1 Introduction

The purpose of this paper is to discuss the time asymptotic behavior of the solution of the following Cauchy problem governed by a transport operator

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\partial \psi }{\partial t}(x,v,t) = A_H\psi (x,v,t) := T_H\psi (x,v,t) + K\psi (x,v,t) \\ \\ \ \ \ \ \ \ \ \ \ \ \ \ \quad = -v\cdot \nabla _x\psi (x,v,t)-\sigma (v)\psi (x,v,t)+\displaystyle \int _{{\mathbb {R}}^N}\kappa (x,v,v^{\prime })\psi (x,v^{\prime },t) dv^{\prime }, \\ \\ \psi (x,v,0) = \psi _0(x,v), \end{array} \right. \end{aligned}$$
(1)

where \((x,v) \in \varOmega \times {\mathbb {R}}^N\) and K is the partial integral part of \(A_H\) and called the collisional operator. Here \(\varOmega \) is a smooth bounded open convex subset of \({\mathbb {R}}^N\). The function \(\psi (x,v,t)\) represents the number (or probability) density of gas particles having the position x and the velocity v at time t. The functions \(\sigma (\cdot )\) and \(\kappa (\cdot ,\cdot ,\cdot )\) are called, respectively, the collision frequency and the scattering kernel.

The boundary conditions are modeled by:

$$\begin{aligned} \psi _-=H(\psi _+) , \end{aligned}$$
(2)

where \(\psi _-\)(resp. \(\psi _+\)) is the restriction of \(\psi \) to \(\varGamma _-\) (resp. \(\varGamma _+\)), with \(\varGamma _-\) (resp. \(\varGamma _+\)) is the incoming (resp. the outgoing ) part of the phase space and H is a linear bounded operator from a suitable space on \(\varGamma _+\) to a similar one on \(\varGamma _-.\)

Let S be the infinitesimal generator of a \(\hbox {C}_0\)-semigroup \((U(t))_{t \ge 0}\) on a Banach space X and let \({\mathcal {L}}(X)\) denote the set of all bounded linear operators on X. If \(B \in {\mathcal {L}}(X)\), then by the classical perturbation theorem, \(S+B\) generates a strongly continuous semigroup \((V(t))_{t\ge 0}\) given by the Dyson-Phyllips expansion, that is

$$\begin{aligned} V(t) = \displaystyle \sum _{j = 0}^{n-1} U_j(t) + R_n(t), \end{aligned}$$
(3)

where \(U_0(t) = U(t)\), \(U_j(t) = \int _0^tU(s)BU_{j-1}(t-s)ds\)\((j \ge 1)\) and the series (3) converges in \({\mathcal {L}}(X)\) uniformly in bounded times and the \(n^{th}\) remainder term \(R_n(t)\) is given by

$$\begin{aligned} R_n(t)= \displaystyle \sum _{j=n}^{\infty }U_j(t) = \displaystyle \int _{s_1 + s_2 +\cdots +s_n \le t, s_i \ge 0}U(s_1) B\cdots U(s_n)BV\left( t-\displaystyle \sum _{i=1}^ns_i\right) ds_1 \cdots ds_n. \end{aligned}$$
(4)

So the Cauchy problem

$$\begin{aligned} \frac{d\psi }{dt}=(S+B)\psi (t),\quad \psi (0)=\psi _0 \end{aligned}$$
(5)

has a unique classical solution given by \(\psi (t)=V(t)\psi _0\) provided that \(\psi _0\) belongs to D(S). In general this results follows from the Hille–Yosida–Phillips theorem (see, for example [9]). This procedure is not constructive, so in order to get more information on the solution, in particular, its behavior for large times, the knowledge of the spectrum of \(S+B\) or \((V(t))_{t\ge 0}\) plays a central role.

Let \(W\in {\mathcal {L}}(X)\). The essential spectral radius of the operator W is defined by (see [34])

$$\begin{aligned}&r_e(W):\\&\ =\sup \big \{\left| \lambda \right| ; \ \lambda \in \sigma (W)\, \text { but } \lambda \text { is not an isolated eigenvalue of finite algebraic multiplicity}\big \}. \end{aligned}$$

Let \(\omega \) be the type of the semigroup \((U(t))_{t \ge 0}\). It follows from Lemma 2.1 in [34] that there exists \(\omega _e\in [-\infty , \omega ]\) such that

$$\begin{aligned} r_e(U(t))=e^{\omega _e t}, \quad \text {for all } t\in [0,+\infty ). \end{aligned}$$

The number \(\omega _e\) is called the essential type of \((U(t))_{t \ge 0}\).

It is well known that, if some remainder term of the Dyson-Phillips expansion \(R_n(t)\) is compact, then the operators U(t) and V(t) have the same essential type [26, 34, 35] and therefore, for all \(t\ge 0\) we have \(r_{e}(U(t)) = r_{e}(V(t))\). Thus, for \(\nu > 0\), \(\sigma (V(t)) \cap {\mathbb {C}}\backslash B(0,r_{e}(U(t)) + \nu )\) consists of, at most, isolated eigenvalues with finite algebraic multiplicities. Assuming the existence of such eigenvalues, the semigroup \((V(t))_{t\ge 0}\) can be decomposed into two parts: the first containing the time development of finitely many eigenmodes, the second being of faster decay. Using the spectral mapping theorem for the point spectrum, we infer that, for any \(\eta >\omega \), \(\sigma (S+B) \cap \lbrace \text{ Re }\lambda \ge \eta \}\) consists of finitely many isolated eigenvalues say \(\{ \lambda _1,\ldots ,\lambda _q\}\). Let \(\beta _1 = \sup \lbrace \text{ Re }\lambda ,\, \lambda \in \sigma (S+B) , \ \text{ Re }\lambda < \omega \rbrace \), and \(\beta _2 = \min \lbrace \text{ Re }{\lambda }_j,\ 1\le j \le q\rbrace \). The solution of the problem (5) satisfies

$$\begin{aligned} \left\| \psi (t) - \displaystyle \sum _{j=1}^q e^{\lambda _{j}t} \, e^{D_{j}t} P_{j}\psi _{0}\right\| = o ( e^{\beta ^* t}) \text{ with } \beta _1< \beta ^* < \beta _2, \end{aligned}$$
(6)

where \(\psi _0\in D(S)\), \(P_{j}\) and \(D_{j}\) denote, respectively, the spectral projection and the nilpotent operator associated with \(\lambda _{j},\)\(j=1,\, 2, \,\ldots ,\, q.\)

In the context of neutron transport theory, these ideas were initiated by Vidav [33] and developed afterwards by Voigt [34], Weis [35], Mokhtar-Kharroubi [25,26,27] and others (see the reference therein). The compactness of some order remainder term of the Dyson Phillips expansion in \(L^p\)-spaces, \(1\le p<+\infty \), was established only for no-reentry boundary conditions (i.e. with zero incoming flux in the spacial domain) [25, 27, 32, 34, 35]. It is clear that the success of this method is related to the possibility of computing some remainder order term of the Dyson-Phillips expansion and to the possibility of discussing their compactness properties. Unfortunately, when dealing with reentry boundary conditions, except the one-dimensional case with reflective or periodic boundary conditions [6, 7], the semigroup generated by the advection operator is not explicit (see, for example, [16,17,18, 20]) and therefore it is difficult to compute \(R_n(t)\) because its expression involves the boundary operator. So, except some simple cases, this approach does not work.

An alternative way to discuss the time structure of \(\psi (t)\) is the so called resolvent approach.

It is based on the following assumption:

$$\begin{aligned} {\left\{ \begin{array}{ll} \exists m\in {\mathbb {N}} \text { such that }[(\lambda - S)^{-1} B]^{m}\, \text{ is } \text{ compact } \text{ for } \text{ all } \lambda \text { such that } \text {Re} \lambda> \omega , \\ \displaystyle \lim _{\left| \text {Im} \lambda \right| \rightarrow +\infty } \left\| [(\lambda - S)^{-1} B]^{m}\right\| = 0 \text { uniformly on }\big \{ \text {Re} \, \lambda \ge \eta , \, \eta > \omega \big \}.\\ \end{array}\right. } \end{aligned}$$
(7)

In [25, Theorem 1.1], it is proved that under the condition (7), the part of the spectrum of \(A:=S+B\) lying in the half plane \( \{ \text{ Re } \lambda \ge \eta \} \) consists of, at most, a finite number of isolated eigenvalues with finite algebraic multiplicity, say, \( \{ \lambda _1,\cdots ,\lambda _n \rbrace \). Further, the Cauchy problem (5) fulfills

$$\begin{aligned} \Vert \psi (t) - \displaystyle \sum _{i=1}^n e^{\lambda _{i}t} \, e^{D_{i}t} P_{i}\psi _{0}\Vert = o ( e^{\beta ^* t}) \text{ with } \beta _1< \beta ^* < \beta _2 \end{aligned}$$
(8)

provided that \(\psi _0\in D(A^2)\). Here \(P_{i}\), \(D_{i}\), \(\beta _1\), \(\beta _2\) and \(\beta ^*\) have the same meaning as above.

Let us recall the streaming operator \(T_H\) (where \(\left\| H\right\| \not = 1\)) generates a strongly continuous semigroup \((U_H(t))_{t\ge 0}\) on \(L^p(\varOmega \times {\mathbb {R}}^N,dxdv)\), \(1\le p<+\infty \) (in fact, when \(\left\| H\right\| <1\) the generation of the \(\hbox {C}_0\)-semigroup is an immediate consequence of Lumer–Phillips’s theorem while the case of the multiplying boundary condition has been investigated in [22, 24] and sufficient condition on H guaranteeing the generation of the \(\hbox {C}_0\)-semigroup were provided). Since the collision operator K is bounded, \(A_H=T_H+K\) generates also a strongly continuous semigroup \((V_H(t))_{t\ge 0}\) on \(L^p(\varOmega \times {\mathbb {R}}^N,dxdv)\), \(1\le p<+\infty \) given by

$$\begin{aligned} V_H(t) = \displaystyle \sum _{j = 0}^{n-1} U_j^H(t) + R_n^H(t). \end{aligned}$$

For the meaning of the different terms appearing in the last equation, we refer to (3).

