Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 46, pp. 1–21. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS FOR SOFT POTENTIALS WITH CUT-OFF YAKUI WU, JIAWEI SUN Abstract. We consider the asymptotic behavior of the linearized Boltzmann equation for soft potentials with cut-off. By introducing a new decomposition of the linearized Boltzmann operator, we analyze the spectrum of the linearized Boltzmann operator and obtain the asymptotic behaviors of the linearized Boltzmann equation for γ ∈ (−3, 0), extending the result in [12] for γ ∈ (−1, 0). 1. Introduction We consider the Boltzmann equation ∂F ∂t + v · ∇xF = Q(F, F ), (1.1) where F = F (t, x, v) is the density distribution function of the particles with (t, x, v) ∈ R+ × R3 × R3, Q(F,G) is a bilinear collision operator given by Q(F,G) = ∫ R3 ∫ S2 q(|u− v|, ω) (F (u′)G(v′)− F (u)G(v)) du dω with u′ = u− [(u−v) ·ω]ω, v′ = v+ [(u−v) ·ω]ω, ω ∈ S2. For the case with inverse power interactions between particles in [4], the collision kernel q(|u− v|, ω) is taken as q(|u− v|, ω) = |u− v|γ | cos θ|−γ ′ q0(θ) (1.2) for γ = 1 − 4 s , γ′ = 1 + 2 s , s > 1, where the function q0(θ) is bounded, q0(θ) 6= 0 near θ = π/2, and cos θ = (u− v) · ω |u− v| . We study the Boltzmann equation (1.1) for soft potentials with cut-off. Namely, the collision kernel q(|u− v|, ω) is chosen as q(|u− v|, ω) = q(θ)|u− v|γ , γ ∈ (−3, 0), (1.3) where q(θ) satisfies 0 < q(θ) ≤ C| cos θ|. Considering the perturbation f of F around the global Maxwellian M as follows F = M +M1/2f, 2010 Mathematics Subject Classification. 76P05, 35B20, 35B40, 35P20. Key words and phrases. Linearized Boltzmann operator; soft potentials; spectrum; semigroup; time decay; estimates. c©2021 Texas State University. Submitted December 9, 2020. Published May 27, 2021. 1 2 Y. WU, J. SUN EJDE-2021/46 where M = M(v) = 1 (2π)3/2 e−|v| 2/2, v ∈ R3, (1.4) then the Boltzmann equation (1.1) for F is reformulated in terms of f into ∂f ∂t = Bf + Γ(f, f), where B the linearized Boltzmann operator B = −v · ∇x + L (1.5) with the linearized collision operator Lf = M−1/2 ( Q(M,M1/2f) +Q(M1/2f,M) ) , (1.6) and the nonlinear term Γ(f, f) is Γ(f, f) = M−1/2Q(M1/2f,M1/2f). There is a much important progress on the time decay estimates based on the spectral analysis for the linearized Boltzmann equation for hard potentials in [11, 13, 14, 15]. There have been a few researches on the time decay estimates with the help of the spectral analysis of the linearized Boltzmann equation for soft potentials with cut-off. The spectrum theory and time decay estimates for the linearized Boltzmann equation for γ ∈ (−1, 0) with cut-off in spatially-periodic case were established in [1, 2]. The asymptotic behaviors of the semigroup based on the spectral analysis of the linearized Boltzmann equation for γ ∈ (−1, 0) with cut-off in whole space were studied in [12]. In this article, we are concerned with the asymptotic behavior of the semigroup based on the spectral analysis of the linearized Boltzmann equation for γ ∈ (−3, 0) with cut-off. The linearized Boltzmann collision operator L defined by (1.6) can be written as L = −ν +K, where the operators ν and K with the kernel k(u, v) are defined by (2.1) and (2.2) respectively. Ukai and Asano applied the upper bound of the kernel k(u, v) for γ ∈ (−1, 0) to obtain the following important estimate, cf. [12],∫ R3 |k(u, v)|2(1 + |u|)−βdu ≤ Cβ(1 + |v|)−(β+1) for any β ≥ 0, which implies that the integral operator K satisfies K ∈ C(L2 θ(R3 v), L 2 ς (R3 v)), if ς > θ + 2 γ . (1.7) The compactness of the integral operator K plays an important role in the spec- tral analysis of the linearized Boltzmann operator. Inspired by the work [5], we introduce a new decomposition of the linearized Boltzmann collision operator L = −ν +Ks︸ ︷︷ ︸ as a whole +Kc, (1.8) where Kc is compact and the norm of Ks is small. In particular, it holds that Kc ∈ C(L2 θ1(R3 v), L 2 θ2(R3 v)) for any θ1, θ2 ∈ R. Under the help of the decomposition, we can establish the time decay estimates of the semigroup etB . EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 3 We take the Fourier transform in (1.5) with respect to x, the linearized Boltz- mann operator B is turned into B̂(ξ) = −iv · ξ + L. (1.9) By the Plancherel theorem, we only need to consider the time decay estimates of the semigroup etB̂(ξ). To this end, we need to establish the estimates of the resolvent (λI − B̂(ξ))−1. We define the operator B̂0(ξ) = −iv · ξ + L− P, where the projection operator P is defined by (2.5). According to the result on the spectral analysis of B̂0(ξ) in Proposition 3.7 and the properties of the resolvent (λI − B̂0(ξ))−1 in Lemma 3.10, we can study the spectrum of the operator B̂(ξ) in L2 θ(R3 v) for any θ ∈ R and ξ 6= 0, and prove that (refer to Proposition 4.2) σ(B̂(ξ)) ∈ C−, σp(B̂(ξ)) ∈ C−, which is different from the