dx ≥ C 1 (3.7) and D v ,m (x; β) 2 dx ≤ C 2 τ -β(m+2) e -2τ β 2 dist (K ,D) . (3.8)
We postpone the proof of Lemma 3.2 to the next section. We continue to prove Theorem 1.1.

First, following (3.3), (3.6) and (3.8), we find

|I ,m (τ )| ≤ τ λ 1 e -2τ α 0 2 dist (K ,D) , (3.9)
where

λ 1 = α 0 + h L ∞ (D) + 2(α 0 -1) -β(m + 2).
From this we obtain lim τ →∞ e α 0 2 T I ,m (τ ) = 0 for T < 2dist (K , D). Next consider the case when α >> α 0 . From (3.4), (3.6) and (3.7) one has

I ,m (τ ) ≥ Cτ λ 2 e -2τ α 0 2 dist (K ,D) , (3.10)
where C is a positive constant independent of τ and

λ 2 = α 0 + 2(α 0 -1) -λ. ( 3

.11)

A combination (3.9) and (3.10) yields (1.12) provided α >> α 0 . Besides, one has

e τ α 0 2 T I ,m (τ ) ≥ C 1 τ λ 2 e τ α 0 2 (T -2 dist (K ,D)) . ( 3
I ,m (τ ) ≤ -C τ λ 3 e -2τ α 0 2 dist (K ,D) ,
(3.13) where C is a positive constant independent of τ and

λ 3 = α 0 -C sup x∈D dist(x, D) γ + 2(α 0 -1) -λ. (3.14)
Note that C above is the same one in the condition (A.II). A combination (3.9) and (3.13) yields (1.12) provided α << α 0 . Besides, one has

e τ α 0 2 T I ,m (τ ) ≤ -C τ λ 3 e τ α 0 2 (T -2 dist (K ,D)) . ( 3
v ext,m (x) = η 2(1+m) • e -τ |x-p| |x -p| a m (τ ), (4.1)
where

a m (τ ) = 1 τ 1 0 s(1 -s 2 ) m sinh(ητ s) ds. (ii) For |x -p| < R 1 , we get v int,m (x) = 2 • sinh τ |x -p| |x -p| b m (τ ), (4.2)
where

b m (τ ) = 1 τ R 2 R 1 s(R 2 2 -s 2 ) m (s 2 -R 2 1
) m e -sτ ds.

Besides, by Theorem 7.1 on p.81 in [START_REF] Olver | Asymptotics and special functions[END_REF], we have, as τ → ∞

a m (τ ) ∼ η 2 m-1 m! e τ η (τ η) m+2 (4.3) and b m (τ ) ∼ 2 m m!R m+1 1 (R 2 2 -R 2 1 ) m e -R 1 τ τ m+2 . (4.4)
4.1. The case when = ext. A combination of (4.1) and (4.3) gives, for all

x ∈ R 3 \ B η v ext,m (x) 2 ≥ C 2 2 τ -β(m+2) e -2τ β 2 (|x-p|-η) |x -p| 2 (4.5) and v ext,m (x) 2 ≤ C 2 3 τ -β(m+2) e -2τ β 2 (|x-p|-η) |x -p| 2 , (4.6)
where C 2 and C 3 are positive constants independent of τ . From (4.6) and the fact that inf x∈D |x -p| -η = dist (B η , D), we obtain (3.8) for = ext. * Therein the case m = 0 is excluded. However, the proof still works also for the case.

From (4.5) one gets

τ β (m+2) D (τ C dist (x,∂D) γ -1)v ext,m (x) 2 dx ≥ C 2 2 D (τ C dist (x,∂D) γ -1) e -2τ β 2 (|x-p|-η) |x -p| 2 dx ≥ C 2 sup x∈D |x -p| 2 D (τ C dist (x,∂D) γ -1) e -2τ β 2 (|x-p|-η) dx. (4.7)
First consider the case when γ = 0. By Lemma A.2 in [START_REF] Ikehata | On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body[END_REF] under the assumption that ∂D is C 2 we have (τ