The resolvent approach was already applied to transport equations with vacuum boundary conditions \((H=0\)) in bounded geometry [25]. In [16], it is shown that this method works also for one-dimensional transport equation for a large class of boundary conditions. The drawback of this method lies in the fact that, unlike (6), the quantity \(\psi (t)- \sum _{i=1}^n e^{\lambda _{i}t} \, e^{D_{i}t} P_{i}\psi _{0}\) can be evaluated only for initial data \(\psi _0\) belonging to \(D(A_H^2)\) (see [30, Theorem 2.1] or [25, Lemma 2.1]). Note however that, using of Wrobel’s result [36, Theorem 3.3] about the asymptotic behavior of \(\hbox {C}_0\)-semigroups on B-convex space and observing that \(L^p\)-spaces, for \(p\in (1,+\infty )\), are B-convex, we see that the condition \(\psi _0\in D(A_H^2)\) can be relaxed and therefore the estimate (8) holds true for all initial data \(\psi _0\) belonging to \(D(A_H)\) (we refer to [36] for general B-convex spaces, to [31] for vacuum boundary conditions and to [18] for one dimensional transport equation with particular noreentry boundary conditions). Unfortunately, except some particular boundary conditions in slab geometry [20], the results available in \(L^1\)-spaces for reentry boundary conditions assume that \(\psi _0\) belongs to \(D(A_H^2)\).

The purpose of this paper is twofold. First, we shall pursue the investigation started in the work [19] concerning the time asymptotic behavior of the solution to the Cauchy problem (1) and (2) in \(L^1\)-spaces for a slightly more general boundary operator than the bounce back one. Our main goal is to show that the estimate (8) is also valid for all initial data \(\psi _0\) lying in \(D(A_H)\). Our approach uses the following results.

Proposition 1

[21, Corollary 1.1] Let T be the generator of a \(\hbox {C}_0\)-semigroup \((U(t))_{t\ge 0}\) on a Banach space X and let B be a bounded linear operator on X. Let \(\omega \) be the type of \((U(t))_{t\ge 0}\). If there exists \(m \in {\mathbb {N}}\) and \( \eta > \omega \) such that

  1. (a)

    \((\lambda -T)^{-1}[B(\lambda -T)^{-1}]^m\) is compact for all \(\lambda \) such that \(\text {Re} \lambda \ge \eta \),

  2. (b)

    \(\lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty }\left| \mathrm{Im} \lambda \right| \left\| (\lambda -T)^{-1}[B(\lambda -T)^{-1}]^m\right\| = 0\) for every \(\lambda \) such that \(\text {Re} \lambda \ge \eta \)

then \(R_{2m+1}(t)\) is compact on X for each \(t>0\).

Proposition 2

[26, Theorem 2.10] Let T be the generator of a \(\hbox {C}_0\)-semigroup \((U(t))_{t\ge 0}\) on a Banach space X and let B be a bounded linear operator on X. If some remainder term \(R_n(t)\)\((n\ge 1)\) of the Dyson-Phillips expansion is compact, then \((U(t))_{t\ge 0}\) and \((V(t))_{t\ge 0}\) have the same essential type.

We shall establish that all order remainder terms \(R_n^H(t)\), with \(n\ge 9\), of the Dyson-Phillips expansion of the transport semigroup are compact on \(L^1(\varOmega \times {\mathbb {R}}^N, dxdv)\) and therefore, according the semigroup approach, we infer that the estimate (8) holds true for all initial data in \(D(A_H)\) which shows that the condition \(\psi _0\in D(A_H^2)\) can be relaxed and replaced by \(\psi _0\in D(A_H)\). It should be noticed that the transport semigroup is not differentiable nor analytic, so the condition \(\psi _0\in D(A_H)\) is optimal and can not be improved.

Secondly, for \(p\in (1, +\infty ),\) using some arguments due to Brendle [3] (which were refined afterwards by Sbihi [28]) we shall prove that the first order remainder term of the Dyson-Phillips expansion, \(R_1^H(t)\) is compact. In our opinion, regardless of its consequences, this result is interesting in itself. The latter implies that, for all \(t\ge 0\), we have \(\sigma _{ess}(V_H(t))=\sigma _{ess}(U_H(t))\) and therefore \(r_{e}(V_H(t))=r_{e}(U_H(t)).\) Accordingly, the estimate (8) holds true for all \(\psi _0\in D(A_H)\).

The layout of this papers is as follows. In Sect. 2, we fix the functional setting of the problem and we derive some preliminary facts concerning the problem. The aim of Sect. 3 is to establish some lemmas required in the proofs of the results presented in Sect. 4. The main result of this paper are given in Sect. 4. Section 4.1 is devoted to the case where the boundary operator H is dissipative (i.e. \(\left\| H\right\| <1\)) . More precisely, we suppose that H is a sum of the bounce-back boundary operator and a compact one. The main result of this subsection is Theorem 3 which asserts that, even in the space \(L^1(\varOmega \times {\mathbb {R}}^N, dxdv)\), the estimate (8) is satisfied for all initial data belonging to \(D(A_H)\). In Sect. 4.2 we discuss the case of multiplying compact boundary operators. We show in Theorem 5 that the estimate (8) holds also true for initial data \(\psi _0\in D(A_H)\) without any restriction on the value of p. Finally, in Appendix, we give some results about regular collision operators.

Notation: Let X be a Banach space and let A be a linear operator on X. As usually we denote by \(\sigma (A)\) and \(\rho (A)\) the spectrum and the resolvent set of A, respectively. If \(A\in {\mathcal {L}}(X)\), we denote by \(r_\sigma (A)\) the spectral radius of A. If A is an unbounded linear operator on X, we call the spectral bound of A the real defined by \(s(A):=\sup \big \{\text {Re} \lambda \, :\, \lambda \in \sigma (A)\big \}.\) And finally, if A is a closed densely defined linear operator on X, then by the essential spectrum of A we mean the set

$$\begin{aligned} \sigma _{ess}(A)=\bigcap _{C\in {\mathcal {K}}(X)}\sigma (A+C), \end{aligned}$$

where \({\mathcal {K}}(X)\) stands for the ideal of compact operators on X (cf. [29, p. 172]). This definition of the essential spectrum suffices for our own use.

2 Preliminaries

The goal of this section is to recall some basic definitions and results for the usual neutron transport equation which we shall use in the sequel.

Let \(\varOmega \) be a smooth bounded open convex subset of \({\mathbb {R}}^N\). The boundary of the phase space \(\varOmega \times {\mathbb {R}}^n\) writes as \(\partial \varOmega \times {\mathbb {R}}^N:= \varGamma _-\cup \varGamma _+\cup \varGamma _0\) where

$$\begin{aligned} \varGamma _\pm =\big \{(x,v)\in \partial \varOmega \times {\mathbb {R}}^N, \pm v.\nu _x\ge 0\big \} \end{aligned}$$

and

$$\begin{aligned} \varGamma _0=\{(x,v)\in \partial \varOmega \times {\mathbb {R}}^N, \pm v.\nu _x=0\}, \end{aligned}$$

with \(\nu _x\) stands for the outer unit normal vector at \(x\in \partial \varOmega \). We shall suppose throughout this paper that \(\varGamma _0\) is of zero measure with respect to \(d\gamma _xdv\) (which is a natural hypothesis), \(d\gamma _x\) being the Lebesgue measure on \(\partial \varOmega \).

We denote by \({\mathsf {d}}\) its diameter, that is

$$\begin{aligned} {\mathsf {d}}:=\sup \Big \{ \left\| x-y\right\| \, :\, x,y\in \varOmega \Big \}. \end{aligned}$$

Let \(p\in [1,+\infty )\) and let \(X_p\) be the space

$$\begin{aligned} X_p:=L^p(\varOmega \times {\mathbb {R}}^N;dx\, dv). \end{aligned}$$

We define the partial Sobolev space \(W_p\) by

$$\begin{aligned} W_p=\big \{\psi \in X_p\hbox { such that }v.\nabla _x \psi \in X_p\big \}. \end{aligned}$$

It is well known [4, 5, 10] that any function in \(W_p\) possesses traces \(\psi ^\pm \) on \(\varGamma _\pm \) belonging to \(L_{p,\text {loc}}^{\pm }(\varGamma _\pm ;|v.\nu _x|d\gamma _xdv)\). It should be noticed that, in applications, suitable \(L^p\)-spaces for the traces are

$$\begin{aligned} L_p^\pm :=L^p(\varGamma _\pm \, :\, \left| v.\nu _x\right| d\gamma _xdv). \end{aligned}$$

Accordingly, we define the set

$$\begin{aligned} {\widetilde{W}}_p=\big \{\psi \in W_p\, :\, \psi ^-\in L_p^-\big \}. \end{aligned}$$

According to [4, 5, 10], if \(\psi \in W_p\), \(1\le p<+\infty \), and \(\psi ^-\in L_p^-\), then \(\psi ^+\in L_p^+\) and vice versa. More precisely we have the identity

$$\begin{aligned} {\widetilde{W}}_p=\big \{\psi \in W_p\, :\, \psi ^-\in L_p^-\big \}= \big \{\psi \in W_p\, :\, \psi ^+\in L_p^+\big \}. \end{aligned}$$

Definition 1

Let \((x,v)\in {\bar{\varOmega }}\times {{\mathbb {R}}^N}\). We set

$$\begin{aligned} t^{\pm }(x,v)&=\sup \{t>0, \, x \pm sv\in \varOmega , 0<s<t\}\\&=\inf \{t>0, \, x\pm tv\notin \varOmega \} \end{aligned}$$

and

$$\begin{aligned} \tau (x,v):=t_-(x,v) + t_+(x,v) \,\, \text{ for } \text{ any } (x,v) \in {\overline{\varOmega }} \times {\mathbb {R}}^N. \end{aligned}$$

\(\square \)

Hence, for \((x,v)\in \varGamma _\pm \), one has \(t^\pm (x,v)=0\) and in all cases \(x\mp t^\mp (x,v)v\in \varGamma _\mp \). The number \(t^\pm (x,v)\) is the time required by a particle having the position \(x\in \varOmega \) and the velocity \(\pm \, v\in {\mathbb {R}}^N\) to go out \(\varOmega \).

