Solutions of a Neutron Transport Equation with a Partly Elastic Collision Operators

Abstract

In this paper, we derive sufficient conditions that guarantee an description of long-time asymptotic behavior of the solution to the Cauchy problem governed by a linear neutron transport equation with a partially elastic collision operator under periodic boundary conditions. Our strategy involves showing that the strongly continuous semigroups ${\left(e^{t(T+K_e)}\right)}_{t\ge 0}$ and ${\left(e^{t(T+K_c+K_e)}\right)}_{t\ge 0}$, generated by the operators $T+K_e$ and $T+K_c+K_e$, respectively, have the same essential type. According to Proposition 1, it is sufficient to show that remainder term in the Dyson–Philips expansion is compact. Our analysis focuses on the compactness properties of the second-order remainder term in the Dyson–Phillips expansion related to the problem. We first show that $R_2\left(t\right)$ is compact on $L^2(\mathsf{\Omega }\times V,dxdv)$, and, using an interpolation argument (see Proposition 2), we establish the compactness of $R_2\left(t\right)$ on $L^p(\mathsf{\Omega }\times V,dxdv)$-spaces for $1<p<+\infty$. To the best of our knowledge, outside the one-dimensional case, this result is known only for vaccum boundary conditions in the multidimensional setting. However, our result, Theorem 1, is new for periodic boundary conditions.

1. Introduction

The purpose of this work is to discuss the long-time asymptotic behavior of the solution to the following Cauchy problem for partly elastic collisions operators, as introduced by E. W. Larsen and P. F. Zweifel [1]:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial \phi }{\partial t}(x,v,t)=-v.\frac{\partial \phi }{\partial x}(x,v,t)}\hfill & {\displaystyle -\sigma \left(v\right)\phi (x,v,t)+{\int }_V{k}_c(x,v,v^{\prime })\phi (x,v^{\prime },t)d{v}^{\prime }}\hfill \\ & {\displaystyle +\sum _{j=1}^l{\int }_{{\mathbb{S}}^{N-1}}{k}_d^j(x,{\rho }_j,\omega ,{\omega }^{\prime })\phi (x,{\rho }_j{\omega }^{\prime },t)d{\omega }^{\prime }}\hfill \\ & {\displaystyle +{\int }_{{\mathbb{S}}^{N-1}}{k}_e(x,\rho ,\omega ,{\omega }^{\prime })\phi (x,\rho {\omega }^{\prime },t)d{\omega }^{\prime }}\hfill \\ \phi (x,v,0)={\phi }_0(x,v).\hfill \end{array}\right.\end{array}$$

(1)

with periodic boundary conditions

$$\phi (x,v,t){\mid }_{x_i=-a_i}=\phi (x,v,t){\mid }_{x_i=a_i}.$$

(2)

where $(x,v)\in \mathsf{\Omega }\times V$, with $\mathsf{\Omega }$ being an open subset of ${\mathbb{R}}^N$ given by

$$\mathsf{\Omega }=\prod _{\begin{array}{c}i=1\end{array}}^N(-a_i,a_i),a_i>0,i=1,···,N$$

where the speed v in V is written in the form $v=\rho \omega$, $\omega \in {\mathbb{S}}^{N-1}$ (${\mathbb{S}}^{N-1}$ is the unit sphere of ${\mathbb{R}}^N$) and $\rho \in I:=[{\rho }_{min},{\rho }_{max}]$, where ${\rho }_{max}<+\infty$. The velocity space is equipped with the Lebesgue measure $dv={\rho }^{N-1}d\rho d\omega$, where $d\omega$ is the Lebesgue measure on the unit sphere ${\mathbb{S}}^{N-1}$. The function $\phi (x,v,t)$ represents the number density of particles with position x and velocity v at time t. The function $0\le \sigma (·)\in L^{\infty }(V,dv)$ is called the collision frequency and the functions $k_c(·,·,·)$, $k_e(·,·,·,·)$ and $k_d^j(·,·,·,·),j=1,···,l,$ denote the scattering kernels of operators $K_c$, $K_e$, and $K_d={\sum }_{j=1}^l{K}_d^j$, which we define below.