spectrum of the linearized Boltzmann operator for hard potentials with cut-off in the case with θ = 0 as [3, 14]. Combining the decompo- sition (λI − B̂(ξ))−1 = (I − (λI − B̂0(ξ))−1P )−1(λI − B̂0(ξ))−1, and the properties of the resolvent (λI − B̂0(ξ))−1 given by Lemma 3.10, we can obtain the properties of the resolvent (λI−B̂(ξ))−1 (refer to Lemma 4.7 for details). By the inverse Laplace transform and the properties of the resolvent (λI−B̂(ξ))−1, we can obtain the time decay estimates of the semigroup etB̂(ξ) for any |ξ| ≥ r and r > 0 in a weighted velocity space, which is described by Theorem 4.8. Using resolvent identity, we have (λI − B̂(ξ))−1 = (λI − B̂0(ξ))−1 + (λI − B̂0(ξ))−1P (I − P (λI − B̂0(ξ))−1P )−1P (λI − B̂0(ξ))−1. Then we analyze the singularities of (λI − B̂(ξ))−1 near ξ = 0, and point out that the singularities of (λI − B̂(ξ))−1 near ξ = 0 arise from (I −P (λI − B̂0(ξ))−1P )−1. We compute the eigenvalues of P (λI − B̂0(ξ))−1P near ξ = 0, and find that the singular points of (I − P (λI − B̂0(ξ))−1P )−1 near ξ = 0 are µj(κ) = σj(κ) + iτj(κ), j = ±1, 0, 2, 3, where σj(κ), τj(κ) ∈ C∞[−r0, r0] for some sufficiently small constant r0 > 0 and κ = |ξ|, which sat- isfy the following asymptotic expansions for any κ ∈ [−r0, r0], σj(κ) = σ (2) j κ2 +O(κ3), j = ±1, 0, 2, 3, τj(κ) = τ (1) j κ+O(κ3), j = ±1, 0, 2, 3, where σ (2) j < 0 and τ (1) j are constants. For more details, we refer to Proposition 4.9. We obtain the time decay estimates of the semigroup etB̂(ξ) near ξ = 0 under the help of the asymptotic analysis of etB̂(ξ) near ξ = 0 given in Theorem 4.10. For any ξ ∈ R3 and θ ∈ R, B̂(ξ) generates a semigroup etB̂(ξ) on L2 θ(R3 v) (refer to Lemma 4.4). Since the absence of the spectral gap for the linearized Boltzmann operator for soft potentials, we obtain the time decay estimates of the semigroup etB in a weighted Sobolev space. We state our main result below. 4 Y. WU, J. SUN EJDE-2021/46 Theorem 1.1. Let γ ∈ (−3, 0). For any p ∈ [1, 6 5 ), n ∈ [1, 3 2 ( 1 p − 1 2 ) + 1 2 ), θ ∈ R and l ∈ N, it holds that ‖etBf0‖l,θ,2 ≤ C ( (1 + t)−n(‖f0‖l,θ−2n,2 + ‖f0‖Lp,2) + (1 + t)− 3 2 ( 1 p− 1 2 )‖Pf0‖Lp,2 ) (1.10) for any t ≥ 0 and f0 ∈ Hl,θ−2n,2 ∩ Lp,2, where P is defined by (2.5). Remark 1.2. If Pf0 = 0, then the time decay rate in (1.10) could reach (1 + t)−n, which is faster than (1 + t)− 3 2 ( 1 p− 1 2 ) for any n ∈ [1, 3 2 ( 1 p − 1 2 ) + 1 2 ) and p ∈ [1, 6 5 ). Notation. We will use C as a general positive constant. Denote 〈·, ·〉 as the inner product on L2(R3 v). We write T ∗ for the adjoint operator of the operator T . B(X,Y ) stands for the class of linear bounded operators defined on the space X with the range in Y , the norm of T ∈ B(X,Y ) is expressed as ‖T‖B(X,Y ), we will use B(X) for B(X,X). C(X,Y ) represents the class of compact operators defined on the space X with the range in Y , we will write C(X) for C(X,X). Let Σ be a metric space and L be a normed space, we define L∞(Σ,L ) and C0(Σ,L ) as follows L∞(Σ,L ) = {f : Σ→ L : sup x∈Σ ‖f‖L <∞}, C0(Σ,L ) = {f : Σ→ L : f is continuous from Σ to L }. We denote by σ(T ), σp(T ) and σe(T ) the spectrum, point spectrum and essential spectrum for the operator T . We denote by %(T ) the resolvent set, and by (λI−T )−1 the resolvent with λ ∈ %(T ). We define C+ = {λ ∈ C : Reλ > 0} and C− = {λ ∈ C : Reλ < 0}. We define the Fourier transform f̂(ξ) of f(x) as f̂(ξ) = 1 (2π)3/2 ∫ R3 e−ix·ξf(x)dx. For θ ∈ R, we define a weighted L2-Lebesgue space L2 θ(R3 v) = {f(v) : νθ/2(v)f(v) ∈ L2(R3 v)} with the norm ‖f‖L2 θ(R3 v) = (∫ R3 ν(v)θ|f(v)|2dv )1/2 , where ν(v) is given by (2.1). For θ ∈ R, we introduce the weighted Sobolev space of the function f(x, v) by Hl,θ,2 = L2 θ(R3 v;H l(R3 x)) with the norm ‖f‖l,θ,2 = (∫ R3 ∫ R3 ν(v)θ(1 + |ξ|)2l|f̂(ξ, v)|2dξdv )1/2 . For p ≥ 1, we also need the space Lp,2 = L2(R3 v;L p(R3 x)) with the norm ‖f‖Lp,2 = (∫ R3 (∫ R3 |f(x, v)|pdx )2/p dv )1/2 . The rest of the paper is organized as follows. In Section 2, we introduce some properties of the linear collision operator. In Section 3, we present the results on spectral analysis of the operator B̂0(ξ) for any ξ ∈ R3 and some properties of the resolvent (λ− B̂0(ξ))−1. In Section 4, we give the spectral analysis of the operator B̂(ξ) for any ξ ∈ R3 and the time decay estimates of the semigroup etB . EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 5 2. Preliminaries In this section, we introduce a new decomposition of the linearized Boltzmann collision operator, then give some properties of the collision operator and some lemmas, which will be used later. The linearized collision operator L defined by (1.6) satisfies (Lf)(v) = −ν(v)f(v) + (Kf)(v), where ν(v) = ∫ R3 ∫ S2 q(|u− v|, ω)M(u) du dω, (2.1) (Kf)(v) = ∫ R3 k(u, v)f(u)du = ∫ R3 ∫ S2 q(|u− v|, ω)M1/2(u) × ( M1/2(u′)f(v′) +M1/2(v′)f(u′)−M1/2(v)f(u) ) du dω. (2.2) We will describe some properties