β 2 ) 2 e 2τ β 2 dist (D,Bη) D e -2τ β 2 (|x-p|-η) dx ≥ C .
Thus, from (4.7) one gets

τ β (m+2)+β-C e 2τ β 2 dist (D,Bη) D (τ C -1) v ext,m (x) 2 dx ≥ C 2 sup x∈D |x -p| 2 C . (4.8)
Next consider the case when γ > 0. We make a reduction to a simple geometry along the lines of the proof of Lemma A.1 in [START_REF] Ikehata | On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body[END_REF]. Let τ ≥ 1. Choose a point q ∈ ∂D such that |q -p| = d ∂D (p). Since ∂D is C 2 , one can find an open ball B with radius δ and centered at q -δν q such that B ⊂ D and ∂B ∩ ∂D = {q}. Then dist (B , B η ) = dist (D, B η ) and dist (x, ∂B ) ≤ dist (x, ∂D) for all x ∈ B . Thus, for all x ∈ B we have τ Cdist (x,∂D) γ ≥ τ Cdist (x,∂B ) γ ≥ 1. Therefore, it suffices to prove (3.7) in the case when D = B .

Set d = d ∂D (p). Let B be the open ball with radius d + δ centered at p. As described in the proof of Lemma A.1 in [START_REF] Ikehata | On finding a cavity in a thermoelastic body using a single displacement measurement over a finite time interval on the surface of the body[END_REF], we have the global parametrization of B ∩ B :

B ∩ B = {Υ(s, r, θ) | 0 < s < δ, 0 < r < (d + s) sin θ(s), θ ∈ [0, 2π[},
where † Υ(s, r, θ) = p -(d + s) 2 -r 2 ν q + r(cos θb + sin θc); b and c are unit vectors chosen in such a way that b

• c = 0 and b × c = -ν q ; θ(s) ∈]0, π 2 [ the unique solution of cos θ = (d + δ) 2 + (d + s) 2 -δ 2 2(d + δ)(d + s) .
We have det Υ (s, r, θ) = r(d + s)

(d + s) 2 -r 2 † One can write Υ(s, r, θ) = p + (d + s)ω, where ω = 1 d + s -(d + s) 2 -r 2 ν q + r(cos θb + sin θc ∈ S 2 . and dist (x, ∂B ) = δ -|x -(p -(d + δ)ν q )| = δ - d + δ -(d + s) 2 -r 2 2 + r 2 ,
where x = Υ(s, r, θ). The change of variables x = Υ(s, r, θ) yields, for all τ ≥ 1

B (τ C dist (x,∂B ) γ -1) e -2τ β 2 (|x-p|-η) dx ≥ B ∩B (τ C (δ-|x-(p-(d+δ)νq)|) γ -1) e -2τ β 2 (|x-p|-η) dx = δ 0 ds (d+s) sin θ(s) 0 dr 2π 0 dθ r(d + s) (d + s) 2 -r 2 e -2τ β 2 (d+s-η) (τ Cf (s,r) -1) = 2πe -2τ β 2 (d-η) δ 0 ds (d+s) sin θ(s) 0 dr r(d + s) (d + s) 2 -r 2 e -2τ β 2 s (τ Cf (s,r) - 1), where f (s
, r) = δ - d + δ -(d + s) 2 -r 2 2 + r 2 γ .
Making the change of a variable given by r = (d + s) sin ξ, ξ ∈ [0, θ(s)], we have

I ≡ δ 0 ds (d+s) sin θ(s) 0 dr r(d + s) (d + s) 2 -r 2 e -2τ β 2 s (τ Cf (s,r) -1) = δ 0 (d + s) 2 e -2τ β 2 s ds θ(s)

0

(τ Cf (s,(d+s) sin ξ) -1) sin ξ dξ.