Let \(H\in {\mathcal {L}}(L_p^+,L_p^-)\) be a boundary operator. The streaming operator \(T_H\) is defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} T_H\, :\, D(T_H)\subseteq X_p\longrightarrow X_p\\ \quad \quad \quad \psi \longrightarrow \displaystyle \,T_H\psi (x,v)= -v.\nabla _x\psi (x,v)-\sigma (v)\psi (x,v) \end{array}\right. } \end{aligned}$$

with domain

$$\begin{aligned} D(T_H)=\Big \{\psi \in {\widetilde{W}}_p\ \text { such that }\ \psi ^{-}=H(\psi ^{+})\Big \}. \end{aligned}$$

In this paper, we suppose that \(\sigma (\cdot )\) is a measurable function belonging to \(L^{\infty }({\mathbb {R}}^N,dv)\) and satisfies

$$\begin{aligned} 0\le \sigma (v) \text { for almost all } v\in {\mathbb {R}}^N. \end{aligned}$$

Let \(\varphi \in X_{p}\) and consider the resolvent equation for \(T_H\)

$$\begin{aligned} (\lambda -T_H)\psi =\varphi \end{aligned}$$
(9)

where \(\lambda \) is a complex number and the unknown \(\psi \) must be sought in \(D(T_H)\). Let \(\lambda ^{*}\) denote the real defined by

$$\begin{aligned} \lambda ^{*}:= \inf _{v \in {\mathbb {R}}^N}\sigma (v). \end{aligned}$$

It is shown in [17] that, for \(\text {Re}\lambda >-\lambda ^{*}\), the invertibility of \((\lambda -T_H)\), reduces to the invertibility of the operator \({\mathcal {P}}_\lambda :=I-M_\lambda H\) where \(M_{\lambda }\) is given by

$$\begin{aligned} L_p^- \ni u \longrightarrow M_\lambda u= u(x-\tau (x,v)v,v)e^{-\tau (x,v)(\lambda +\sigma (v))}\in L_p^+. \end{aligned}$$

More precisely following [17], if \({\mathcal {P}}_\lambda \) is invertible, then the solution of (9) writes as

$$\begin{aligned} \psi =B_\lambda H{\mathcal {P}}_\lambda ^{-1}G_\lambda \varphi +C_\lambda \varphi \end{aligned}$$

and therefore the resolvent of \(T_H\) is given by

$$\begin{aligned} (\lambda -T_H)^{-1}=B_\lambda H{\mathcal {P}}_\lambda ^{-1}G_\lambda +C_\lambda . \end{aligned}$$
(10)

Here

$$\begin{aligned}&X_p \ni \varphi \longrightarrow C_\lambda \varphi :=(\lambda -T_0)^{-1}\varphi = \displaystyle \int _0^{t^-(x,v)} e^{- s(\lambda +\sigma (v)) }\varphi (x-sv,v)ds\in X_p,\\&X_p \ni \varphi \longrightarrow G_\lambda \varphi :={C_\lambda \varphi }_{\vert \varGamma ^+}= \displaystyle \int _0^{\tau (x,v)}e^{- s(\lambda +\sigma (v)) }\varphi (x-sv,v)ds\in L_p^+ \end{aligned}$$

and

$$\begin{aligned} L_p^- \ni u \longrightarrow B_\lambda u:=u(x-t^{-} (x,v)v,v)e^{- t^-(x,v)(\lambda +\sigma (v))}\in X_p. \end{aligned}$$

Remark 1

These operators are bounded on their respective spaces. In fact, the norms of \(B_{\lambda }\) and \(C_{\lambda }\) are bounded above, respectively, by \( [p(Re \lambda +\lambda ^*)]^{- {1\over p}}\) and \((Re \lambda +\lambda ^*)^{-1}\) where q denotes the conjugate of p. Moreover, the operator \(M_{\lambda }u = [B_{\lambda }u]_{\mid {\varGamma _+}}\) is a contraction and \(G_{\lambda }\varphi =[C_{\lambda }\varphi ]_{\mid {\varGamma _+}}\) is bounded with norm less than \([q(Re \lambda +\lambda ^*)]^{-{1\over q}}\). \(\square \)

Note that in neutron transport theory, in general, the collision operator K has the form

$$\begin{aligned} K\, :\, \varphi \in X_p \longrightarrow K\varphi (x,v)=\displaystyle \int _{{\mathbb {R}}^N}\kappa (x,v,v')\varphi (x,v')dv', \end{aligned}$$
(11)

where \(\kappa (\cdot ,\cdot ,\cdot )\) is a non-negative measurable function. It is a partial integral operator.

Remark 2

In the remainder of the paper, we will assume that K is a regular collision operator in the sense of [23, 25] and refer the reader to Appendix for formal definition and mains properties of this class of operator.

We close this section to state the following compactness result which will play a fundamental role in this paper.

Theorem 1

Let K be a regular operator and let \(\lambda \) be such that \(\text {Re}\lambda >s(T_H)\).

  1. (1)

    If \(p \in (1,\infty )\), then \((\lambda -T_H)^{-1}K\) and \(K(\lambda -T_H)^{-1}\) are compact on \(X_p\).

  2. (2)

    If \(p=1\), then \(K(\lambda -T_H)^{-1}K\) is weakly compact on \(X_1\).

Proof

The first item of the theorem was established in [17]. So it suffices to prove the second assertion.

Recall that

$$\begin{aligned} K(\lambda -T_H)^{-1}K = KB_{\lambda }H(I-M_{\lambda })^{-1}G_{\lambda }K + KC_{\lambda }K. \end{aligned}$$

Note however that, \(C_{\lambda }\) is nothing else but the resolvent of the streaming operator with vacuum boundary condition \(T_0\). So, we know from [25] that, if K is a regular collision operator on \(X_1\), then the operator \(KC_{\lambda }K\) is weakly compact operator on \(X_1\). Thus, in order to prove the weakly compactness of \(K(\lambda -T_H)^{-1}K\), it suffices to show that the operator \(KB_{\lambda }\) is weakly compact. Since K is a regular collision operator, according to Lemma 11, it suffices to establish the result for a collision operator of the form

$$\begin{aligned} \varphi \in X_1\rightarrow f(v)\int _{{\mathbb {R}}^N}\varphi (x,v') dv' \end{aligned}$$

where \(f(\cdot ) \in L^1({\mathbb {R}}^N;dv)\). For \(\varphi \in L^{1,-}\), one can write

$$\begin{aligned} KB_{\lambda }\varphi (x,v) := f(v)\displaystyle \int _{{\mathbb {R}}^N} B_{\lambda }\varphi (x,v')dv', \end{aligned}$$

where \(B_{\lambda }\varphi (x,v) = \varphi (x-t^-(x,v)v,v)e^{-t^-(x,v)(\lambda +\sigma (v))}\) and \(\left\| B_{\lambda }\right\| \le (\text {Re} \lambda + \lambda ^*)^{-1}\).

This yields that

$$\begin{aligned} \left\| KB_{\lambda }\varphi (x,v)\right\| _{X_1}&\le \displaystyle \int _{{\mathbb {R}}^N} |f(v)| dv\displaystyle \int _{\varOmega \times {\mathbb {R}}^N}| B_{\lambda }\varphi (x,v')|dxdv' \\ \\&\le \left\| f(\cdot )\right\| _{L^1({\mathbb {R}}^N)}\left\| B_{\lambda }\varphi (\cdot ,\cdot )\right\| _{X_1} \\ \\&\le (\text {Re} \lambda + \lambda ^*)^{-1} \left\| f(\cdot )\right\| _{L^1({\mathbb {R}}^N)} \left\| \varphi \right\| _{L^{1,-}}. \end{aligned}$$

So, we conclude that

$$\begin{aligned} \left\| KB_{\lambda }\right\| \le (\text {Re} \lambda + \lambda ^*)^{-1} \left\| f(\cdot )\right\| _{L^1({\mathbb {R}}^N)} . \end{aligned}$$
(12)

The estimate (25) shows that \(KB_{\lambda }\) depends continuously (for the uniform topology) on \(f(\cdot ) \in L^1({\mathbb {R}}^N)\). So, by approximating \(f(\cdot )\) (in the \(L^1\)-norm) by bounded functions, \(KB_{\lambda }\) is a limit (for operator topology) of integral operators with bounded kernel. Hence, \(KB_{\lambda }\) is weakly compact on \(X_1\) (cf., [9, Corollary 11, p. 294]). \(\square \)

Remark 3

We point out that the assertion (2) of Theorem 1 is new and, as in slab geometry [15, Theorem 2.1], the weak compactness of \(K(\lambda -T_H)^{-1}K\) does not depend on the boundary operator (see also Remarks 3.1 and 3.2 in [17]). We note however that this result is proved only for Lebesgue measure on the velocity space. In fact, the same arguments of the proof of this result work for general positive radon measures on the velocity space.\(\square \)

3 Preparatory results

Our aim in this section is to establish some technical lemmas required in the sequel. We define the full transport operator \(A_H \) by \(A_H= T_H + K\). Since the collision operator K is bounded, \(A_H\) is a bounded perturbation of \(T_H\).