In this model, the collision operator consists of three terms:

$$\begin{array}{c}K=K_c+K_e+K_d.\end{array}$$

The collision operator $K_c$ denotes the classical collision operator involved in the classical neutron transport theory (see, for example, [2]); this can be expressed by

$${\displaystyle K_c\phi (x,v)={\int }_V{k}_c(x,v,v^{\prime })\phi (x,v^{\prime })d{v}^{\prime }}$$

and it corresponds physically to fission, high energy elastic slowing down, and thermal inelastic scattering. $K_e$ is an elastic operator for low energy neutrons describing microscopic events in which the kinetic energy is conserved and velocities are changed only in their direction. It is given by

$${\displaystyle K_e\phi (x,v)={\int }_{{\mathbb{S}}^{N-1}}{k}_e(x,\rho ,\omega ,{\omega }^{\prime })\phi (x,\rho {\omega }^{\prime })d{\omega }^{\prime }}$$

Finally, $K_d$ represents high energy inelastic scattering and is described by a downshift operator of the form

$${\displaystyle K_d\phi (x,v)=\sum _{j=1}^l{K}_d^j\phi (x,v)=\sum _{j=1}^l{\int }_{{\mathbb{S}}^{N-1}}{k}_d^j(x,{\rho }_j,\omega ,{\omega }^{\prime })\phi (x,{\rho }_j{\omega }^{\prime })d{\omega }^{\prime },}$$

where $K_d^j$$(j=1,\dots ,l)$ describes an event in which a discrete energy $E_j$ is lost by a neutron at x with an initial speed $e_j\left(\rho \right)$ and final speed $\rho$. The speed $e_j\left(\rho \right)$ is defined by

$${\displaystyle E_j=\frac{1}{2}M{e}_j^2\left(\rho \right)-\frac{1}{2}M{\rho }^2,}$$

where M is the mass of neutrons.

Finally, by $K_e$, we denote the operator

$${\displaystyle K_e\phi (x,v)={\int }_{{\mathbb{S}}^{N-1}}{k}_e(x,\rho ,\omega ,{\omega }^{\prime })\phi (x,\rho {\omega }^{\prime })d{\omega }^{\prime }}$$

called the elastic collision operator. It describes the collisions, which do not vary the kinetic energy of the neutrons, they act only through the angular part of the variable of velocity.

Let us notice that the spectral properties of the operator $T+K_c+K_e+K_d$ was first studied by E. W. Larsen and P. F. Zweifel in [1] in the space $L^1$. It was considered afterwards by M. Sbihi [3,4] for vacuum boundary conditions (zero incoming flow) and for reentry boundary conditions in [5]. This work intends to investigate the properties of solutions to Problem (1) with periodic boundary conditions in a box of finite measure. Our approach is based on the spectral theory of the semigroup generated by the operator $T+K_c+K_e+K_d$. Due to certain technical difficulties, we restrict our analysis to partially elastic collision operators. So, we suppose that $K_d=0$; therefore, the collision operator is written as $K=K_c+K_e$. Hence, the Cauchy problem (1) and (2) may be written abstractly on the spaces $L^p(\mathsf{\Omega }\times V,dxdv)$, $p\in (1,+\infty )$ in the form

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \phi }{\partial t}(t,x,v)=(T+K_c+K_e)\phi (t,x,v),}\hfill \\ \phi (0,x,v)={\phi }_0(x,v),\hfill \\ \phi (x,v,t){\mid }_{x_i=-a_i}=\phi (x,v,t){\mid }_{x_i=a_i}\hfill \end{array}\right.$$

(3)

where ${\phi }_0$ denotes the initial data.