Using (d + δ) 2 + (d + s) 2 = δ 2 + 2(d + s)(d + δ) cos θ(s),
we have

f (s, (d + s) sin ξ) = δ -((d + δ) -(d + s) cos ξ) 2 + (d + s) 2 sin 2 ξ γ = δ -(d + δ) 2 + (d + s) 2 -2(d + δ)(d + s) cos ξ γ = δ -δ 2 + 2(d + δ)(d + s) cos θ(s) -2(d + δ)(d + s) cos ξ γ = δ -δ 2 -2(d + δ)(d + s)(cos ξ -cos θ(s)) γ = 2(d + δ)(d + s)(cos ξ -cos θ(s)) δ + δ 2 -2(d + δ)(d + s)(cos ξ -cos θ(s)) γ ≥ d + δ δ (d + s)(cos ξ -cos θ(s)) γ ≥ d(d + δ) δ γ (cos ξ -cos θ(s)) γ .
This yields θ(s) 0 (τ Cf (s,(d+s) sin ξ) -1) sin ξ dξ ≥ θ(s) 0 (τ C (cos ξ-cos θ(s)) γ -1) sin ξ dξ

= 1-cos θ(s) 0 (τ C σ γ -1)dσ,
where

C = C d(d + δ) δ γ .
Here we have

1 -cos θ(s) = s(2δ -s) 2(d + s)(d + δ) ≥ δs 2(d + δ) 2 .

Thus one gets

(τ C σ γ -1)dσ = - 1 2τ β 2 [e -2τ β 2 s C s 0 (τ C σ γ -1)dσ] s=δ s=0 -C δ 0 e -2τ β 2 s (τ C 3 s γ -1)ds = C 2τ β 2 δ 0 e -2τ β 2 s (τ C 3 s γ -1)ds - e -2τ β 2 δ 2τ β 2 C δ 0 (τ C σ γ -1)dσ. Set C 3 = C (C ) γ .
Since we have

C δ 0 (τ C σ γ -1)dσ = O(τ C 3 δ γ ), one gets 2τ β 2 I ≥ C δ 0 e -2τ β 2 s (τ C 3 s γ -1)ds + O(τ -∞ ).
For simplicity, we write C 3 = C. We have

δ 0 e -2τ β 2 s (τ C 3 s γ -1)ds = - 1 2τ β 2 δ 0 (e -2τ β 2 s ) (τ Cs γ -1)ds = Cγ 2 τ -β 2 log τ δ 0 e -2τ β 2 s τ Cs γ s γ-1 ds + O(τ -β 2 e -2τ β 2 δ (τ Cδ γ -1)) = Cγ 2 τ -β 2 log τ δ 0 e -2τ β 2 s τ Cs γ s γ-1 ds + O(τ -∞ ) = Cγ 2 τ -β 2 log τ K(τ ) + O(τ -∞ ) ,
where

K(τ ) ≡ δ 0 e -2τ
β 2 s τ Cs γ s γ-1 ds.

Let τ ≥ 1. Since τ Cs γ ≥ 1, we have

K p (τ ) ≥ 2 -γ τ -β 2 γ 2τ β 2 δ 0 e -t t γ-1 dt. This yields lim inf τ -→∞ τ β 2 γ K(τ ) ≥ 2 -γ ∞ 0 e -t t γ-1 dt = Γ(γ) 2 γ . Therefore we obtain lim inf τ -→∞ τ β 2 (γ+1) (log τ ) -1 δ 0 e -2τ β 2 (τ C 3 s γ -1)ds ≥ C 3 Γ(γ + 1) 2 γ+1
, Thus, we obtain lim inf

τ -→∞ 2τ β 2 (γ+2) (log τ ) -1 I ≥ C C 3 Γ(γ + 1) 2 γ+1 .
Summing up, from this together with (4.8) we see that the λ on (3.7) should be

λ =      β(m + 3) -C if γ = 0, β(m + 2) + β 2 (γ + 2) + if 0 < γ,
where is an arbitrary positive number.

4.2.

The case when = int. A combination of (4.2) and (4.4) gives, for all

x ∈ B R 1 v int,m (x) 2 ≥ C 2 3 τ -2(m+2) sinh τ |x -p| |x -p| 2 e -2R 1 τ (4.9)
and 2 and centered at q -δν q such that B ⊂ D and

v int,m (x) 2 ≤ C 2 4 τ -2(m+2) sinh τ |x -p| |x -p| 2 e -2R 1 τ , (4.10
∂B ∩ ∂D = {q}. Then dist (B R 2 \ B R 1 , B ) = R 1 -R D (p) = dist (B R 2 \ B R 1 , D
) and dist (x, ∂B ) ≤ dist (x, ∂D) for all x ∈ B . Thus, for all x ∈ B we have τ Cdist (x,∂D) γ ≥ τ Cdist (x,∂B ) γ ≥ 1. Therefore it suffices to prove (3.7) in the case when D = B . Up to this point, it is the same as above.