We suppose that the boundary operator H has the form

$$\begin{aligned} {\left\{ \begin{array}{ll} H\, :\, L_p^+\longrightarrow L_p^-\\ \quad \quad \psi ^-\longrightarrow \displaystyle \,H\psi ^+=\alpha I_1\psi ^+ + \beta I_2\psi ^+, \ \ \alpha , \beta \in [0,+\infty ), \end{array}\right. } \end{aligned}$$
(13)

where \(I_1\) is a compact operator and \(I_2\) is defined by

$$\begin{aligned} I_2 u(x,v) = u(x,-v) \end{aligned}$$

(\(I_2\) is the so called bounce-back boundary operator). The constants \(\alpha \) and \(\beta \) are chosen so that

$$\begin{aligned} \left\| H\right\| < 1. \end{aligned}$$
(14)

For the sake of simplicity and in order to avoid some technical difficulties, we shall assume in this section that the collision frequency is an even function on \({\mathbb {R}}^N\), that is,

$$\begin{aligned} \sigma (-v) = \sigma (v), \quad \forall v \in {\mathbb {R}}^N. \end{aligned}$$

Since \(\left\| H\right\| < 1\), \({\mathcal {P}}_\lambda \) is invertible and Eq. (10) (see also [13, 17]) shows that, for all \(\lambda \in {\mathbb {C}}\) such that \(\text {Re} \lambda + \lambda ^* > 0\), the resolvent of the operator \(T_H\) is given by

$$\begin{aligned} R(\lambda ,T_H) = \sum _{n \ge 0} B_{\lambda }H(M_{\lambda }H)^{n}G_{\lambda }\psi + C_{\lambda }\varphi . \end{aligned}$$

This shows that

$$\begin{aligned} \Big \{ \lambda \in {\mathbb {C}}: \text {Re} \lambda > -\lambda ^* \Big \} \subseteq \rho (T_H) \end{aligned}$$

and, for any \(\lambda \in {\mathbb {C}}\) such that \(\text {Re} \lambda > -\lambda ^*\), we have

$$\begin{aligned} \left\| KR(\lambda ,T_H)K\right\| \le \sum _{n \ge 0} \left\| KB_{\lambda }H(M_{\lambda }H)^{n}G_{\lambda }K\right\| + \left\| KC_{\lambda }K\right\| . \end{aligned}$$

Let \(\omega >0\) and denote by \(\varGamma _{\omega }\) the set

$$\begin{aligned} \varGamma _{\omega } = \Big \{ \lambda \in {\mathbb {C}}: \text {Re} \lambda \ge -\lambda ^* + \omega \Big \}. \end{aligned}$$

Lemma 1

If K is a regular collision operator, then, for all \(r \in [0,1]\), we have

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty } \left| \mathrm{Im} \lambda \right| ^r \left\| KC_{\lambda }K\right\| = 0 \quad \text {uniformly on } \varGamma _{\omega }. \end{aligned}$$

Proof

According to Remark 8, we may take K in the form (24). So, \(KC_{\lambda }K\) writes in the form

$$\begin{aligned} (KC_{\lambda }K\varphi )(x,v) = \displaystyle \int _{{\mathbb {R}}^N}f(v)h(v^{\prime })dv^{\prime } \displaystyle \int _0^{t^{-}(x,v^{\prime })}e^{-(\lambda + \sigma (v^{\prime }))t}dt \displaystyle \int _{{\mathbb {R}}^N}g(v^{\prime \prime })\varphi (x-tv^{\prime },v^{\prime \prime })dv^{\prime \prime }, \end{aligned}$$

where \(h(v^{\prime }) := g(v^{\prime })f(v^{\prime }) \in L^1({\mathbb {R}}^N)\) with \(f_i(\cdot ) \in L_p({\mathbb {R}}^N;dv)\) and \(g_i(\cdot ) \in L_q({\mathbb {R}}^N;dv)\). Here for simplicity we take \(\alpha (\cdot )=\chi _\varOmega (\cdot )\) (the function characteristic of the set \(\varOmega \)).

This yields the factorization \(KC_{\lambda }K := A_3A_{2}(\lambda )A_1\) with

$$\begin{aligned}&A_1 : X_p \rightarrow L^p(\varOmega ), \quad \psi \rightarrow \displaystyle \int _{{\mathbb {R}}^N} g(v)\psi (x,v) dv,\\&A_2(\lambda ): L^p(\varOmega ) \rightarrow L^p(\varOmega ), \varphi \rightarrow \displaystyle \int _{{\mathbb {R}}^N} h(v) \int _0^{t^-(x,v)}e^{-(\lambda + \sigma (v))t} \varphi (x-tv) dt dv \end{aligned}$$

and

$$\begin{aligned} A_3 :L^p(\varOmega ) \rightarrow X_p,\quad \varphi \rightarrow \varphi (x)f(v). \end{aligned}$$

Since \(A_1\) and \(A_3\) are bounded operators and independent of the parameter \(\lambda \), it suffices to establish the result for \(A_2(\lambda )\). To this end, let \(\varepsilon >0\) and define the operator

$$\begin{aligned} {\left\{ \begin{array}{ll} A^\varepsilon _2\, :\, L^p(\varOmega )\longrightarrow L^p(\varOmega )\\ \varphi \rightarrow \displaystyle \int _{{\mathbb {R}}^N} h(v) \displaystyle \int _{\varepsilon }^{t^-(x,v)}e^{-(\lambda + \sigma (v))t} \varphi (x-tv) dt dv. \end{array}\right. } \end{aligned}$$

Since \(\lim _{\varepsilon \rightarrow 0}\left\| A_2^{\varepsilon }(\lambda )-A_2(\lambda )\right\| _{{\mathcal {L}}(L^p(\varOmega ))}=0\) uniformly on \(\varGamma _{\omega }\), so it suffices to establish the lemma for the operator \(A_2^\varepsilon (\lambda )\). Now, using the convexity of \(\varOmega \), one sees that, for all \((x,v) \in {\overline{\varOmega }} \times {\mathbb {R}}^N\),

$$\begin{aligned} t \in (0 , t^{-}(x,v)) \Leftrightarrow y = x -tv \in \varOmega . \end{aligned}$$

Hence, using the change of variable \(y= x- tv\), we get

$$\begin{aligned} (A_2^\varepsilon (\lambda )\varphi )(x) = \displaystyle \int _{\varOmega } \varphi (y)dy\displaystyle \int _{\varepsilon }^{\infty } h\left( \frac{x-y}{t} \right) e^{-\left( \lambda + \sigma \left( \frac{x-y}{t}\right) \right) t} \frac{dt}{t^N}. \end{aligned}$$

Let us recall that

$$\begin{aligned} \lambda ^{*}:= \inf _{v \in {\mathbb {R}}^N}\sigma (v). \end{aligned}$$

Note that if \(\lambda \in \varGamma _{\omega }\), then

$$\begin{aligned} \mathrm{Re}\lambda +\lambda ^*-\frac{\omega }{2}\ge \frac{\omega }{2}>0. \end{aligned}$$
(15)

For all \(v\in {\mathbb {R}}^N\), we have

$$\begin{aligned} \sigma (v)-\lambda ^*+\frac{\omega }{2}\ge \frac{\omega }{2}>0. \end{aligned}$$
(16)

Without loss of generality we may suppose that \(h(\cdot )\) is a simple measurable function. For \(x\in \varOmega \), consider the function \(\psi _x(\cdot )\) defined by

$$\begin{aligned} (\varepsilon ,+\infty )\ni t \longmapsto \psi _x(t)=\frac{1}{t^N}e^{-(\sigma (\frac{x}{t})-\lambda ^*+\frac{\omega }{2})t}. \end{aligned}$$

It is clear that (16) implies \(0\le \psi _x(\cdot )\in L^1(\varepsilon ,+\infty ).\) We denote by \((l_{x,i}(\cdot ))_{i\in {\mathbb {N}}}\) a sequence of non negative step functions which converges to \(\psi _x(\cdot )\) almost everywhere and satisfying

$$\begin{aligned} 0\le l_{x,i}(\cdot ) \le \psi _x(\cdot ) \le \frac{1}{\varepsilon ^N} \quad \text {for all } i\in {\mathbb {N}}. \end{aligned}$$
(17)

Let \(A_{2,i}^\varepsilon (\lambda )\) be the sequence of operators defined, for all \(i\in {\mathbb {N}}\), by

$$\begin{aligned} {\left\{ \begin{array}{ll} A_{2,i}^\varepsilon (\lambda )\, :\, L^p(\varOmega )\longrightarrow L^p(\varOmega )\\ \varphi \longmapsto \displaystyle \int _{\varOmega } \varphi (y)dy\displaystyle \int _{\varepsilon }^{\infty } h\left( \frac{x-y}{t} \right) e^{-\left( \lambda + \lambda ^*-\frac{\omega }{2}\right) t}l_{x-y,i}(t)dt \end{array}\right. } \end{aligned}$$

We claim that, for any \(\varepsilon >0\), \((A_{2,i}^\varepsilon (\lambda ))_{i\in {\mathbb {N}}}\) converges uniformly on \(\varGamma _{\omega }\) to \(A_{2}^\varepsilon (\lambda )\) in \({\mathcal {L}}(L^p(\varOmega )).\) Indeed, for \(\varphi \in L^p(\varOmega )\), we have

$$\begin{aligned}&\left\| (A_{2,i}^\varepsilon (\lambda )-A_{2}^\varepsilon (\lambda ))\varphi \right\| _{L^p(\varOmega )}^p \\&\quad = \displaystyle \int _\varOmega dx \left| \displaystyle \int _{\varOmega } \varphi (y)dy\displaystyle \int _{\varepsilon }^{\infty } h\left( \frac{x-y}{t} \right) e^{-\left( \lambda + \lambda ^*-\frac{\omega }{2}\right) t}\{l_{x-y,i}(t)-\psi _{x-y}(t)\}dt\right| ^p. \end{aligned}$$

Applying the Hölder inequality, we get

$$\begin{aligned}&\left\| (A_{2,i}^\varepsilon (\lambda )-A_{2}^\varepsilon (\lambda ))\varphi \right\| _{L^p(\varOmega )}^p \\&\quad \le \left| \varOmega \right| ^{\frac{p}{q}} \displaystyle \int _\varOmega dx \displaystyle \int _{\varOmega }dy\left| \displaystyle \int _{\varepsilon }^{\infty } \varphi (y)h\left( \frac{x-y}{t}\right) e^{-\left( \lambda + \lambda ^*-\frac{\omega }{2}\right) t}\{l_{x-y,i}(t)-\psi _{x-y}(t)\}dt\right| ^p \end{aligned}$$

where \(\left| \varOmega \right| \) denotes the Lebesgue measure of \(\varOmega \). It follows from Fubini’s theorem and the change of variable \(x'=x-y\), that

$$\begin{aligned}&\left\| (A_{2,i}^\varepsilon (\lambda )-A_{2}^\varepsilon (\lambda ))\varphi \right\| _{L^p(\varOmega )}^p \\&\quad \le \sup \left| h(\cdot )\right| ^p \left| \varOmega \right| ^{\frac{p}{q}} \displaystyle \int _\varOmega dx'\Big (\displaystyle \int _{\varepsilon }^{\infty } e^{-\Big (\lambda + \lambda ^*-\frac{\omega }{2}\Big )t}\{l_{x',i}(t)-\psi _{x'}(t)\}dt\Big )^p\left\| \varphi \right\| _{L^p(\varOmega )}^p. \end{aligned}$$

Hence

$$\begin{aligned} \left\| A_{2,i}^\varepsilon (\lambda )-A_{2}^\varepsilon (\lambda )\right\| ^p \le \sup \left| h(\cdot )\right| ^p \left| \varOmega \right| ^{\frac{p}{q}} \displaystyle \int _\varOmega dx'\Big (\displaystyle \int _{\varepsilon }^{\infty } e^{-(\mathrm{Re}\lambda + \lambda ^*-\frac{\omega }{2})t}\{l_{x',i}(t)-\psi _{x'}(t)\}dt\Big )^p. \end{aligned}$$

Now, using Eqs. (15)–(17), we get

$$\begin{aligned} e^{-(\mathrm{Re}\lambda + \lambda ^*-\frac{\omega }{2})t}\{l_{x',i}(t)-\psi _{x'}(t)\}\le \frac{1}{\varepsilon }e^{-\frac{\omega }{2}t}. \end{aligned}$$

By the continuity of the norm and the Lebesgue dominated convergence theorem, we conclude \(\lim _{i\rightarrow \infty }\left\| A_{2,i}^\varepsilon (\lambda )-A_{2}^\varepsilon (\lambda )\right\| =0\) uniformly on \(\varGamma _{\omega }\). This proves our claim.