of the operator L. For more details, we refer to [5]. The null space N0 of the operator L is a subspace spanned by the orthonormal basis {Mj , j = 0, 1, 2, 3, 4} with M0 = M1/2, Mj = vjM 1/2 (j = 1, 2, 3), M4 = (|v|2 − 3)√ 6 M1/2, (2.3) where M is defined by (1.4). The operator −L is nonnegative and self-adjoint on L2(R3 v), and satisfies 〈−Lf, f〉 ≥ δ‖(I − P )f‖2L2 1 (2.4) for some constant δ > 0, where the projection operator P is defined in L2(R3 v) as Pf = 4∑ j=0 〈f,Mj〉Mj . (2.5) ν(v) is called the collision frequency, and satisfies C1(1 + |v|)γ ≤ ν(v) ≤ C2(1 + |v|)γ (2.6) for γ ∈ (−3, 0) and some constants C1, C2 > 0. We use a crucial decomposition of the operator K introduced by [5]. For con- venience to the readers, we write it here. For any ε > 0, define a smooth cut-off function χε(r) satisfying χε(r) = 1 for r ≥ 2ε, χε(r) = 0 for r ≤ ε. (2.7) The operator K is decomposed as follows K = Kc +Ks, Kc = K2c −K1c, Ks = K2s −K1s = (K1−χ 2s +Kχ 2s)− (K1−χ 1s +Kχ 1s), (2.8) where K1cf = ∫ |u|+|v|≤m ∫ S2 |u− v|γχε(|u− v|)q(θ)M1/2(u)M1/2(v)f(u) du dω, K1−χ 1s f = ∫ R3 ∫ S2 |u− v|γ{1− χε(|u− v|)}q(θ)M1/2(u)M1/2(v)f(u) du dω, 6 Y. WU, J. SUN EJDE-2021/46 Kχ 1sf = ∫ |u|+|v|≥m ∫ S2 |u− v|γχε(|u− v|)q(θ)M1/2(u)M1/2(v)f(u) du dω, K1−χ 2s f = ∫ R3 ∫ S2 |u− v|γ{1− χε(|u− v|)}q(θ)M1/2(u) × ( M1/2(u′)f(v′) +M1/2(v′)f(u′) ) du dω, K2cf = 4 ∫ |v|+|v+u‖|≤m 1 |u‖| e− 1 4 |u‖| 2−|ζ‖|2f(v + u‖)k(u‖, ζ⊥)du‖, Kχ 2sf = 4 ∫ |v|+|v+u‖|≥m 1 |u‖| e− 1 4 |u‖| 2−|ζ‖|2f(v + u‖)k(u‖, ζ⊥)du‖ with k(u‖, ζ⊥) = ∫ R2 e−|u⊥+ζ⊥|2 [|u‖|2 + |u⊥|2] γ−1 2 χ (√ |u‖|2 + |u⊥|2 ) q(θ) | cos θ| du⊥, and the integration variables u‖ = (u · ω)ω, u⊥ = u− (u · ω)ω, (2.9) ζ‖ + ζ⊥ = 1 2 (2v + u‖), ζ‖‖u‖, ζ⊥‖u⊥. (2.10) We list some properties of the operators K and P , which will be used later. For the simplicity of expression, for any θ ∈ R, we write L2 θ for L2 θ(R3 v). Lemma 2.1 ([5]). For θ ∈ R, it holds that |〈νθKf, g〉| ≤ C‖νθ/2f‖L2 1 ‖νθ/2g‖L2 1 , (2.11) where ν(v) is given by (2.1). Lemma 2.2 ([5]). It holds that |〈νθKsf, g〉| ≤ η‖νθ/2f‖L2 1 ‖νθ/2g‖L2 1 (2.12) for any θ ∈ R and η > 0. Lemma 2.3. For P defined by (2.5), we have P ∈ C(L2 θ1 , L 2 θ2) (2.13) for any θ1, θ2 ∈ R. A proof of the above lemma can be found in [12, Lemma 4.3], we omit it here. Lemma 2.4. Let γ ∈ (−3, 0). We have (i) For any θ ∈ R, it holds that K ∈ B(L2 θ, L 2 θ−2). (2.14) (ii) For any θ ∈ R and η > 0, it holds that ‖Ks‖B(L2 θ,L 2 θ−2) ≤ η. (2.15) (iii) For any θ1, θ2 ∈ R, it holds that Kc ∈ C(L2 θ1 , L 2 θ2). (2.16) EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 7 Proof. (i) From (2.11), for any θ ∈ R, we have |〈ν−1/2νθ/2(v)Kf, ν1/2νθ/2(v)g〉| ≤ C‖ν1/2νθ/2(v)f‖L2‖ν1/2νθ/2(v)g‖L2 , which implies that ‖Kf‖L2 θ−1 ≤ C‖f‖L2 θ+1 . Thus, we can get (2.14). (ii) By (2.12), for any θ ∈ R and η > 0, we have |〈ν−1/2νθ/2(v)Ksf, ν 1/2νθ/2(v)g〉| ≤ η‖ν1/2νθ/2(v)f‖L2‖ν1/2νθ/2(v)g‖L2 , which implies that ‖Ksf‖L2 θ−1 ≤ η‖f‖L2 θ+1 . Thus, we can obtain (2.15). (iii) Since 1 |u‖| ∈ L2 loc(R3), the kernel k(u‖, ζ⊥) is bounded for the chosen ε > 0 and any given m > 0. The Hilbert-Schmidt theorem clearly shows that Kc is a compact operator from L2 θ1 to L2 θ2 for any θ1, θ2 ∈ R. Thus, we have proved (2.16). The proof of Lemma 2.4 is complete. � 3. Spectrum and resolvent To analyze the spectrum of the operator B̂(ξ) on L2 θ for any ξ 6= 0 and θ ∈ R, we introduce some auxiliary operators as follows Â0(ξ) = −iv · ξ, (3.1) Â(ξ) = −iv · ξ − ν(v), (3.2) Âs(ξ) = −iv · ξ − ν(v) +Ks, (3.3) B̂0(ξ) = B̂(ξ)− P = Âs(ξ) +K0 (3.4) with K0 = Kc − P, (3.5) where P , Ks, and Kc are defined by (2.5) and (2.8). Let D(T (ξ)) = {f ∈ L2 θ : v · ξf(v) ∈ L2 θ}, (3.6) where T (ξ) = Â0(ξ), Â(ξ), Âs(ξ), B̂0(ξ) or B̂(ξ) for any θ ∈ R and ξ ∈ R3. It is obvious that D(B̂(ξ)) = D(B̂0(ξ)) = D(Âs(ξ)) = D(Â(ξ)) = D(Â0(ξ)). Lemma 3.1. The operator Â(ξ) generates a strongly continuous contraction semi- group on L2 θ for any θ ∈ R and ξ ∈ R3. Proof. It holds for f ∈ D(Â(ξ)) that Re〈νθÂ(ξ)f, f〉 = Re〈νθ(−iv · ξ − ν)f, f〉 = 〈νθ(−ν)f, f〉 ≤ 0, Re〈νθÂ∗(ξ)f, f〉 = Re〈νθ(iv · ξ − ν)f, f〉 = 〈νθ(−ν)f, f〉 ≤ 0, which implies that the operators Â(ξ) and Â∗(ξ) are dissipative on L2 θ. Since D(Â∗(ξ)) and D(Â(ξ)) are dense in L2 θ, then Â(ξ) is a densely defined closed op- erator on L2 θ by [9, Theorem VIII.1]. Thus, with the help of Corollary 4.4 on p.15 of [8], we obtain that the operator Â(ξ) generates a strongly continuous contraction semigroup on L2 θ. The proof is complete. � 8 Y. WU, J. SUN EJDE-2021/46 Let Σ = Σ(λ,ξ) = C+ × R3. (3.7) Based on Lemma 3.1, we can obtain the following properties for the resolvent (λI − Â(ξ))−1. Lemma 3.2. Let γ ∈ (−3, 0). For any θ ∈ R, the following statements hold. (i) (λI − Â(ξ))−1 ∈ L∞(Σ, B(L2 θ−2, L 2 θ)). (ii) (λI − Â(ξ))−1 ∈ C0(Σ, B(L2 θ−2−ζ , L 2 θ)) for any ζ > 0. (iii) For any fixed r > 0 and f ∈ L2 θ−2, it holds that sup λ∈C+,|λ|≥a,|ξ|≤r ‖(λI − Â(ξ))−1f‖L2 θ → 0, as a→∞. (iv) Write