It is well known that T generates a $C_0$-semigroup. As $K_c$ and $K_e$ are bounded operatiors, the operator $T+K_c+K_e$ also generates a $C_0$-semigroup, denoted by ${\left(e^{t(T+K_c+K_e)}\right)}_{t\ge 0}$. This semigroup provides a solution to the Cauchy problem (3) for any initial data ${\phi }_0$ that belongs to $D\left(T\right)$. So, in order to obtain more information about the solution, in particular about its behavior over long times ($t\to +\infty$), knowledge of the spectral properties of ${\left(e^{t(T+K_c+K_e)}\right)}_{t\ge 0}$ plays a central role. We point out that the spectral properties of ${\left(e^{t(T+K_c)}\right)}_{t\ge 0}$ and its generator are, for now, multidimensional bounded geometry that is well understood for absorbing boundary conditions (see, for example, [6], Chapter 4 and the references therein). The case of no-reentry boundary conditions has been considered in particular cases, especially in slab geometry, and a few papers have dealt with higher dimensions (see, for example, [7]).

We first recall the concept of essential type for a strongly continuous semigroup on a Banach space. Let X be a Banach space and let B be an operator belonging to $\mathcal{L}\left(X\right)$. We define the essential resolvent set of B by

$${\rho }_{ess}\left(B\right)=\rho \left(B\right)\cup \{\lambda \in \sigma \left(B\right);\lambda \mathrm{is}\mathrm{an}\mathrm{isolated}\mathrm{eigenvalue}\mathrm{of}\mathrm{finite}\mathrm{multiplicity}\}$$

where $\rho \left(B\right)$ is the resolvent set of B. The essential spectral radius of B is defined by

$$r_{ess}\left(B\right)=sup\left\{\right|\lambda |;\lambda \notin {\rho }_{ess}\left(B\right)\}$$

where $r_{ess}(·)$ denotes the essential spectral radius. Let A be the generator of a strongly continuous semigroup ${\left(U\left(t\right)\right)}_{t\ge 0}$ on X with type ω (see, for example, [8], Definition 5.6, p. 40). Following Voigt ([9], Lemma 2.1, p. 157) there exists a real ${\omega }_{ess}\in [-\infty ,\omega ]$ (called the essential type) such that

$$r_{ess}\left(U\left(t\right)\right)=e^{{\omega }_{ess}t}\mathrm{for}\mathrm{all}t\ge 0.$$

Because $B\in \mathcal{L}\left(X\right)$, following the classical perturbation theory (see, for example, [8], Section 1, p. 157), the operator $A+B$ generates also a strongly continuous semigroup on X. It is given by the Dyson-Philips expansion, that is,

$$e^{t(A+B)}=\sum _{j=0}^{\infty }{U}_j\left(t\right),$$

where

$${\displaystyle U_0=e^{tA}\phantom{aaa}\mathrm{and}\phantom{aaaaz}{U}_{j+1}\left(t\right)={\int }_0^t{U}_0\left(t\right)B{U}_{j-1}(t-s)ds}$$

and the remainder term of order n, $R_n\left(t\right)$, of the Dyson–Philips expansion is given by

$${\displaystyle R_n\left(t\right)={\int }_{s_1+s_2+...+s_n\le t,s_i\ge 0}U\left(s_1\right)B...BU\left(s_n\right)BV\left(t-\sum _{i=1}^n{s}_i\right)d{s}_1...d{s}_n.}$$

According to Proposition 1, if some remainder term of the Dyson–Philips expansion $R_n\left(t\right)$$(n\ge 1)$ for all $t\ge 0$ is compact, then the operators $U\left(t\right)$ and $V\left(t\right)$ have the same essential type and, therefore, the same spectral radius, that is, $r_{ess}\left(V\left(t\right)\right)=r_{ess}\left(U\left(t\right)\right)$. We note that the stability of the essential type via the compactness of the remainder terms of the Dyson–Philips expansion $R_n\left(t\right)$ was extensively investigated, in particular, in neutron transport theory (see, for example, [3,9,10,11,12] and the references therein). Hence, $\sigma \left(V\left(t\right)\right)\cap \left\{\mu \in \mathbb{C}:\left|\mu \right|>r_{ess}\left(U\left(t\right)\right)\right\}$ consists primarily of finitely many eigenvalues. If such eigenvalues exist, then, according to the spectral decomposition theorem, the semigroup ${\left(V\left(t\right)\right)}_{t\ge 0}$ can be decomposed into two parts: the first part captures the time development of finitely many eigenmodes, while the second part exhibits faster decay. Using the spectral mapping theorem for the point spectrum, we infer that for any $\zeta >\omega$, the set $\sigma (A+B)\cap \{\mathrm{Re}\lambda \ge \zeta \}$ consists of finitely many isolated eigenvalues, say $\{{\lambda }_1,\cdots {\lambda }_q\}.$ Let ${\beta }_1=sup\{\mathrm{Re}\lambda ,\lambda \in \sigma (A+B),\mathrm{Re}\lambda <\omega \}$, and ${\beta }_2=min\{\mathrm{Re}{\lambda }_j,1\le j\le q\}$. The solution of the Cauchy problem satisfies

