Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients

We consider damped $s$-fractional Klein--Gordon equations on $\mathbb{R}^d$, where $s$ denotes the order of the fractional Laplacian. In the one-dimensional case $d = 1$, Green (2020) established that the exponential decay for $s \geq 2$ and the polynomial decay of order $s/(4-2s)$ hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the $o(1)$ energy decay is also equivalent to these conditions in the case $d=1$. Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the $o(1)$ decay, and the thickness of the damping coefficient are equivalent for $s \geq 2$. In addition, we also prove that the exponential decay holds for $0<s<2$ if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition.

We recast the equation (1.1) as an abstract first-order equation for U = (u, u t ): then A γ generates a C 0 -semigroup (e tAγ ) t≥0 on H s/2 (R d ) × L 2 (R d ) (see [4]).Here the Sobolev space H r (R d ) is defined by In this paper, we discuss the decay rate of the energy E(t) := e tAγ (u(0), u t (0)) By standard calculus, we have E(t) = E(0) if γ ≡ 0 and the exponential energy decay if γ ≡ C > 0. In recent works, the intermediate case, that is, the case that γ = 0 on a large set is studied: Definition 1.1.We say that Ω ⊂ R d satisfies the Geometric Control Condition (GCC) if there exist L > 0 and 0 < c ≤ 1 such that for any line segments l ∈ R d of length L, the inequality holds, where H 1 denotes the one-dimensional Hausdorff measure.
Burq and Joly [2] proved that if γ is uniformly continuous and {γ ≥ ε} satisfies (GCC) for some ε > 0, then we have the exponential energy decay in the nonfractional case s = 2.After that, Malhi and Stanislavova [6] pointed out that (GCC) is also necessary for the exponential decay in the one-dimensional case d = 1: Then the following conditions are equivalent: (1.3)There exists ε > 0 such that the upper level set {γ ≥ ε} satisfies (GCC).
In another paper [7], Malhi and Stanislavova introduced the fractional equation (1.1) and showed that if γ is periodic, continuous and not identically zero, then we have the exponential decay for any s ≥ 2 and the polynomial decay of order s/(4 − 2s) for any 0 < s < 2 in the case d = 1.
Remark.Nonzero periodic functions satisfy (GCC) in the case d = 1, but it is not true in the higher-dimensional case d ≥ 2. Wunsch [10] showed that continuous periodic damping gives the polynomial energy decay of order 1/2 for the nonfractional equation in the case d ≥ 2. In addition, recently another proof and an extension to fractional equations of Wunsch's result were obtained by Täufer [9] and Suzuki [8], respectively.Note that these results for periodic damping are established by reducing to estimates on the torus T d .Indeed, there are numerous studies on bounded domains; see references in [2] and [3], for example.
Green [4] improved results of Malhi and Stanislavova as follows:

Then the following conditions are equivalent:
1 To be precise, the exponential decay estimate given in [6, Theorem 1] is a little weaker: However, this is because the Gearhart-Prüss theorem in their paper ([6, Theorem 2]) is stated incorrectly.Using the theorem correctly (see Theorem 3.1), one can obtain the exponential decay estimate E(t) ≤ C exp(−ωt)E(0) as in (1.4).
In comparison with the result of [7], which states that (1.6) holds if γ is periodic, continuous and not identically zero, Theorem 1.3 refines this result by giving a necessary and sufficient condition for (1.6).Furthermore, Theorem 1.3 also improves the (1.3) ⇐⇒ (1.4) part of Theorem 1.2 by extending it to fractional equations and removing the continuity of γ, but on the other hand, it lacks the (1.5) =⇒ (1.3), (1.4) part.One of our goal is to recover this part for fractional equations: Then the following conditions are equivalent: (1.3)There exists ε > 0 such that the upper level set {γ ≥ ε} satisfies (GCC).
(1.6)There exist C, ω > 0 such that whenever (u(0), We also give the following result, which says that we cannot expect the exponential decay for 0 < s < 2 under (GCC).
Note that the "if" part easily follows by reducing to the constant damping case, so we will prove the "only if" part.Furthermore, we extend Theorem 1.4 to the higher-dimensional case d ≥ 2 using a notion of thickness, which is equivalent to (GCC) in the case d = 1: (1.8)There exists ε > 0 such that the upper level set {γ ≥ ε} is thick.
The implication (1.8) =⇒ (1.9) is a generalization of the result given by Burq and Joly [2].They established (1.9) under the so-called network control condition, which is stronger than (1.8).Also, similarly to the case d = 1, the condition (1.8) is equivalent to that there exists R > 0 such that Finally, we explain the organization of this paper.In Sections 2, 3, and 4, we will give proofs of Theorems 1.4, 1.5, and 1.7, respectively.To prove these theorems, we use a kind of uncertainty principle and results of the C 0 semigroup theory.