λ = σ + iτ with σ, τ ∈ R and let f ∈ L2 θ−1. Then sup σ≥0,ξ∈R3 ∫ +∞ −∞ ‖((σ + iτ)I − Â(ξ))−1f‖2L2 θ dτ ≤ C‖f‖2L2 θ−1 . Here Σ is defined by (3.7). A proof of the above lemma can be found in [12, Lemma 5.1], we omit it here. Remark 3.3. Since L2 θ1 is dense in L2 θ2 for θ1, θ2 ∈ R and θ1 < θ2, by the aid of (i) and (ii) in Lemma 3.2, it holds that (λI − Â(ξ))−1f ∈ C0(Σ, L2 θ) for any f ∈ L2 θ−2. Thanks to (2.15), we can analyze the resolvent set of the operator Âs(ξ) on L2 θ for any ξ ∈ R3 and θ ∈ R. Lemma 3.4. Let γ ∈ (−3, 0). We have %(Âs(ξ)) ⊃ C+, σ(Âs(ξ)) ⊂ C−. (3.8) Proof. For λ ∈ %(Â(ξ)), we decompose λI − Âs(ξ) as follows (λI − Âs(ξ)) = (λI − Â(ξ))(I − (λI − Â(ξ))−1Ks). (3.9) Combining (i) in Lemma 3.2 and (2.15), and choosing sufficiently small η, we are able to show that ‖(λI − Â(ξ))−1Ks‖B(L2 θ) ≤ sup (λ,ξ)∈C+×R3 ‖(λI − Â(ξ))−1‖B(L2 θ−2,L 2 θ) · ‖Ks‖B(L2 θ,L 2 θ−2) ≤ 1 2 , which implies that ‖(I − (λI − Â(ξ))−1Ks) −1‖B(L2 θ) ≤ 2. (3.10) According to Lemma 3.1, (3.9) and the Hille-Yosida theorem, we have C+ ⊂ %(Â(ξ)) ⊂ %(Âs(ξ)). The proof is complete. � For any λ ∈ %(Âs(ξ)) ∩ %(Â(ξ)), we have (λI − Âs(ξ))−1 = (I − (λI − Â(ξ))−1Ks) −1(λI − Â(ξ))−1. (3.11) From Lemma 3.4, we can obtain the following properties for the resolvent of (λI − Âs(ξ)) −1. EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 9 Lemma 3.5. Let γ ∈ (−3, 0). For any θ ∈ R, the following statements hold. (i) (λI − Âs(ξ))−1 ∈ L∞(Σ, B(L2 θ−2, L 2 θ)). (ii) (λI − Âs(ξ))−1 ∈ C0(Σ, B(L2 θ−2−ζ , L 2 θ)) for any ζ > 0. (iii) For any fixed r > 0 and f ∈ L2 θ−2, it holds that sup λ∈C+,|λ|≥a,|ξ|≤r ‖(λI − Âs(ξ))−1f‖L2 θ → 0, as a→∞. (iv) Write λ = σ + iτ with σ, τ ∈ R and let f ∈ L2 θ−1. Then sup σ≥0,ξ∈R3 ∫ +∞ −∞ ‖((σ + iτ)I − Âs(ξ))−1f‖2L2 θ dτ ≤ C‖f‖2L2 θ−1 . Here Σ is defined by (3.7). Proof. By (3.10), it holds that (I − (λI − Â(ξ))−1Ks) −1 ∈ L∞(Σ, B(L2 θ)). (3.12) Combining (3.11) and (3.12), we can respectively obtain (i), (iii) and (iv) from (i), (iii) and (iv) in Lemma 3.2. We next prove (ii). Let S1(λ, ξ) = (λI − Âs(ξ))−1. For any (λ0, ξ0), (λ1, ξ1) ∈ Σ, we have ‖S1(λ1, ξ1)f − S1(λ0, ξ0)f‖L2 θ ≤ ‖S1(λ1, ξ1)χm(|v|)f − S1(λ0, ξ0)χm(|v|)f‖L2 θ + ‖S1(λ1, ξ1){1− χm(|v|)}f − S1(λ0, ξ0){1− χm(|v|)}f‖L2 θ =: I1 + I2, (3.13) where χm(|v|) is defined by (2.7). For any ε > 0 and ζ > 0, it holds that I1 ≤ 2 sup (λ,ξ)∈C+×R3 ‖S1(λ, ξ)χm(|v|)f‖L2 θ ≤ C(1 +m) ζγ 2 ‖f‖L2 θ−2−ζ < ε, (3.14) where m > 0 is chosen large enough. For any ε > 0, assuming |λ1 − λ0| < ε and |ξ1 − ξ0| < ε, we have I2 = ‖(λ1I + iv · ξ1 + ν(v)−Ks) −1(λ0 − λ1 + iv · ξ0 − iv · ξ1) × (λ0I + iv · ξ0 + ν(v)−Ks) −1{1− χm(|v|)}f‖L2 θ ≤ C‖(λ1I + iv · ξ1 + ν(v)−Ks) −1{1− χm(|v|)}‖B(L2 θ)(|λ0 − λ1| +m|ξ0 − ξ1|)‖(λ0I + iv · ξ0 + ν(v)−Ks) −1{1− χm(|v|)}f‖L2 θ ≤ Cε‖f‖L2 θ , (3.15) which together with (3.13) and (3.14) yields (ii). The proof is complete. � For B̂0(ξ), we have the following similar result to [12, Lemma 5.2]. Lemma 3.6. Let γ ∈ (−3, 0). For any ξ ∈ R3, B̂0(ξ) generates a strongly contin- uous contraction semigroup on L2. 10 Y. WU, J. SUN EJDE-2021/46 Proof. Since D(B̂∗0(ξ)) = D(B̂0(ξ)) is dense in L2, it holds that B̂0(ξ) is a densely defined closed operator on L2 by [9, Theorem VIII.1]. Thanks to (2.4), for any f ∈ D(B̂0(ξ)), it holds that Re〈B̂0(ξ)f, f〉 = Re〈B̂∗0(ξ)f, f〉 = 〈(L− P )f, f〉 = −(〈−Lf, f〉+ 〈Pf, f〉) ≤ −(δ‖(I − P )f‖2L2 1 + ‖Pf‖2L2) < 0, (3.16) which implies that B̂0(ξ) and B̂∗0(ξ) are dissipative operators on L2. Thus, with the help of [8, Corollary 4.4 on p.15], the operator B̂0(ξ) generates a strongly continuous contraction semigroup on L2. The proof is complete. � Based on Lemma 3.6, we can analyze the spectrum of the operator B̂0(ξ) in L2 θ for any ξ ∈ R3 and θ ∈ R. Proposition 3.7. Let γ ∈ (−3, 0). We have the following results. (i) σ(B̂0(ξ)) ⊂ C−, %(B̂0(ξ)) ⊃ C+. (ii) σe(B̂0(ξ)) = σe(Âs(ξ)). (iii) σp(B̂0(ξ)) ⊂ C−. Proof. According to Lemma 2.3 and (2.16), we know that the operator K0 : L2 θ → L2 θ is compact. By [6, Theorem 5.35 on p.244], we have σe(B̂0(ξ)) = σe(Âs(ξ)). Thus, we have proved (ii). By Lemma 3.4, we have σe(Âs(ξ)) ⊂ σ(Âs(ξ)) ⊂ C−. Combining this and (ii), (iii), we can gain (i). We next prove (iii). Let λ ∈ σp(B̂0(ξ)), there exists f ∈ D(B̂0(ξ)) and f 6= 0, we have λf = B̂0(ξ)f. (3.17) For θ ≤ 0, then f ∈ L2, we can apply (3.16) to derive Reλ < 0. For θ > 0, assume Reλ ≥ 0. According to Lemma 2.3, (2.16) and (3.5), K0 is bounded from L2 θ to L2 −2, which together with Lemma 3.5 leads to λf = B̂0(ξ)f ⇒ f = (λI − Âs(ξ))−1K0f ∈ L2. Then, by (3.16), we have λ ∈ C−, which is a contraction to the assumption. Thus, we have proved (iii). The proof is complete. � For any λ ∈ %(B̂0(ξ)) ∩ %(Âs(ξ)), we have (λI − B̂0(ξ))−1 = (I − (λI − Âs(ξ))−1K0)−1(λI − Âs(ξ))−1. (3.18) Let M(λ, ξ) = (λI − Âs(ξ))−1K0, (3.19) where Âs(ξ) and K0 is defined by (3.3) and (3.5) respectively. We state some properties of M(λ, ξ) below. Lemma 3.8. Let