Now we have only to prove the lemma for \(A_{2,i}^{\varepsilon }(\lambda )\). Note that easy calculations show that

$$\begin{aligned} \left\| A_{2,i}^{\varepsilon }\right\| ^p \le \left| \varOmega \right| ^{\frac{p}{q}} \displaystyle \int _\varOmega dx\left| \displaystyle \int _{\varepsilon }^{\infty } e^{-(\lambda + \lambda ^*-\frac{\omega }{2})t}h\left( \frac{x}{t}\right) l_{x,i}(t)dt\right| ^p. \end{aligned}$$

Let x be a fixed real in \(\varOmega \). Clearly, the map \(G_x(\cdot ): (\varepsilon ,\infty ) \rightarrow {\mathbb {R}}^N,\)\(t\longmapsto e^{-\lambda ^4t}h(\frac{x}{t})l_{x,i}(t)\) is a simple function. Let \((t_j)_{1\le j \le m}\) denote a subdivision of its support such that

$$\begin{aligned} \forall j \in \{1,\ldots , m-1\}, \quad G_x(t)=G_x(t_j) \quad \forall t\in [t_j,t_{j+1}[. \end{aligned}$$

Hence

$$\begin{aligned} \displaystyle \int _{\varepsilon }^\infty e^{-\left( -\lambda ^4+\lambda + \lambda ^*-\frac{\omega }{2}\right) t} G_x(t)dt&= \displaystyle \sum _{j=1}^{m-1}G_x(t_j) \displaystyle \int _{t_j}^{t_{j+1}}e^{-\left( -\lambda ^4+\lambda + \lambda ^*-\frac{\omega }{2}\right) t}dt\\&= \frac{1}{-\lambda ^4+\lambda + \lambda ^*-\frac{\omega }{2}}\\&\quad \times \displaystyle \sum _{j=1}^{m-1}G_x(t_j) \Big (e^{-\left( -\lambda ^4+\lambda + \lambda ^*-\frac{\omega }{2}\right) t_j}-e^{-\left( -\lambda ^4+\lambda + \lambda ^*-\frac{\omega }{2}\right) t_{j+1}}\Big ) \end{aligned}$$

and consequently

$$\begin{aligned} \left| \displaystyle \int _{\varepsilon }^\infty e^{-\left( -\lambda ^4+\lambda + \lambda ^*-\frac{\omega }{2}\right) t} G_x(t)dt\right| ^p \le \frac{e^{-p\mathrm{Im}\lambda ^4\left( 1+\frac{\text {Re}\lambda }{\text {Im}\lambda }-\frac{6\text {Re}\lambda ^2}{\text {Im}\lambda ^2}\right) }\sup \left| h(\cdot )\right| ^p(2(m-1))^p}{\left| \text {Im}\lambda \right| ^{4p}\varepsilon ^{Np}}. \end{aligned}$$

Therefore

$$\begin{aligned} \left| \text {Im}\lambda \right| ^r\left\| A_{2,i}^{\varepsilon }\right\| \le \frac{2(m-1)\left| \varOmega \right| ^pe^{-\mathrm{Im}\lambda ^4\left( 1+\frac{\text {Re}\lambda }{\text {Im}\lambda }-\frac{6\text {Re}\lambda ^2}{\text {Im}\lambda ^2}\right) }\sup \left| h(\cdot )\right| }{\left| \text {Im}\lambda \right| ^{4-r}\varepsilon ^{N}}. \end{aligned}$$

This ends the proof because, for any \(r\in [0,1]\), we have

$$\begin{aligned} \displaystyle \lim _{\left| \text {Im}\right| \lambda \rightarrow \infty } \frac{2(m-1)\left| \varOmega \right| ^pe^{-\mathrm{Im}\lambda ^4\left( 1+\frac{\text {Re}\lambda }{\text {Im}\lambda }-\frac{6\text {Re}\lambda ^2}{\text {Im}\lambda ^2}\right) }\sup \left| h(\cdot )\right| }{\left| \text {Im}\right| ^{4-r}\varepsilon ^{N}} =0. \end{aligned}$$

\(\square \)

Our next task is to estimate the quantity \(\left| \mathrm{Im} \lambda \right| ^r\left\| KB_{\lambda }H(I-M_{\lambda }H)^{-1}G_{\lambda }K\right\| \) as \(\left| \mathrm{Im} \lambda \right| \) goes to \(+\infty \) for all \(r \in [0,1]\). Set \(H(\lambda ) := M_{\lambda }H\). Since \(\left\| H\right\| <1\), the use of Remark 1 and Eq. (10) imply that

$$\begin{aligned} \left\| KB_{\lambda }H(I-H(\lambda ))^{-1}G_{\lambda }K\right\| \le \displaystyle \sum _{n\ge 0} \left\| KB_{\lambda }H(H(\lambda ))^nG_{\lambda }K\right\| . \end{aligned}$$

Further, according to (13), one sees that the operator \(H(\lambda )\) may be expressed as

$$\begin{aligned} H(\lambda ) = H^1(\lambda ) + H^{2}(\lambda ), \end{aligned}$$

where \(H^{1}(\lambda ) := \alpha M_{\lambda }I_1\) et \(H^{2}(\lambda ) := \beta M_{\lambda }I_2\). Since the operators \(H^i(\lambda )\), \(i = 1,2\), do not commute, the operator \((H(\lambda ))^n = (H^1(\lambda ) + H^2(\lambda ))^n\) is a sum of \(2^n\) different terms, that is, \((H(\lambda ))^n = \sum _{j=1}^{2^n}P_j\) where each \(P_j\) is a product of n factors formed from the operators \(H^i(\lambda ), \ i=1,2\). These \(2^n\) terms \(P_j\) may be divided into two classes \(C_1\) and \(C_2\). The class \(C_i\) consists of those \(P_j\), \(1 \le j \le (2)^n\), for which the last factor (on the right hand side) is \(H^i(\lambda )\), \(i=1,2\). It follows that, for each integer n, we have

$$\begin{aligned} \left\| KB_{\lambda }H(H(\lambda ))^nG_{\lambda }K\right\| \le \displaystyle \sum _{j=1}^{2^n}\left\| KB_{\lambda }HP_jG_{\lambda }K\right\| , \end{aligned}$$

where j is an integer belonging to the set \(\big \{1,2,\ldots , 2^n\big \}\).

Lemma 2

Let K be a regular collision operator. If the boundary operator H is in the form (13) and satisfies (14), then

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty }\left| \mathrm{Im} \lambda \right| ^r\left\| KB_{\lambda }H(I-H(\lambda ))^{-1}G_{\lambda }K\right\| = 0, \quad \text {uniformly on } \varGamma _{\omega }, \end{aligned}$$

for all \(r \in [0,1]\).

Before proving this lemma, let us establish two preliminary results.

Lemma 3

If the hypotheses of the Lemma 2 hold true, then, for all \(r \in [0,1]\), we have

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty }\left| \mathrm{Im} \lambda \right| ^r\left\| I_1G_{\lambda }K\right\| = 0 \quad \text {uniformly on } \varGamma _{\omega }. \end{aligned}$$

Proof

Since \(I_1\) is a compact operator, as in Remark 8, we can establish the result for an operator of rank one, that is

$$\begin{aligned}&I_1: L_p^+ \rightarrow L_p^-, \varphi \rightarrow I_1 \varphi (x,v) = \theta _1(x,v)\displaystyle \int _{\varGamma _+} \theta _2(x^{\prime },v^{\prime })\varphi (x^{\prime },v^{\prime })|v^{\prime }.\nu _{x^{\prime }}|d\gamma _{x^{\prime }}dv^{\prime },\\&\quad \forall (x,v) \in \varGamma _-, \end{aligned}$$

where \(\theta _1 \in L_p^-\) and \(\theta _2 \in L_q^+\). So, the operator \(I_1G_{\lambda }K\) writes in the form

$$\begin{aligned} (I_1G_{\lambda }K\varphi )(x,v)&= \theta _1(x,v)\displaystyle \int _{\varGamma _+}\theta _2(x^{\prime },v^{\prime })|v^{\prime }. \nu _{x^{\prime }}|d\gamma _{x^{\prime }}dv^{\prime } \displaystyle \int _0^{\tau (x^{\prime },v^{\prime })}e^{-(\lambda + \sigma (v)) t}f(v^{\prime })dt \\&\quad \times \displaystyle \int _{{\mathbb {R}}^N} g(v^{\prime \prime })\varphi (x^{\prime }-tv^{\prime },v^{\prime \prime })dv^{\prime \prime }. \end{aligned}$$