Set ρ = R D (p) and let 0 < δ < δ and B be the open ball with radius ρ -δ centered at p. We make use of the parametrization of the set B \ B which is essentially same as that used in [START_REF] Ikehata | The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data[END_REF]:

B \ B = ∪ 0<s<δ {p + (ρ -s)ω | ω ∈ S(s)} , where S(s) = ω ∈ S 2 | ω • ν q > cos θ(s) and the θ(s) ∈ ]0, π 2 [ is the unique solution of the equation cos θ = (ρ -δ ) 2 + (ρ -s) 2 -δ 2 2(ρ -δ )(ρ -s) .
Choose two linearly independent unit vectors b and c in such a way that b • c = 0 and b × c = ν q . Then we have the expression

B \ B = {Υ(s, r, θ) | 0 < s < δ , 0 ≤ r < (ρ -s) sin θ(s), 0 ≤ θ < 2π} , where Υ (s, r, θ) = p + (ρ -s) 2 -r 2 ν q + r(cos θb + sin θc). We have det Υ (s, r, θ) = - r(ρ -s) (ρ -s) 2 -r 2 and dist (x, ∂B ) = δ -|x -(p + (ρ -δ )ν q )| = δ - ρ -δ -(ρ -s) 2 -r 2 2 + r 2 ,
where x = Υ(s, r, θ). The change of variables x = Υ(s, r, θ) yields, for all τ ≥ 1

B (τ C dist (x,∂B ) γ -1) sinh τ |x -p| |x -p| 2 e -2R 1 τ dx ≥ B \B (τ C (δ -|x-(p+(ρ-δ )νq)|) γ -1) sinh τ |x -p| |x -p| 2 e -2R 1 τ dx = δ 0 ds (ρ-s) sin θ(s) 0 dr 2π 0 dθ r(ρ -s) (ρ -s) 2 -r 2 τ (ρ -s) ρ -s 2 e -2R 1 τ (τ Cf (s,r) -1) = 2πe -2(R 1 -ρ)τ δ 0 ds (ρ-s) sin θ(s) 0 dr r (ρ -s) 2 -r 2 e -2ρτ sinh 2 τ (ρ -s) ρ -s (τ Cf (s,r) -1), (4.11) where f (s, r) = δ - ρ -δ -(ρ -s) 2 -r 2 2 + r 2 γ .
Making the change of variable r = (ρ -s) sin ξ, 0 < ξ < θ(s),

we have

I ≡ δ 0 ds (ρ-s) sin θ(s) 0 dr r (ρ -s) 2 -r 2 e -2ρτ sinh 2 τ (ρ -s) ρ -s (τ Cf (s,r) -1) = δ 0 dse -2ρτ sinh 2 τ (ρ -s) θ(s) 0
(τ Cf (s,(ρ-s) sin ξ) -1) sin ξ dξ.

Here using

(ρ -δ ) 2 + (ρ -s) 2 = δ 2 + 2(ρ -s)(ρ -δ ) cos θ(s),
we have

f (s, (ρ -s) sin ξ) = δ -((ρ -δ ) -(ρ -s) cos ξ) 2 + (ρ -s) 2 sin 2 ξ γ = δ -(ρ -δ ) 2 + (ρ -s) 2 -2(ρ -δ )(ρ -s) cos ξ γ = δ -δ 2 + 2(ρ -δ )(ρ -s) cos θ(s) -2(ρ -δ )(ρ -s) cos ξ γ = δ -δ 2 -2(ρ -δ )(ρ -s)(cos ξ -cos θ(s)) γ = 2(ρ -δ )(ρ -s)(cos ξ -cos θ(s)) δ + δ 2 -2(ρ -δ )(ρ -s)(cos ξ -cos θ(s)) γ ≥ ρ -δ δ (ρ -s)(cos ξ -cos θ(s)) γ ≥ (ρ -δ ) 2 δ γ (cos ξ -cos θ(s)) γ .
This yields θ(s) 0 (τ Cf (s,(ρ-s) sin ξ) -1) sin ξ dξ ≥ θ(s) 0 (τ C (cos ξ-cos θ(s)) γ -1) sin ξ dξ