Proof of Theorem 1.4
To prove this theorem, we use the following result by Batty, Borichev, and Tomilov [1]: Let A be a generator of a bounded C 0 -semigroup (e tA ) t≥0 on a Banach space X, and λ ∈ ρ(A).Then the following are equivalent: In the case A = A γ , for λ ∈ ρ(A γ ), the map (λ  A γ ).This implies that for each λ ∈ R, there exists some c 0 > 0 such that Now we consider the case λ = 1.Let u ∈ H s/2 (R) satisfy supp u ⊂ [−D, D] for some D > 0, which is chosen later.For such u, we have Hence, taking D > 0 small enough, we get some c > 0 such that for each a ∈ R, so that f a (ξ) = e iaξ f (ξ).Then, for each a ∈ R and R > 0, we have The second integral goes to 0 as R → +∞ since γ is bounded and |f a | 2 is integrable, and this convergence is uniform with respect to a. Furthermore, for the first integral, we have

Proof of Theorem 1.5
This section is based on the proof of Theorem 2 in Green [4].To prove this theorem, we use the classical semigroup result by Gearhart, Prüss, and Huang: Theorem 3.1 (Gearhart-Prüss-Huang).Let X be a complex Hilbert space and let (e tA ) t≥0 be a bounded C 0 -semigroup on X with infinitesimal generator A. Then there exist C, ω > 0 such that e tA ≤ C exp(−ωt) holds for any t ≥ 0 if and only if iR ⊂ ρ(A) and sup λ∈R (iλ − A) −1 B(X) < ∞.Proof of Theorem 1.5.We will prove the contraposition of the "only if" part of Theorem 1.5, that is, if the energy decays exponentially and ess inf x∈R d γ(x) = 0 holds, then s ≥ 2. By the Gearhart-Prüss-Huang theorem and the exponential decay, there exists c 0 > 0 such that for some K, which is chosen later.For such u, we have Hence, taking K > 0 small enough, we get some c > 0 such that We prove s ≥ 2 by contradiction.Assume that s < 2. In this case, the thickness of the annulus A λ (K) is unbounded with respect to λ: Thus, the inequality (3.1) holds for any u ∈ L 2 (R d ) such that supp u is compact.To see this, notice that there exist a ∈ R d and λ ∈ R satisfying a + supp u ⊂ A λ (K) for such u.Therefore, letting u a (x) := e ia•x u(x), we have since supp u a = a + supp u ⊂ A λ (K).Now note that E ε := {x ∈ R d : γ(x) < ε} has a positive measure for any ε > 0, since ess inf x∈R d γ(x) = 0.For each ε > 0, we take a subset F ε ⊂ E ε such that 0 < m d (F ε ) < ∞.Take R, ε > 0 arbitrarily and set where χ Ω denotes the indicator function of Ω ⊂ R d .By the definition, we have supp g R,ε ⊂ B(0, R) and g R,ε → f ε as R → ∞ in L 2 (R d ).Therefore, applying the inequality (3.1) to g R,ε , we get Taking the limit as R → +∞, we obtain This is a contradiction since ε > 0 is arbitrary.

Proof of Theorem 1.7
The proof of (1.10) =⇒ (1.8) is similar to that of (1.7) =⇒ (1.3) in Section 2, and the implication (1.9) =⇒ (1.10) is trivial.Therefore, we will show that (1.8) =⇒ (1.9).We use a kind of the uncertainty principle to obtain a certain resolvent estimate for the fractional Laplacian: Let Ω ⊂ R d be thick.Then there exist a constant C > 0 such that for each R > 0, the inequality In order to obtain the logarithmic energy decay (1.9), we use the following result.
where m d denotes the d-dimensional Lebesgue measure.Then we have the following result: Theorem 1.7.Let d ≥ 2, let s ≥ 2, and let 0 ≤ γ ∈ L ∞ (R d ).Then the following conditions are equivalent: since γ and f are bounded and f a L ∞ = f L ∞ .Thus, there exists R > 0 such that inf a∈R a+R a−R γ(x) dx > 0 holds, which is equivalent to(1.3).