Hence, it can be factorized as

$$\begin{aligned} I_1G_{\lambda }K \,=\, A_3\, A_{2}(\lambda )\, A_1, \end{aligned}$$

where

$$\begin{aligned}&A_1: \psi \in X_p \rightarrow \displaystyle \int _{{\mathbb {R}}^N} g(v) \psi (x,v) dv \in L_p(\varOmega ),\\&A_{2}(\lambda )\, :\, \varphi \in L_p(\varOmega ) \rightarrow \displaystyle \int _{\varGamma _+}\theta _2(x,v)f(v)|v.\nu _{x}|d\gamma _{x}dv \displaystyle \int _0^{\tau (x,v)}e^{-(\sigma (v) + \lambda ) t}\varphi (x-tv) dt \in {\mathbb {R}}\end{aligned}$$

and

$$\begin{aligned} A_3 \, :\, \gamma \in {\mathbb {R}}\rightarrow \gamma \theta _1(x,v) \in L_p^-. \end{aligned}$$

Since \(A_1\) and \(A_3\) are bounded linear operators independent of the parameter \(\lambda \), it suffices to establish the lemma for \(A_2(\lambda )\). Let \(\varepsilon >0\) and define the operator \(A_{2}^\varepsilon (\lambda )\) by

$$\begin{aligned} {\left\{ \begin{array}{ll} A_{2}^\varepsilon (\lambda )\, :\, L^p(\varOmega )\longrightarrow {\mathbb {R}}\\ \varphi \rightarrow \displaystyle \int _{\varGamma _+}\theta _2(x,v)f(v)|v.\nu _{x}|d\gamma _{x}dv\displaystyle \int _{\varepsilon }^{\tau (x,v)}e^{-(\sigma (v) + \lambda ) t}\varphi (x-tv)dt. \end{array}\right. } \end{aligned}$$

Since \(A_2^{\varepsilon }(\lambda )\rightarrow A_2(\lambda )\), in the operator norm, as \(\varepsilon \rightarrow 0\), it suffices to show the lemma for the operator \(A_2^\varepsilon (\lambda )\).

Let \((x,v) \in \varGamma _+\). We know from Sect. 2 that \(\tau (x,v) = t^-(x,v)\), so using the convexity of \(\varOmega \), we get the following equivalence

$$\begin{aligned} 0< t < t^-(x,v) \Longleftrightarrow y = x - tv \in \varOmega \end{aligned}$$

and therefore the change of variable \(y = x - tv\) leads to

$$\begin{aligned} A_2^\varepsilon (\lambda )\varphi = \displaystyle \int _{\varOmega } \varphi (y) F_{\lambda ,\varepsilon }(y)dy, \end{aligned}$$

where

$$\begin{aligned} F_{\lambda ,\varepsilon }(y) = \displaystyle \int _{\varepsilon }^{\infty } \frac{dt}{t^N}\displaystyle \int _{\partial {\varOmega ^+}} \theta _2\left( x,\frac{x-y}{t}\right) f\left( \frac{x-y}{t}\right) |\frac{x-y}{t}.\nu _{x}|e^{-\left( \lambda + \sigma \left( \frac{x-y}{t}\right) \right) t}d\gamma _{x} . \end{aligned}$$

Now arguing as in the last part of the proof of Lemma 1 we reach the desired result. \(\square \)

4 Main results

It should be noticed that the streaming operator \(T_H\) with conservative boundary conditions does not generate a strongly continuous but it possesses an extension \({\widetilde{T}}_H\) which generates a strongly continuous semigroup. So, in this work we consider only dissipative boundary conditions \(\left\| H\right\| <1\) and multiplying boundary conditions \(\left\| H\right\| >1\). A natural hypothesis (from physical view point) is that the boundary operator H is positive in the lattice sense.

4.1 Dissipative boundary conditions

As in the previous section, the collision frequency \(\sigma (\cdot )\) is also assumed to be an even on \({\mathbb {R}}^N\) in this subsection, that is,

$$\begin{aligned} \sigma (v)=\sigma (-v), \quad \text {for all} \, v \in {\mathbb {R}}^N. \end{aligned}$$

Now we ready to prove the following result.

Theorem 2

Let K be a regular collision operator and let H be a boundary operator satisfying (13) and (14). If the hypotheses of Theorem 1 hold true, then, \([(\lambda - T_H)^{-1}K]^n\) is compact on \(X_p\)\((1 \le p < \infty )\) for \(n \ge 4\). Further, for all \(r \in [0,1]\), we have

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty }\left| \mathrm{Im} \lambda \right| ^{r}\left\| K(\lambda - T_H)^{-1}K\right\| = 0 \ \ \text {uniformly on } \ \ \varGamma _{\omega }. \end{aligned}$$

Proof

If \(p\in (1,+\infty )\), then, according to Theorem 1(1), the operator \((\lambda -T_H)^{-1}K\) is compact on \(X_p\). If \(p=1\), then, by Theorem 1(2), we infer that \(K(\lambda -T_H)^{-1}K\) weakly compact on \(X_1\) and therefore \((\lambda -T_H)^{-1}K(\lambda -T_H)^{-1}K\) is weakly compact on \(X_1\). Next, this together with the fact that \(X_1\) admits the Dunford–Pettis property, implies that \([(\lambda -T_H)^{-1}K(\lambda -T_H)^{-1}K]^2= [(\lambda -T_H)^{-1}K]^4\) is compact on \(X_1\) which proves the first assertion. The second statement follows immediately from Lemmas 1 and 2. \(\square \)

The remainder of the subsection is devoted to give a spectral decomposition of the solution to Problem (1). Before going further, we first define the set \(\sigma _{as}(A_H)\) (the asymptotic spectrum of the operator \(A_H\)) by

$$\begin{aligned} \sigma _{as}(A_H) := \sigma (A_H) \cap \{ \lambda \in {\mathbb {C}}: \text {Re} \lambda > -\lambda ^* \}. \end{aligned}$$

Lemma 4

If the hypotheses of Theorem 2 hold true, then

  1. (1)

    \(\sigma _{as}(A_H)\) consists of, at most, isolated eigenvalues with finite algebraic multiplicities;

  2. (2)

    if \(\varepsilon > 0\) then \(\sigma (A_H)\cap \{ \lambda \in {\mathbb {C}}: \text {Re} \lambda > -\lambda ^*+ \varepsilon \}\) is finite;

  3. (3)

    if \(\varepsilon > 0\) then \(\left\| (\lambda - A_H)^{-1}\right\| \) is uniformly bounded in \(\{ \lambda \in {\mathbb {C}}: \text {Re} \lambda > -\lambda ^*+ \varepsilon \}\) for large \(\left| \mathrm{Im} \lambda \right| .\)

Proof

Let \(\lambda \) be such that \(\text {Re} \lambda > -\lambda ^*\). Since K is a regular collision operator, according to Theorem 1 with \(n=4\), the operator \([(\lambda - T_H)^{-1}K]^4\) is compact on \(X_p\), \(1 \le p \le \infty \). Next, applying Theorem 2 with \(r=0\), we get

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty } \left\| [(\lambda - T_H)^{-1}K]^4\right\| = 0 \quad \text {uniformly in} \, \varGamma _{\omega }. \end{aligned}$$

Now, the result follows from [25, Lemma 1.1]. \(\square \)

Assume that K is a regular collision operator on \(X_p\) and H satisfies the hypotheses (13) and (14). Then, by Lemma 4, the spectrum of \(A_H\) in the half plane \(\{ \lambda \in {\mathbb {C}}: \text {Re} \lambda > -\lambda ^*+ \varepsilon \}\) consists of, a finite number of isolated eigenvalues with finite algebraic multiplicity which we denote \(\lambda _1,\lambda _2,\ldots ,\lambda _n\). Let \(P_i\) and \(D_i\) denote, respectively, the spectral projection and the nilpotent operator associated with \(\lambda _i, i=1,2,\ldots ,n\). Then \(P:= P_1+\cdots +P_n\) is the spectral projection of the compact set \(\{ \lambda _1, \lambda _2,\ldots , \lambda _{n} \}\). According to the spectral decomposition theorem corresponding to the compact set \(\{ \lambda _1, \lambda _2,\ldots , \lambda _{n} \}\) (see, for example, [12]), we may write

$$\begin{aligned} V_H(t) = {\tilde{V}}_H(t) + \sum _{i=1}^ne^{\lambda _it}e^{D_it}P_i \end{aligned}$$

where \({\tilde{V}}_H(t) := V_H(t)(I-P)\) is the \(C_0\)-semigroup on the Banach space \((I-P)X_p\) with generator \({\tilde{A}}_H := A_H(I-P)\) (\({\tilde{A}}_H\) is the part of \(A_H\) on the closed subspace \((I-P)X_p\)).

Now, we are in a position to give the main result in this paper.

Theorem 3

Let \(p=1\) and K be a regular collision operator. Assume that the boundary operator satisfies (13) and (14). Then, there exists \(\varepsilon > 0\) small enough and \(M>0\) such that

$$\begin{aligned} \left\| V_H(t)\psi _0- \displaystyle \sum _{i=1}^{n}e^{(\lambda _i + D_i)t}P_i\psi _0\right\| _{X_1} \le Me^{(-\lambda ^*+ \varepsilon )t}, \quad \forall t>0, \end{aligned}$$

where \(\psi _0 \in D(A_H)\).

Proof

Let \(\lambda \) be a complex number satisfying \(\text {Re}\lambda > -\lambda ^*\). We know from Theorem 1 that if K is a regular collision operator, then \((\lambda -T_H)^{-1}[K(\lambda -T_H)^{-1}]^4\) is compact on \(X_1.\)

So, applying Theorem 2 with \(r=1\), we get

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty } \left| \mathrm{Im} \lambda \right| \left\| (\lambda -T_H)^{-1}[K(\lambda -T_H)^{-1}]^4\right\| = 0. \end{aligned}$$

Now, according to Proposition 1, we get the compactness of \(R_n^H(t)\) for \(n\ge 9\). Then by Proposition 2, the \(\hbox {C}_0\)-Semigoups \(U_H(t)\) and \(V_H(t)\) have the same essential type, that is, for all \(t\ge 0\) we have

$$\begin{aligned} r_{e}(U_H(t)) = r_{e}(V_H(t)) \le e^{-\lambda ^*t}. \end{aligned}$$

This implies that, outside the spectral disc \(|\mu | \le e^{(\varepsilon -\lambda ^*)t}\), the spectrum of \(V_H(t)\) consists of, at most, a finite number of isolated eigenvalues with finite algebraic multiplicity. On other hand, all points \(\mu '\) satisfying \(\left| \mu '\right| \ge e^{(\varepsilon -\lambda ^*)t}\) belong to the resolvent set of \({\tilde{V}}_H(t)\) and consequently,

$$\begin{aligned} \left\| {\tilde{V}}_H(t)\right\| < e^{(\varepsilon -\lambda ^*)t}. \end{aligned}$$

Hence, \(\left\| {\tilde{V}}_H(t)\right\| = o(e^{(-\lambda ^*+ \varepsilon )t})\) as \(t \rightarrow \infty \). This complete the proof. \(\square \)

In the remainder of this subsection, our aim is to establish a similar result to that of Theorem 3 for \(p\in (1,+\infty )\). In fact, we have the following result.