= 1-cos θ(s) 0 (τ C σ γ -1)dσ,
where

C = C (ρ -δ ) 2 δ γ .
Here we have

1 -cos θ(s) = s(2δ -s) 2(ρ -s)(ρ -δ ) ≥ δ s 2ρ(ρ -δ ) .
Thus one gets

θ(s) 0 (τ Cf (s,(ρ-s) sin ξ) -1) sin ξ dξ ≥ C s 0 (τ C σ γ -1)dσ, where C = δ 2ρ(ρ -δ )
.

Therefore, we obtain

I ≥ e -2ρτ δ 0 ds sinh 2 τ (ρ -s) C s 0 (τ C σ γ -1)dσ. (4.12)
Since we have

sinh 2 x dx = e 2x -e -2x 8 - x 2 ,
integration by parts yields It is easy to see that the proof of Proposition 1.1 still works also for this case since the concrete form of Φ is not used therein.

e -2ρτ δ 0 ds sinh 2 τ (ρ -s) C s 0 (τ C σ γ -1)dσ = -τ -1 e -2ρτ    e 2τ (ρ-s) -e -2τ (ρ-s) 8 - τ (ρ -s) 2 C s 0 (τ C σ γ -1)dσ s=δ s=0 - δ 0 e 2τ (ρ-s) -e -2τ (ρ-s) 8 - τ (ρ -s) 2 C (τ C s γ -1) ds = -τ -1 e -2ρτ
Then, one can define a new indication function by trivial replacements of w 0 ,m and u ,m on (1.8). We see that Lemma 3.1 where the old indicator function replaced with this new one, is valid by replacing w 0 ,m on (3.1) and (3.2) with w 0 ,m 1 ,m 2 . Besides, by virtue of assumption (5.1), for τ > 1, we have w 0

,m 1 (x) ≤ w 0 ,m 1 ,m 2 (x) ≤ w 0 ,m 2 (x) a.e.x ∈ R 3 . Thus, the estimates for the new indicator function corresponding to (3.3), (3.4)

  where dist(p, D) = inf x∈D |x -p| and R D (p) = sup x∈D |x -p|. Thus knowing the value of dist (K , D) is equivalent to that of dist (p, D)/R D (p) if = ext/int. Note that the sphere |x -p| = dist (p, D) is the largest one whose exterior contains D: the sphere |x -p| = R D (p) is the smallest one whose interior contains D (see Figure 1 for more detail).

Figure 1 .

 1 Figure 1. The sets Ω, D and dist (K , D).

1 ) m e -τ β 2

 12 |x-y| |x -y| dy, if = int and the point p denotes the center of B η when = ext; the common center of B R 1 and B R 2 when = int.

. 12 )This yields lim τ →∞ e α 0 2 T

 122 I ,m (τ ) = ∞ for T > 2dist (K , D) provided α >> α 0 .The proof in the case α << α 0 can be done similarly as follows. From (3.5), (3.6) and (3.7) we obtain

. 15 )This yields lim τ →∞ e α 0 2 T

 152 I ,m (τ ) = -∞ for T > 2dist (K , D) provided α << α 0 .This completes the proof of Theorem 1.1.

4 . 2 Set τ = τ β 2

 422 Proof of Lemma 3.and v ,m (x; β) = v ,m (x). By Proposition 3.1 in [20] * we obtain the following: (i) For |x -p| > η, we have

  Cf (s,(d+s) sin ξ) -1) sin ξ dξ≥ C s 0 (τ C σ γ -1)dσ, where C = δ 2(d + δ) 2 .