γ ∈ (−3, 0). For any θ ∈ R, the following statements hold. (i) M(λ, ξ) ∈ L∞(Σ, C(L2 θ)). (ii) M(λ, ξ) ∈ C0(Σ, C(L2 θ)). (iii) For any r > 0, it holds that sup λ∈C+,|λ|≥a,|ξ|≤r ‖M(λ, ξ)‖B(L2 θ) → 0, as a→∞. EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 11 (iv) It holds that sup λ∈C+,|ξ|≥r ‖M(λ, ξ)‖B(L2 θ) → 0, as r →∞. Here Σ is defined by (3.7). Proof. (i) According to Lemma 2.3, (2.16) and (3.5), we have K0 ∈ C(L2 θ, L 2 θ−2). (3.20) Combining (i) in Lemma 3.5 and (3.20), we can obtain (i). (ii) By Lemma 2.3, (2.16) and (3.5), for any ζ > 0, it holds that K0 ∈ C(L2 θ, L 2 θ−2−ζ), (3.21) which together with (ii) in Lemma 3.5 and (3.21) leads to (ii). (iii) For any f ∈ L2 θ, it holds that ‖M(λ, ξ)‖B(L2 θ) = sup ‖f‖ L2 θ =1 ‖M(λ, ξ)f‖L2 θ . (3.22) Combining (iii) in Lemma 3.5, (3.20) and (3.22), we can get (iii). (iv) For any f ∈ L2 θ, we have ‖M(λ, ξ)f‖L2 θ ≤ ‖(λI − Âs(ξ))−1χm(|v|)K0f‖L2 θ + ‖(λI − Âs(ξ))−1{1− χm(|v|)}K0f‖L2 θ =: J1 + J2, where χm(|v|) is defined by (2.7). By (3.21), it holds for any ε > 0 and ζ > 0 that J1 ≤ C(1 +m) ζδ 2 ‖(λI − Âs)−1‖B(L2 θ−2,L 2 θ)‖K0‖B(L2 θ,L 2 θ−2−ζ)‖f‖L2 θ ≤ ε, (3.23) where m > 0 is chosen large enough. We next estimate J2. Write S1 = {v ∈ R3 : |v| ≤ m, | Imλ + v · ξ| ≤ |ξ|√ r }, S2 = {v ∈ R3 : |v| ≤ m}\S1. We use ξ |ξ| , ξ1, ξ2 as an orthonormal basis, then v = 〈v, ξ |ξ| 〉 ξ |ξ| + ( v − 〈v, ξ |ξ| 〉 ξ |ξ| ) = L ξ |ξ| + L1ξ1 + L2ξ2. It holds that measS1 = ∫ S1 1dv ≤ ∫ m −m dL1 ∫ m −m dL2 ∫ 1√ r − Imλ |ξ| − 1√ r − Imλ |ξ| dL ≤ 8m2 √ r . For any λ ∈ C+ and |ξ| ≥ r, we have ‖(λI − Â(ξ))−1{1− χm(|v|)}K0f‖2L2 θ = ∫ S1 νθ(v) 1 |Reλ+ ν(v)|2 + | Imλ+ v · ξ|2 {1− χm(|v|)}|K0f |2dv + ∫ S2 νθ(v) 1 |Reλ+ ν(v)|2 + | Imλ+ v · ξ|2 {1− χm(|v|)}|K0f |2dv ≤ C‖f‖2L2 θ(S1) + 1 r ‖f‖2L2 θ → 0, as r →∞. (3.24) 12 Y. WU, J. SUN EJDE-2021/46 Combining (3.24) and (3.10) yields that J2 = ‖(I − (λI − Â(ξ))−1Ks) −1(λI − Â(ξ))−1{1− χm(|v|)}K0f‖L2 θ ≤ ‖(I − (λI − Â(ξ))−1Ks) −1‖B(L2 θ) · ‖(λI − Â(ξ))−1{1− χm(|v|)}K0f‖L2 θ → 0, as r →∞, which together with (3.23) verifies (iv). The proof is complete. � Lemma 3.9. Let γ ∈ (−3, 0). For any θ ∈ R, we have the following results. (i) 1 ∈ %((λI − Âs(ξ))−1K0) for any (λ, ξ) ∈ Σ. (ii) (I − (λI − Âs(ξ))−1K0)−1 ∈ C0(Σ, B(L2 θ)). (iii) (I − (λI − Âs(ξ))−1K0)−1 ∈ L∞(Σ, B(L2 θ)). Here Σ is defined by (3.7). Proof. From (i) in Lemma 3.8, (λI− Âs(ξ))−1K0 : L2 θ → L2 θ is compact. By the aid of the spectral theory of the compact operator, if 1 ∈ σ((λI − Âs(ξ))−1K0), then 1 ∈ σp((λI − Âs(ξ))−1K0). There exists f ∈ L2 θ and f 6= 0, it holds that (λI − Âs(ξ))−1K0f = f ⇒ B̂0(ξ)f = λf, which implies that λ ∈ σp(B̂0(ξ)). It is a contradiction to (iii) in Proposition 3.7. Thus, we have proved (i). Combining (i), (ii) in Lemma 3.8 and (i) in Lemma 3.9, it holds that (I−(λI−Âs(ξ))−1K0)−1 ∈ C0(Σ, B(L2 θ)). Thus, we obtain (ii). Making use of (iii), (iv) in Lemma 3.8, there exists a constant r0 which is large enough, it holds for (λ, ξ) ∈ Σ and |λ|+ |ξ| ≥ r0 that ‖(λI − Âs(ξ))−1K0‖B(L2 θ) ≤ 1 2 . Then ‖(I − (λI − Âs(ξ))−1K0)−1‖B(L2 θ) ≤ 2. (3.25) In view of (ii) in Lemma 3.9, we know that (I − (λI − Âs(ξ))−1K0)−1 is uniformly bounded for (λ, ξ) ∈ Σ and |λ| + |ξ| ≤ r0. Combining this and (3.25), we have proved (iii). The proof is complete. � With the help of (3.18), Lemma 3.5, Lemma 3.8, and Lemma 3.9, we can obtain the following properties of the resolvent (λI− B̂0(ξ))−1. The proof is omitted here. Lemma 3.10. Let γ ∈ (−3, 0). For any θ ∈ R, the following statements hold. (i) (λI − B̂0(ξ))−1 ∈ L∞(Σ, B(L2 θ−2, L 2 θ)). (ii) (λI − B̂0(ξ))−1 ∈ C0(Σ, B(L2 θ−2−ζ , L 2 θ)) for any ζ > 0. (iii) For any fixed r > 0 and f ∈ L2 θ−2, it holds that sup λ∈C+,|λ|≥a,|ξ|≤r ‖(λI − B̂0(ξ))−1‖L2 θ → 0, as a→∞. (iv) Write λ = σ + iτ with σ, τ ∈ R and let f ∈ L2 θ−1. Then sup σ≥0,ξ∈R3 ∫ +∞ −∞ ‖((σ + iτ)I − B̂0(ξ))−1f‖2L2 θ dτ ≤ C‖f‖2L2 θ−1 . Here Σ is defined by (3.7). EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 13 4. Decay estimates of semigroup In this section, we give the spectrum structure of the operator B̂(ξ) on L2 θ for any ξ ∈ R3 and θ ∈ R, and some properties of the resolvent (λI− B̂(ξ))−1. Finally, we obtain the time decay estimates of the semigroup etB on the space Hl,θ,2. 