Theorem 4

Let \(p \in (1,\infty )\) and let K be a regular collision operator. Assume that the boundary operator satisfies (13) and (14). Then, there exists \(\varepsilon > 0\) small enough and \(M>0\) such that

$$\begin{aligned} \left\| V_H(t)\psi _0- \displaystyle \sum _{i=1}^{n}e^{(\lambda _i + D_i)t}P_i\psi _0\right\| _{X_p} \le Me^{(-\lambda ^*+ \varepsilon )t}, \quad \forall t>0, \end{aligned}$$

where \(\psi _0 \in D(A_H)\).

The proof of the Theorem 4 is based on the following result.

Proposition 3

Assume that the hypotheses of Theorem 4 hold true, then the first order remainder term of the Dyson-Phillips expansion, \(R_1^H(t)\), is compact on \(X_p\).

As an immediate consequence of Proposition 3 we have

Corollary 1

Assume that the hypotheses of Theorem 4 hold true, then

$$\begin{aligned} \sigma _{ess}(V_H(t)) = \sigma _{ess}(U_H(t)) \text { for any } t > 0. \end{aligned}$$

Proof of Proposition 3

Making use Eq. (4), we have

$$\begin{aligned} R_1^H(t) = \displaystyle \int _0^t U_H(s)KV_H(t-s)ds. \end{aligned}$$

Thus, it is obvious that \(R_1^H(t)\) depends linearly and continuously, in the norm operator topology, on the collision operator K. On the other hand, according to Remark 8K has the following form

$$\begin{aligned} K : X_p \rightarrow X_p, \varphi \rightarrow K\varphi (x,v) = \alpha (x)\displaystyle \int _{{\mathbb {R}}^N} f(v)g(v^{\prime })\varphi (x,v^{\prime })dv^{\prime }, \, p\in (1,\infty ) \end{aligned}$$

where f and g are continuous functions with compact supports. Hence, \(R_1^H(t)\) maps \(X_q\) into itself for all \(q \in (1,\infty )\). Taking into account of Krasnoselskii interpolation [14, Theorem 3.10, p. 57], we may restrict ourselves to the Hilbert space \(X_2\). On the other hand, let \(\lambda \) be a complex number such that \(\text {Re} \lambda > -\lambda ^*\). According to Theorem 1(1), we obtain the compactness of \((\lambda - T_H)^{-1}K\). Furthermore, using Theorem 2 with \(r=0\), we get

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty } \left\| K(\lambda - T_H)^{-1}K]\right\| = 0 \quad \text {uniformly in} \, \varGamma _{\omega }. \end{aligned}$$

Finally, the dissipativity of \(T_H\) (because H satisfies the condition (14)) together with [28, Theorem 2.2, Corollary 2.1] give the desired result. \(\square \)

Proof of Theorem 4

It is clear that, according to Corollary 1, we have

$$\begin{aligned} r_{ess}(V_H(t)) = r_{ess}(U_H(t)), \quad \text {for all } t>0. \end{aligned}$$

Now, arguing as in the last part of the proof of the Theorem 3, we obtain the desired result. \(\square \)

Remark 4

(a) It should be noticed that, by using the so called resolvent approach, one can give a different proof of Theorems 3 and 4 based on the Riesz-Thorin interpolation theorem. One of the main steps in the proofs consists in showing that the asymptotic spectrum of the operator \(A_H\) remains unchanged in all \(X_p\) for \(p\in [1,\infty )\). Unlike our proofs, this approach uses many complex computations (cf. [20, Section 5]).

(b) It seems that the compactness in the space \(L^1(\varOmega \times {\mathbb {R}}^N,dxdv)\) of the first order remainder term of the Dyson-Phillips expansion, \(R_1^H(t)\), is an open question.

4.2 Multiplying compact boundary conditions

The aim of this subsection is to extend the results of the last subsection to the case of multiplying compact boundary conditions. Here we follow the same approach as in the preceding subsection.

We now introduce the following two assumptions required in the sequel. we shall assume that following condition is satisfied:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {Assume that } \, \varOmega \, \text {is a smooth open subset of }\, {\mathbb {R}}^N\, \text {for which }\\ \text { there exists} \, \delta>0 \, \text {such that } \, \displaystyle \text { ess inf }_{(x,v)\in \varGamma _+} \tau (x,v)>\delta . \end{array}\right. } \end{aligned}$$
(18)

Note that, in general, the sojourn time \(\tau (x,v)\) may be arbitrary big as well as arbitrary small. Our assumption means that the boundary operator H vanish on the set \(\big \{(x,v)\in \varGamma _+\, :\, \tau (x,v)\le \delta \big \}\) (the tangential velocities are not taken into account by the boundary operator regardless of its norm). We suppose further that the boundary operator satisfies

$$\begin{aligned} \left\| H\right\| > 1. \end{aligned}$$
(19)

Remark 5

Note that, for streaming operator \(T_H\) in multidimensional geometry with a positive boundary operator satisfying (19), it is established in [22, Theorem 5.2] that, under the condition (18), \(T_H\) generates a strongly continuous semigroup on the space \(L^1(\varOmega \times V, dxd\mu (v))\) where \(V\subset {\mathbb {R}}^N\) is the space of admissible velocities and \(\mu (\cdot )\) a positive Radon measure on V. Using the same idea and a renormalized argument it is proved that this result holds also true for \(L^p\)-spaces (see [2, 24]).

Set

$$\begin{aligned} \lambda _0 = -\lambda ^* + \displaystyle \frac{1}{\delta }\ln (\left\| H\right\| ). \end{aligned}$$

As in Sect. 2, in order to derive the expression of \((\lambda - T_H)^{-1}\), we consider the equation \( (\lambda - T_H)\psi = \varphi , \) where \(\varphi \) is a given function in \(X_p\), \(\lambda \) is a complex number. The unknown \(\psi \) must be sought in \(D(T_H)\). For \(\text {Re}\lambda >-\lambda ^{*}\), the invertibility of \((\lambda -T_H)\), reduces to the invertibility of the operator \({\mathcal {P}}_\lambda :=I-M_\lambda H\) (cf. [17]). In particular, if \(\lambda \) is such that \(\text {Re}\lambda > \lambda _0\), then \(\left\| M_{\lambda }H\right\| < 1\). Let \(\varGamma _{\lambda _0}\) be the set defined by

$$\begin{aligned} \varGamma _{\lambda _0} := \{ \lambda \in {\mathbb {C}}\, \text { such that } \text {Re} \lambda > \lambda _0 \}. \end{aligned}$$

Clearly, if \(\lambda \in \varGamma _{\lambda _0}\), the operator \({\mathcal {P}}_\lambda \) is invertible and

$$\begin{aligned} ({\mathcal {P}}_\lambda )^{-1} = \sum _{n \ge 0} (M_\lambda H)^n. \end{aligned}$$
(20)

Next, substituting (20) in the Eq. (10), we get

$$\begin{aligned} (\lambda -T_H)^{-1} = \displaystyle \sum _{n \ge 0} B_{\lambda }H(M_{\lambda }H)^nG_{\lambda } +C_{\lambda }. \end{aligned}$$
(21)

Unlike in Sect. 4.1, here the strip \(\{\lambda \in {\mathbb {C}}\, \text { such that }-\lambda ^* < \text {Re} \lambda \le \lambda _0 \}\) is not necessary contained in \(\rho (T_H).\) However, with additional hypotheses (for example, the compactness of H), we can obtain more information about its structure.

As before, we denote by G the set \( \{\lambda \in {\mathbb {C}}\, \text { such that } \text {Re}\lambda > -\lambda ^* \}\).

Lemma 5

Let \(\lambda \in G\). If H is a compact operator satisfying (19), then there exists a discrete subset S of G such that \(G\backslash S \subset \rho (T_H)\).

Proof

Let \(\lambda \in G\). It is easy to see that \(M_\lambda \rightarrow 0\), as \(\text {Re} \lambda \rightarrow \infty \), in the strong operator topology. Using the compactness of H together with [12, Lemma 3.7, p. 151], we infer that \(M_\lambda H \rightarrow 0\), as \(\hbox {Re}\lambda \rightarrow \infty \), in the operator topology. Applying Gohberg-Smul’yan’s theorem [11, Theorem 11.4] we deduce that \((I-M_{\lambda }H)\) is invertible for all \(\lambda \in G\backslash S\) where S is a discrete subset of G. \(\square \)

Lemma 6

Let H be a boundary operator satisfying (19). Then the following holds

$$\begin{aligned} \displaystyle \lim _{\mathrm{Re} \lambda \rightarrow + \infty } \left\| (\lambda - T_H)^{-1}\right\| = 0. \end{aligned}$$

Proof

This follows from the fact that \(T_H\) is the generator of a \(\hbox {C}_0\)-semigroup. \(\square \)

Lemma 7

If H is a compact boundary operator satisfying (19), then \((I-M_{\lambda }H)^{-1}\) exists for \(\lambda \in G\) such that \(\left| \mathrm{Im} \lambda \right| \) is large enough.