) where C 3

 3 and C 4 are positive constants independent of τ . From (4.10) and R 1 -R D (p) = dist (B R 2 \ B R 1 , D) we obtain (2.16) for = int. We make a reduction to a simple geometry along the lines of the proof of Lemma 4.3 in [19]. Let τ ≥ 1. The D is contained in the open ball B R D (p) centered at p with radius R D (p) and B R D (p) ⊂ B R 1 . Choose a point q ∈ ∂D such that |q -p| = R D (p). Since ∂D is C 2 , one can find an open ball B with radius δ < ρ

0 ee 8 δ 0 e 0 e 15 ) 2 ( 1 δ 0 e 3 Γ 2 ( 5 .

 080015210325 e 2τ (ρ-δ ) -e -2τ (ρ-δ ) σ γ -1)dσ +τ -1 e -2ρτ C δ 2τ (ρ-s) -e -2τ (ρ-s) 8 -τ (ρ -s) 2 (τ C s γ -1) ds= O(τ -1 e -2τ δ + τ -1 e -2τ (2ρ-δ ) + e -2ρτ ) -2τ s (τ C s γ -1) ds + O(τ -1 e -2τ (2ρ-δ ) + e -2τ ρ ) σ γ -1)dσ = O(τ C 4 ) and δ 0 (τ C s γ -1)ds = O(τ C 5 ),whereC 4 = C (C δ ) γ , C 5 = C (δ ) γ . 20Therefore from this together with (4.12) and (4.13) we obtainI ≥ τ -1 C -2τ s (τ C s γ -1) ds + O(τ -∞ ). (4.14)Consider the case when γ = 0. From (4.14) we haveI ≥ τ -1 C 8 (τ C -1) δ -2τ s ds + O(τ -∞ )Next let γ > 0. It follows from the case when = ext and γ > 0 one has lim infτ -→∞ τ β γ+1) (log τ ) --2τ s (τ C s γ -1)ds ≥ C γ+2) (log τ ) -1 I ≥ C C 3 Γ(γ + 1) 2 γ+1 . (4.16) Noting R 1 -ρ = dist (K int , D), from (4.9), (4.11), (4.15) and (4.16) we see that the λ on (2) + if 0 < γ, where is an arbitrary positive number. Some additional remark (1) It seems that the growth order of the absolute value of the function e α 0 2 T I ,m (τ ) in Theorem 1.1 as τ → ∞ for T > 2dist (K , D) becomes worse if γ or m is large. See (3.12) with (3.11), (3.15) with (3.14) and λ with β = α 0 in the end of Subsection 4.1 in the case = ext and Subsection 4.2 in the case = int. (2) Replace g ,m in Proposition 1.1 with g ,m 1 ,m 2 given byg ,m 1 ,m 2 (x, t) = e t 2π ∞ -∞ e its (1 + is) -5 w 0 ,m 1 ,m 2 (x, 1 + is)ds, x ∈ ∂Ω, t ∈ [0, ∞[, where the function w 0 ,m 1 ,m 2 ∈ H 2 (R 3 ) is the unique solution of the equation (∆ -τ α 0 )w 0 ,m 1 ,m 2 + τ α 0 -1 Φ (x) = 0, x ∈ R3 , and Φ an arbitrary fixed measurable function of x ∈ R 3 such that C -1 Ψ ,m 1 (x) ≤ Φ (x) ≤ C Ψ ,m 2 (x) a.e. x ∈ R 3 (5.1) with some m 1 , m 2 = 0, 1, • • • and C > 0.

  and (3.5) are valid under the replacements: m of W 0 ,m on (3.3) with m 2 and (3.4) and (3.5) with m 1 . Hereafter the proof of Theorem 1.1 works also for the new indicator function and one gets the corresponding result to Theorem 1.1.

(∂Ω)) and, for all τ > 1, e -τ t u ∈ L 1 (R + ; L 2 (Ω)). Moreover, we will show

Acknowledgments

MI was partially supported by the Grant-in-Aid for Scientific Research (C)(No. 17K05331) and (B)(No. 18H01126) of Japan Society for the Promotion of Science. The work of YK was partially supported by the French National Research Agency ANR (project Multi-Onde) grant ANR-17-CE40-0029.