4.1. Estimates at high frequency. We recall the definition of B̂(ξ) given by (1.9) and (3.4) B̂(ξ) = −iv · ξ + L = B̂0(ξ) + P. (4.1) We first state the result on the spectrum of the operator B̂(ξ) on L2 for any ξ ∈ R3. Lemma 4.1. Let γ ∈ (−3, 0). For any ξ ∈ R3, the following statements hold. (i) B̂(ξ) generates a strongly continuous contraction semigroup on L2. Conse- quently, %(B̂(ξ)) ⊃ C+, σ(B̂(ξ)) ⊂ C−. (4.2) (ii) σp(B̂(ξ)) ∩ {Reλ = 0} = { ∅, if ξ 6= 0, {0}, if ξ = 0. (4.3) Proof. Since D(B̂∗(ξ)) = D(B̂(ξ)) is dense in L2, it holds that B̂(ξ) is a densely defined closed operator on L2 by [9, Theorem VIII.1]. Thanks to (2.4), for any f ∈ D(B̂(ξ)), we have Re〈B̂(ξ)f, f〉 = Re〈B̂∗(ξ)f, f〉 = 〈Lf, f〉 ≤ −(δ‖(I − P )f‖2L2 1 ) ≤ 0, (4.4) which implies that B̂(ξ) and B̂∗(ξ) are dissipative operators on L2. Thus, with the help of Corollary 4.4 on p.15 of [8], the operator B̂(ξ) generates a strongly continuous contraction semigroup on L2. Thus, we have proved (i). Let λ ∈ σp(B̂(ξ)), there exists f ∈ L2 and f 6= 0, it holds that B̂(ξ)f = λf. (4.5) By (4.4), we have Reλ〈f, f〉 = Re〈B̂(ξ)f, f〉 = 〈Lf, f〉 ≤ 0, (4.6) which implies Reλ ≤ 0. If Reλ = 0, it holds from (4.6) that 〈Lf, f〉 = 0, which implies that f ∈ kerL. Then (4.5) is turned into (Imλ+ v · ξ)Pf = 0, which is impossible for f 6= 0 unless Imλ = 0 and ξ = 0. Thus, we have proved (ii). The proof is complete. � We have the following results about the spectrum of the operator B̂(ξ) on L2 θ for any ξ ∈ R3 and θ ∈ R. Proposition 4.2. Let γ ∈ (−3, 0). The following statements hold. (i) σ(B̂(ξ)) ⊂ C−, %(B̂(ξ)) ⊃ C+. (ii) σe(B̂(ξ)) = σe(B̂0(ξ)). (iii) σp(B̂(ξ)) ⊂ C− for ξ 6= 0. 14 Y. WU, J. SUN EJDE-2021/46 Proof. We only give the proof of (iii). The proof of (i) and (ii) can be given using arguments similar to those in (i) and (ii) of Proposition 3.7. For any ξ 6= 0, let λ ∈ σp(B̂(ξ)), there exists f ∈ D(B̂(ξ)) and f 6= 0. It holds that λf = B̂(ξ)f. For θ ≤ 0, then f ∈ L2, by (ii) in Lemma 4.1, we have Reλ < 0 for ξ 6= 0. For θ > 0, assume Reλ ≥ 0. By Lemma 2.3 and applying the boundness of the operator P from L2 θ to L2 −2 and (i) in Lemma 3.10, we have λf = B̂(ξ)f ⇒ f = (λI − B̂0(ξ))−1Pf ∈ L2. By (4.3), λ ∈ C− for ξ 6= 0, which is a contradiction to the assumption. Thus, we have proved (iii). The proof is complete. � Remark 4.3. By Lemma 4.1 and applying similar arguments to those in the proof of Proposition 4.2, we have the following result about the spectrum of the operator B̂(ξ) on L2 θ for any θ ∈ R and ξ ∈ R3, σp(B̂(ξ)) ⊂ C− ∪ {0}. Lemma 4.4. B̂(ξ) generates a strongly continuous semigroup on L2 θ for any θ ∈ R with ‖etB̂(ξ)‖B(L2 θ) ≤ e t‖K‖ B(L2 θ ) . (4.7) Proof. Based on Lemma 3.1, Â(ξ) generates a strongly continuous contraction semi- group on L2 θ for any θ ∈ R and ξ ∈ R3, which implies that ‖etÂ(ξ)‖B(L2 θ) ≤ 1. By (2.14), we have K ∈ B(L2 θ). By the theory of the bounded perturbation of semi- group in [8], we obtain that B̂(ξ) = Â(ξ) + K generates a strongly continuous semigroup on L2 θ and etB̂(ξ) satisfies (4.7). The proof is complete. � For any λ ∈ %(B̂0(ξ)) ∩ %(B̂(ξ)), we have (λI − B̂(ξ))−1 = (I − (λI − B̂0(ξ))−1P )−1(λI − B̂0(ξ))−1. (4.8) We define the set Σr = {(λ, ξ) ∈ C+ × R3 : |λ|+ |ξ| ≥ r} (4.9) for any r > 0. Let M1(λ, ξ) = (λI − B̂0(ξ))−1P. (4.10) Then we obtain a similar results as in Lemma 3.8. Lemma 4.5. Let γ ∈ (−3, 0). For any θ ∈ R, we have the following results. (i) M1(λ, ξ) ∈ L∞(Σ, C(L2 θ)). (ii) M1(λ, ξ) ∈ C0(Σ, C(L2 θ)). (iii) For any r > 0, it holds that sup λ∈C+,|λ|≥a,|ξ|≤r ‖M1(λ, ξ)‖B(L2 θ) → 0, as a→∞. (iv) It holds that sup λ∈C+,|ξ|≥r ‖M1(λ, ξ)‖B(L2 θ) → 0, as r →∞. Here Σ is defined by (3.7). EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 15 Proof. It holds that M1(λ, ξ) = (I − (λI − Âs(ξ))−1K0)−1(λI − Âs(ξ))−1P for any λ ∈ %(Âs(ξ)). Thus, under the help of Lemma 2.3, Lemma 3.8 and Lemma 3.9, we can prove Lemma 4.5. We omit the details here. � Similar to Lemma 3.9, we have the following results. Lemma 4.6. Let γ ∈ (−3, 0). For any θ ∈ R, the following statements hold. (i) 1 ∈ %((λI − B̂0(ξ))−1P ) for any (λ, ξ) ∈ Σr. (ii) (I − (λI − B̂0(ξ))−1P )−1 ∈ C0(Σr, B(L2 θ)). (iii) (I − (λI − B̂0(ξ))−1P )−1 ∈ L∞(Σr, B(L2 θ)). Here Σr is defined by (4.9). Proof. The proof is similar to the one of Lemma 3.9. We just sketch it. In terms of the compactness of the operator (λI− B̂0(ξ))−1P and the spectrum of the operator B̂(ξ) stated in Proposition 4.2 and Remark refrem41, we can prove (i). By the aid of (i), (ii) in Lemma 4.5 and (i) in Lemma 4.6, we can obtain the continuity of (I − (λI − B̂0(ξ))−1P )−1 on Σr. Finally, combining this and (iii), (vi) in Lemma 4.5, we obtain the uniformly boundness of (I − (λI − B̂0(ξ))−1P )−1 on Σr. � According to Lemma 3.10, Lemma 4.5, Lemma 4.6, Proposition 4.2, and (4.8), we can obtain the following properties of the resolvent of (λI−B̂(ξ))−1. The details are omitted here. Lemma 4.7. For any γ ∈ (−3, 0) and θ ∈ R, the following statements hold. (i) (λI − B̂(ξ))−1 ∈ L∞(Σr, B(L2 θ−2, L 2 θ)). (ii) (λI − B̂(ξ))−1 ∈ C0(Σr, B(L2 θ−2−ζ , L 2 θ)) for any ζ > 0. (iii) For any fixed r > 0 and f ∈ L2 θ−2, it holds that sup λ∈C+,|λ|≥a,|ξ|≤r ‖(λI − B̂(ξ))−1f‖L2 θ → 0, as a→∞. (iv) Write λ = σ + iτ with σ, τ ∈ R and let f ∈ L2 θ−1. It holds for any r > 0 that sup σ≥0,|ξ|≥r ∫ +∞ −∞ ‖((σ + iτ)I − B̂(ξ))−1f‖2L2 θ dτ ≤ C‖f‖2L2 θ−1 with a constant C > 0 depending on r. Here Σr is defined by (4.9). With the help of Lemma 4.7, we can evaluate the time decay estimates of the semigroup etB̂(ξ) for any |ξ| ≥ r and r > 0. Theorem 4.8. Let γ ∈ (−3, 0). For any |ξ| ≥ r, r > 0, θ ∈ R and n ≥ 1, it holds that ‖etB̂(ξ)‖B(L2 θ−2n,L 2 θ) ≤ C(1 + t)−n (4.11) for any t ≥ 0. 