Before giving the proof, we first recall the following lemma established in [13]

Lemma 8

Let H be a compact boundary operator satisfying (19). If \(\lambda \in G\), then

$$\begin{aligned} \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow +\infty }r_{\sigma }(M_{\lambda }H) =0. \end{aligned}$$

Proof of Lemma 7

The result is clear if \(\lambda \in \varGamma _{\lambda _0}\). Next, let \(\lambda \in G\backslash \varGamma _{\lambda _0}\). According to Lemma 8, there exists \(M > 0\) such that, for \(\left| \mathrm{Im} \lambda \right| > M\), we have \(r_{\sigma }(M_{\lambda }H) < 1\) which conclude the proof. \(\square \)

Now, we are ready to state one of the main result of this section. Before going further, we first recall that the set \(\varGamma _{\omega }\) is defined by

$$\begin{aligned} \varGamma _{\omega } = \Big \{ \lambda \in {\mathbb {C}}: \text {Re} \lambda \ge -\lambda ^* + \omega \Big \}. \end{aligned}$$

where \(\omega >0\) is small enough.

Proposition 4

Let K be a regular collision operator. If H is a compact boundary operator satisfying (19), then

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty }\left| \mathrm{Im} \lambda \right| ^r \left\| K(\lambda - T_H)^{-1}K\right\| = 0 \ \ \text {uniformly on } \ \ \varGamma _{\omega }, \end{aligned}$$

where \(r \in [0,1]\).

Proof

According to Remark 8, it suffices to establish the result for a one rank collision operator which we define by

$$\begin{aligned} K : X_p \rightarrow X_p, \varphi \rightarrow K\varphi (x,v) = \displaystyle \int _{{\mathbb {R}}^N} f(v)g(v^{\prime })\varphi (x,v^{\prime })dv^{\prime }, \end{aligned}$$

where \(f(\cdot ) \in L_p({\mathbb {R}}^N)\) et \(g(\cdot ) \in L_q({\mathbb {R}}^N)\).

Since, \(K(\lambda -T_H)^{-1}K = KB_{\lambda }H(I-M_{\lambda }H)^{-1}G_{\lambda }K + KC_{\lambda }K\) (use Eq. (21)), it follows from Lemma 1, that \(\lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty } \left| \mathrm{Im} \lambda \right| ^r\left\| KC_{\lambda }K\right\| = 0 \ \ \text {uniformly on } \ \ \varGamma _{\omega }\). Hence, it suffices to establish the result for the operator \(KB_{\lambda }H(I-M_{\lambda }H)^{-1}G_{\lambda }K\). To do so, we shall proceed in two steps.

(i) We will first prove that the family of operators \(\{ H(I-M_{\lambda }H)^{-1}G_{\lambda }K, -\lambda ^* + \omega \le \text {Re} \lambda \le \lambda _0 \}\) is collectively compact. Let B be the unit ball of \(X_p\) and let \((x_n)_{n \in {\mathbb {N}}}\) be a sequence in \(\cup _{\lambda }H(I-M_{\lambda }H)^{-1} G_{\lambda }KB, \ \lambda \in \{ \lambda , \ -\lambda ^* + \omega \le \text {Re} \lambda \le \lambda _0 \}\). Then there exists \((q_n)_{n \in {\mathbb {N}}}\) in B such that \(x_n = H(I-M_{\lambda }H)^{-1} G_{\lambda }Kq_n\)\(n= 1,2,\ldots \) It is clear that the sequence \((y_n = (I-M_{\lambda }H)^{-1}G_{\lambda }Kq_n)_{n \in {\mathbb {N}}}\) is bounded in \(L_p^+\). So, it follows from the compactness of H that \((x_n = Hy_n)_{n \in {\mathbb {N}}}\) has converging subsequence in \(\overline{\displaystyle \cup _{\lambda }H(I-M_{\lambda }H)^{-1}G_{\lambda }KB}\).

(ii) Now we shall establish that \(\{ \left| \mathrm{Im} \lambda \right| ^r KB_{\lambda }, -\lambda ^* + \omega \le \text { Re} \lambda \le \lambda _0 \}\) converges strongly to zero as \(\left| \mathrm{Im} \lambda \right| \) goes to infinity. For \(\varphi \in L_p^-\) and \(\lambda = \eta + i\, \text {Im} \lambda \), we have

$$\begin{aligned} (KB_{\lambda }\varphi )(x,v) = f(v) \displaystyle \int _{{\mathbb {R}}^N} g(v^{\prime })e^{-i \mathrm{Im}\lambda t^-(x,v^{\prime })}B_{\eta }\varphi (x,v^{\prime }) dv^{\prime }. \end{aligned}$$

So, it can be decomposed as \(KB_\lambda := A_2(\mathrm{Im}\lambda )A_1(\eta )\), where

$$\begin{aligned} A_1(\eta ): L_p^+ \rightarrow X_p, \psi \rightarrow (A_1(\eta )\psi )(x,v) = B_\eta \psi (x,v) \end{aligned}$$

and

$$\begin{aligned} A_2(\mathrm{Im}\lambda ): X_p \rightarrow X_p, \varphi \rightarrow (A_2(\mathrm{Im}\lambda )\varphi )(x,v) = f(v) \displaystyle \int _{{\mathbb {R}}^N} g(v^{\prime })e^{-i \mathrm{Im}\lambda t^-(x,v^{\prime })}\varphi (x,v^{\prime }) dv^{\prime }. \end{aligned}$$

Note that \(A_1(\eta )\) independent of \(\mathrm{Im}\lambda \) and \(t^-(x,v') \in (\delta ,{\mathsf {d}})\) (\({\mathsf {d}}\) stands for the diameter of \(\varOmega \)). Now, arguing as in the last part of the proof of the Lemma 1, we get

$$\begin{aligned} \displaystyle \lim _{\left| \mathrm{Im} \lambda \right| \rightarrow \infty } \left| \mathrm{Im} \lambda \right| ^r\left\| A_2(\mathrm{Im}\lambda )\varphi \right\| _{X_p} = 0. \end{aligned}$$

Now, according to (i), (ii) and [12, Lemma 3.7, p.151], we get the desired result. \(\square \)

Let \({\overline{\lambda }}\) be an element of S with the greater real part (S is the subset obtained in Lemma 5).

Lemma 9

Let K be a regular collision operator and let H be compact boundary operator satisfying (19). Then

  1. (1)

    \(\sigma (A_H)\cap \{ \lambda \in {\mathbb {C}}: \mathrm{Re} \lambda > \mathrm{Re}{\overline{\lambda }} \}\) consists of, at most, isolated eigenvalues with finite algebraic multiplicities,

  2. (2)

    if \(\varepsilon > 0\) then \(\sigma (A_H)\cap \{ \lambda \in {\mathbb {C}}: \mathrm{Re} \lambda > \mathrm{Re}{\overline{\lambda }} + \varepsilon \}\) is finite,

  3. (3)

    if \(\varepsilon > 0\) then \(\left\| (\lambda - A_H)^{-1}\right\| \) is uniformly bounded in \(\{ \lambda \in {\mathbb {C}}: \mathrm{Re} \lambda > \mathrm{Re}{\overline{\lambda }} + \varepsilon \}\) for large \(\left| \mathrm{Im} \lambda \right| .\)

Proof

The proof is similar to that of Lemma 4 and so it is omitted. \(\square \)

Assume that K is a regular collision operator on \(X_p\) and H is an arbitrary compact boundary operator satisfying (19). Then, by Lemma 9, the spectrum of \(A_H\) in the half plane \(\{ \lambda \in {\mathbb {C}}: \text {Re} \lambda > \text {Re}{\overline{\lambda }}+ \varepsilon \}\) consists of, a finite number of isolated eigenvalues with finite algebraic multiplicity which we denote \(\lambda _1,\lambda _2,\ldots ,\lambda _n\). Let \(P_i\) and \(D_i\) denote, respectively, the spectral projection and the nilpotent operator associated with \(\lambda _i, i=1,2,\ldots ,n\). Then \(P:= P_1+\cdots +P_n\) is the spectral projection of the compact set \(\{ \lambda _1, \lambda _2,\ldots , \lambda _{n} \}\). According to the spectral decomposition theorem corresponding to the compact set \(\{ \lambda _1, \lambda _2,\ldots , \lambda _{n} \}\) (see, for example, [12]), we may write

$$\begin{aligned} V_H(t) = {\tilde{V}}_H(t) + \sum _{i=1}^ne^{\lambda _it}e^{D_it}P_i \end{aligned}$$

where \({\tilde{V}}_H(t) := V_H(t)(I-P)\) is the \(C_0\)-semigroup on the Banach space \((I-P)X_p\) with generator \({\tilde{A}}_H := A_H(I-P)\) (\({\tilde{A}}_H\) is the part of \(A_H\) on the closed subspace \((I-P)X_p\)).

We are in a position to state the main result of this subsection.

Theorem 5

Assume that \(\lambda ^*=\inf _{v \in V} \sigma (v)>0\) and \( \delta \ge \displaystyle \frac{\ln (\left\| H\right\| )}{\lambda ^*}\). If the hypotheses of Lemma 9 hold true, then, there exists \(\varepsilon > 0\) small enough and \(M>0\) such that

$$\begin{aligned} \left\| V_H(t)\psi _0 - \displaystyle \sum _{i=1}^{n}e^{(\lambda _i + D_i)t}P_i\psi _0\right\| _{X_p} \le Me^{(\varepsilon + \lambda _0)t}\quad \forall t>0 \end{aligned}$$

where \(\psi _0 \in D(A_H)\) and \(p \in [1,\infty )\).

Proof

According to Proposition 4 and Theorem 1, \((\lambda -T_H)^{-1}K\) is power compact. Further, for all \(r \in [0,1]\), we have \(\left| \mathrm{Im} \lambda \right| ^r \left\| K(\lambda -T_H)^{-1}K\right\| \rightarrow 0\) as \(\left| \mathrm{Im} \lambda \right| \rightarrow \infty \). Now the rest of the proof may modeled in a similar way to those of Theorems 3 and 4. \(\square \)

Remark 6

We close this section by noticing that it is not difficult to prove that the first order remainder term of the Dyson-Phillips, \(R_1^H(t)\), is compact on \(X_p\) for \(p\in (1,+\infty )\). The proof may be derived in the same way as that of Proposition 3. However, the weak compactness of \(R_1^H(t)\) on \(X_1\), even for dissipative boundary conditions, is still an open problem.