16 Y. WU, J. SUN EJDE-2021/46 Proof. Denote the semigroup etB̂(ξ) by the inverse Laplace transform of the resol- vent (λI − B̂(ξ))−1 as follows etB̂(ξ)f0 = lim a→∞ 1 2πi ∫ σ+ia σ−ia eλt(λI − B̂(ξ))−1f0dλ (4.12) for any f0 ∈ D(B̂(ξ)), where σ > 0 can be chosen arbitrarily. Let S2(λ, ξ) = (λI − B̂(ξ))−1 and λ = s+ iτ . According to Proposition 4.2 and Lemma 4.7, we can use the Cauchy’s theorem in (4.12) to shift the path of the integration from s = σ to s = 0 and obtain etB̂(ξ)f0 = lim a→∞ 1 2πi ∫ +ia −ia eλt(λI − B̂(ξ))−1f0dλ + lim a→∞ 1 2πi (∫ 0 σ e(s−ia)tS2(s− ia, ξ)f0ds+ ∫ σ 0 e(s+ia)tS2(s+ ia, ξ)f0ds ) . (4.13) From (iii) in Lemma 4.7, for any f0 ∈ L2 θ−2, we have ‖S2(s∓ ia, ξ)f0‖L2 θ → 0, as a→∞. (4.14) Thus, the last two terms on the right-hand side of (4.13) vanish, and (4.13) is reduced to etB̂(ξ)f0 = lim a→∞ 1 2π ∫ a −a eiτt(iτI − B̂(ξ))−1f0dτ, (4.15) where we make the variable substitution λ = iτ . Applying the integration by parts on the right-hand side of (4.15) yields etB̂(ξ)f0 = lim a→∞ 1 2π ∫ a −a eiτt(iτI − B̂(ξ))−1f0dτ = lim a→∞ ( 1 2π n∑ k=1 eiτt (k − 1)! itk (iτI − B̂(ξ))−kf0 )∣∣∣τ=a τ=−a + lim a→∞ 1 2π n! tn ∫ a −a eiτt(iτI − B̂(ξ))−(n+1)f0dτ (4.16) for any f0 ∈ L2 θ−2(n+1), where we have used dl dsl S2(s+ iτ, ξ)f0 = 1 il dl dτ l S2(s+ iτ, ξ)f0 = (−1)ll!S2(λ, ξ)l+1f0, which is valid at s = 0 for any f0 ∈ L2 θ−2(l+1) from (i) in Lemma 4.7. Owing to (iii) in Lemma 4.7, the first term on the right-hand side of (4.16) tends to 0 as a→∞, (4.16) is reduced to etB̂(ξ)f0 = lim a→∞ 1 2π n! tn ∫ a −a eiτt(iτI − B̂(ξ))−(n+1)f0dτ. (4.17) For any f0 ∈ D(B̂(ξ)) ∩L2 θ−2(n+1) and g ∈ L2 θ, by (4.17) and (iv) in Lemma 4.7, it holds that |〈νθetB̂(ξ)f0, g〉| = ∣∣∣ lim a→∞ ∫ R3 νθ n! 2πtn g ∫ a −a eiτt(iτI − B̂(ξ))−(n+1)f0dτdv ∣∣∣ EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 17 ≤ C tn ∫ ∞ −∞ |〈νθ(iτI − B̂(ξ))−(n+1)f0, g〉|dτ ≤ C tn ∫ ∞ −∞ |〈νθ(iτI − B̂(ξ))−nf0, (−iτI − B̂(−ξ))−1g〉|dτ ≤ C tn ∫ ∞ −∞ ‖(iτI − B̂(ξ))−nf0‖L2 θ−1 ‖(−iτI − B̂(−ξ))−1g)‖L2 θ+1 dτ ≤ C tn (∫ ∞ −∞ ‖(iτI − B̂(ξ))−1f0‖2L2 θ−2n+1 dτ )1/2 × (∫ ∞ −∞ ‖(−iτI − B̂(−ξ))−1g)‖2L2 θ+1 dτ )1/2 ≤ C tn ‖f0‖L2 θ−2n ‖g‖L2 θ , which implies that ‖etB̂(ξ)‖B(L2 θ−2n,L 2 θ) ≤ Ct−n. (4.18) By Lemma 4.4 and (4.18), we can obtain for any n ∈ N∗ and t ≥ 0 that ‖etB̂(ξ)‖B(L2 θ−2n,L 2 θ) ≤ C(1 + t)−n. (4.19) By applying the interpolation theorem, we can obtain (4.19) for any n ≥ 1. The proof is complete. � 4.2. Estimates at low frequency. In this subsection, we analyze the singularities of (λI − B̂(ξ))−1 near ξ = 0. We decompose (λI − B̂(ξ))−1 as follows (λI − B̂(ξ))−1 = (λI − B̂0(ξ))−1 + (λI − B̂0(ξ))−1(I − P (λI − B̂0(ξ))−1)−1P (λI − B̂0(ξ))−1. (4.20) We will check that (I − P (λI − B̂0(ξ))−1)−1Pf = P (I − P (λI − B̂0(ξ))−1P )−1Pf. (4.21) Write g = (I − P (λI − B̂0(ξ))−1)−1Pf. (4.22) By (4.22), it holds that g = P (λI − B̂0(ξ))−1g + Pf ∈ kerL, which, from (4.22), implies Pg = P (λI − B̂0(ξ))−1Pg + Pf. Thus, we obtain g = Pg = P (I − P (λI − B̂0(ξ))−1P )−1Pf. (4.23) Substituting (4.21) into (4.20), we have (λI − B̂(ξ))−1 = (λI − B̂0(ξ))−1 + (λI − B̂0(ξ))−1P (I − P (λI − B̂0(ξ))−1P )−1P (λI − B̂0(ξ))−1. (4.24) Combining (i) in Lemma 3.10 and (i) in Lemma 4.5, we obtain that the singularities of the resolvent (λI − B̂(ξ))−1 near ξ = 0 arise from (I − P (λI − B̂0(ξ))−1P )−1. 18 Y. WU, J. SUN EJDE-2021/46 We next analyze the singularities of (I − P (λI − B̂0(ξ))−1P )−1 near ξ = 0. Write λ = σ + iτ , |ξ| = κ, let W (σ, τ, ξ) = P (λI − B̂0(ξ))−1P. (4.25) By using the C∞ extension theorem in [10], we can make C∞ extension of W (σ, τ, ξ) for σ ≤ 0, which is still written as W (σ, τ, ξ) for the simplicity. Denote respectively the eigenvalues and the corresponding eigenfunctions of the operator W (σ, τ, ξ) by µj(σ, τ, |ξ|) and φj(σ, τ, |ξ|), and the point spectrum of the operator W (σ, τ, ξ) by σp(W (σ, τ, ξ)). We have the following results on the spectral analysis for the operator W (σ, τ, ξ) near ξ = 0. Proposition 4.9. There exists a constant r0 > 0 and functions µj(σ, τ, |ξ|), j = ±1, 0, 2, 3 defined on Σ0 = {(σ, τ, ξ) ∈ R × R × R3 : |σ| + |τ | ≤ r0, |ξ| ≤ r0}, and functions σj(κ), τj(κ), j = ±1, 0, 2, 3 defined on I0 = [−r0, r0], such that (i) (a) σp(W (σ, τ, ξ)) = {µj(σ, τ, |ξ|), j = ±1, 0, 2, 3}, (σ, τ, ξ) ∈ Σ0. (b) µj ∈ C∞(Σ0), −1 ≤ j ≤ 3. (c) µj(0, 0, 0) = 1, −1 ≤ j ≤ 3. (d) ∂µj ∂σ (0, 0, 0) = 1 i ∂µj ∂τ (0, 0, 0) = −1, −1 ≤ j ≤ 3. (ii) (a) σj(κ), τj(κ) ∈ C∞(I0), −1 ≤ j ≤ 3. (b) µj(σj(κ), τj(κ), κ) ≡ 1, κ ∈ I0 and −1 ≤ j ≤ 3. (c) σj(κ), τj(κ) satisfy the following asymptotic expansions for κ ∈ I0 and −1 ≤ j ≤ 3, σj(κ) = σ (2) j κ2 +O(κ3), (4.26) τj(κ) = τ (1) j κ+O(κ3), (4.27) where the constants σ (2) j < 0, and τ (1) j with explicit expression as σ (2) j =  3 5 〈L −1P⊥(v1µ4), P⊥(v1µ4)〉, if j = 0, 1 2 〈L −1P⊥(v1µ1), P⊥(v1µ1)〉 + 1 5 〈L −1P⊥(v1µ4), P⊥(v1µ4)〉, if j = ±1, 〈L−1v1µj , v1µj〉, if j = 2, 3 where P⊥ = I − P , P is defined by (2.5), and τ (1) j = { 0, if j = 0, 2, 3, ∓ √ 5 3 , if j = ±1. Moreover, the eigen-projections Pj(σ, τ, ξ), −1 ≤ j ≤ 3 defined by Pj(σ, τ, ξ)f = 〈f, φj(σ, τ, κ)〉φj(σ, τ, κ) for any f ∈ L2 θ (4.28) satisfy (iii) (a) Pj ∈ C∞(Σ0, B(L2 θ)), −1 ≤ j ≤ 3. (b) ∑3 j=−1 Pj(0, 0, 0) = P . We omit the proof of the above proposition. We mention that the method of the asymptotic analysis for σj and τj ,−1 ≤ j ≤ 3 is different from that in [7]. For more details, please refer to [12]. EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 19 Thanks to Proposition 4.9, for any (σ, τ, ξ) ∈ Σ0 it holds that [I −W (σ, τ, ξ)]−1P = 3∑ j=−1 1 1− µj(σ, τ, κ) Pj(σ, τ, ξ). (4.29) Substituting (4.29) into (4.24), it holds for any (σ, τ, ξ) ∈ Σ0 that (λI − B̂(ξ))−1 = (λI − B̂0(ξ))−1 + 3∑ j=−1 (1− µj(σ, τ, κ))−1Uj(σ, τ, ξ), (4.30) where Uj(σ, τ, ξ) = (λI − B̂0(ξ))−1Pj(σ, τ, ξ)(λI − B̂0(ξ))−1. Taking the derivation with respect to τ on (4.30), we obtain (λI − B̂(ξ))−(n+1) = (λI − B̂0(ξ))−(n+1) + 3∑ j=−1 n∑ m=0 (1− µj(σ, τ, κ))−(m+1)U (n) j,m(σ, τ, ξ) (4.31) for any n ∈ N∗, where we have used dm dσm ((σ + iτ)I − B̂(ξ))−1 = dm imdτm ((σ + iτ)I − B̂(ξ))−1 = (−1)mm!((σ + iτ)I − B̂(ξ))−(m+1) for any 0 ≤ m ≤ n, and U (n) j,m(σ, τ, ξ) are given as the linear combinations of products of µj , Uj and their derivatives and satisfy U (n) j,m(σ, τ, ξ) ∈ C∞(Σ0, C(L2 θ)) (4.32) for any θ ∈ R. In particular, U (n) j,n = i−n( ∂µj ∂τ )nUj . By using (i)(d) in Proposition 4.9, it holds that U (n) j,n (0, 0, 0) = Pj(0, 0, 0). (4.33) With the help of Proposition 4.9, (4.31), (4.32), (4.33) and by repeating the similar arguments as proving Theorem 7.1 in [12], we have the following asymptotic behavior of etB̂(ξ) near ξ = 0. Theorem 4.10. Let γ ∈ (−3, 0). Then there exist two constants r1 > 0 and η0 > 0, such that for any |ξ| ≤ r1, θ ∈ R, n ≥ 1 and t ≥ 0 it holds ‖etB̂(ξ)f0‖L2 θ ≤ C ( (1 + t)−n(‖f0‖L2 θ−2n + ρn− 1 2 (κ)‖f0‖L2) + e−η0κ 2t‖Pf0‖L2 ) , (4.34) where ρn− 1 2 (κ) = |κ|−2(n− 1 2 ) and P is defined by (2.5). By Theorem 4.8 and Theorem 4.10, we can obtain the time decay estimates on etB on the space Hl,θ,2. 20 Y. WU, J. SUN EJDE-2021/46 Proof of Theorem 1.1. It holds that ‖etBf0‖2l,θ,2 = ∫ R3 (1 + |ξ|)2l‖etB̂(ξ)f̂0‖2L2 θ dξ = ∫ |ξ|≥r1 (1 + |ξ|)2l‖etB̂(ξ)f̂0‖2L2 θ dξ + ∫ |ξ|≤r1 (1 + |ξ|)2l‖etB̂(ξ)f̂0‖2L2 θ dξ =: I1 + I2, (4.35) where r1 is given by Theorem 4.10. Applying Theorem 4.8, we have I1 ≤ C(1 + t)−2n‖f0‖2l,θ−2n,2. (4.36) Substituting (4.34) in Theorem 4.10 into I2, we obtain I2 ≤ C((1 + t)−2n(‖f0‖2l,θ−2n,2 + ∫ |ξ|≤r1 ρn− 1 2 (|ξ|)2‖f̂0(ξ)‖2L2dξ) + ∫ |ξ|≤r1 e−2η0|ξ|2t‖P f̂0(ξ)‖2L2dξ) ≤ C((1 + t)−2n(‖f0‖2l,θ−2n,2 + ( ∫ |ξ|≤r1 ρn− 1 2 (|ξ|)2q′dξ) 1 q′ ‖f̂0(ξ)‖2L2q,2) + ( ∫ |ξ|≤r1 e−2q′η0|ξ|2tdξ) 1 q′ ‖P f̂0(ξ)‖2L2q,2) ≤ C((1 + t)−2n(‖f0‖2l,θ−2n,2 + ‖f0‖2Lp,2) + (1 + t)−3( 1 p− 1 2 )‖Pf0‖2Lp,2), (4.37) where we have used the Hölder inequality and the Hausdorff-Young inequality with 1 q + 1 q′ = 1, 1 p + 1 2q = 1, q ≥ 1, p ∈ [1, 6/5), and∫ |ξ|≤r1 ρn− 1 2 (|ξ|)2q′dξ <∞, if n < 3 4q′ + 1 2 = 3 2 ( 1 p − 1 2 ) + 1 2 , and(∫ |ξ|≤r1 e−2q′η0|ξ|2tdξ )1/q′ ≤ C(1 + t)−3( 1 p− 1 2 ), and ‖f̂0‖2L2q,2 ≤ C‖f0‖2Lp,2 , ‖P f̂0‖2L2q,2 ≤ C‖Pf0‖2Lp,2 . Then combining (4.35), (4.36) and (4.37), we can obtain (1.10). The proof is complete. � Acknowledgments. The authors are grateful to the anonymous referees for their valuable comments and suggestions that improved the presentation of this article. The authors also would like to thank Professor Hailiang Li for helpful discussions and comments. This research was supported by the National Natural Science Foun- dation of China (Nos. 11931010, 11871047, 11671384, and 11861040), by the key research project of Academy for Multidisciplinary Studies, Capital Normal Uni- versity, by the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds (No. 007/20530290068), and by Science and Technology Project of Education Department of Jiangxi Province, China (GJJ201814). EJDE-2021/46 ASYMPTOTIC BEHAVIOR OF LINEARIZED BOLTZMANN EQUATIONS 21 References [1] R. E. Caflisch; The Boltzmann equation with a soft potential. I. Linear, spatially- homogeneous, Comm. Math. Phys., 74 (1980), no. 1, 71–95. [2] R. E. Caflisch; The Boltzmann equation with a soft potential. II. 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Yakui Wu School of Mathematical Sciences, Capital Normal University, Beijing 100048, China. College of Science, Jiujiang University, Jiangxi 332005, China Email address: 6070010@jju.edu.cn Jiawei Sun (corresponding author) Department of Mathematics, Shandong Normal University, Jinan 250014, China Email address: sunjiawei0122@163.com 1. Introduction 2. Preliminaries 3. Spectrum and resolvent 4. Decay estimates of semigroup 4.1. Estimates at high frequency 4.2. Estimates at low frequency Acknowledgments References