$$∥\phi \left(t\right)-\sum _{j=1}^q{e}^{{\lambda }_{j}t}{e}^{D_{j}t}{P}_j{\phi }_0∥=o\left(e^{{\beta }^{*}t}\right)\mathrm{with}{\beta }_1<{\beta }^{*}<{\beta }_2.$$

(4)

where ${\phi }_0\in D\left(A\right)$, and $P_j$, and $D_j$ denote the spectral projection and the nilpotent operator, respectively, associated with ${\lambda }_j$, $j=1,\cdots ,q.$

The purpose of this paper is to discuss the time asymptotic behavior of the solution to Problems (1) and (2) in bounded geometry via the spectral properties of the strongly continuous semigroup generated by the operator $T+K_c+K_e$. Our approach consists of discussing the compactness properties of the second-order remainder term of the Dyson–Phillips expansion, $R_2\left(t\right)$, which yields a comparison of the essential types of the two semigroups ${\left(e^{t(T+K_e)}\right)}_{t\ge 0}$ and ${\left(e^{t(T+K_c+K_e)}\right)}_{t\ge 0}$. This provides a description of the time asymptotic behaviour of the solution to Problems (1) and (2) via standard arguments (see the paragraphs above).

The outline of this work is as follows. In Section 2, we introduce the functional setting of the problem and fix the different notations. We also gather some results needed in the sequel. Our main result, Theorem 1, is stated and established in Section 3. This result is new, as it is known only for the vacuum boundary (see [3]). We close this section by stating two open problems related to the description of the time asymptotic behaviour of the solution to Problems (1) and (2) on the space $L^1(\mathsf{\Omega }\times V,dxdv)$, even for vacuum boundary conditions.

Notation: Throughout this paper, if X is a Banach space, then $\mathcal{L}\left(X\right)$ stands for the set of all bounded linear operators on X.

2. Preliminaries

Let $\mathsf{\Omega }={\prod }_{\begin{array}{c}i=1\end{array}}^N(-a_i,a_i),a_i>0,i=1,\cdots ,N,$ be an open set im ${\mathbb{R}}^N$. Suppose that the collision frequency $\sigma (·)$ depends only on the velocity variable and satisfies

$$\begin{array}{c}\hfill 0\le \sigma (·)\in L^{\infty }(V,dv).\end{array}$$

The velocity space

$$V=\{v=\rho \omega ;\omega \in {\mathbb{S}}^{N-1},{\rho }_{min}\le \rho \le {\rho }_{max}<\infty \}=:I\times {\mathbb{S}}^{N-1}$$

is endowed with the Lebesgue measure $dv={\rho }^{N-1}d\rho d\omega$, where $d\omega$ is the Lebesgue measure on the unit sphere ${\mathbb{S}}^{N-1}$ of ${\mathbb{R}}^n$.

We define the periodic streaming operator T by

$$\left\{\begin{array}{c}\mathcal{D}\left(T\right)=W^p(\mathsf{\Omega }\times V,dxdv),\hfill \\ {\displaystyle T\phi (x,v)=-v.\frac{\partial \phi }{\partial x}(x,v)-\sigma \left(v\right)\phi (x,v)}\hfill \end{array}\right.$$

(5)

where

$${\displaystyle W^p(\mathsf{\Omega }\times V,dxdv)=\{\phi \in L^p(\mathsf{\Omega }\times V,dxdv):\phantom{a}v.\frac{\partial \phi }{\partial x}\in L^p(\mathsf{\Omega }\times V,dxdv)\mathrm{and}}$$

$$\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\phi {\mid }_{x_i=-a_i}=\phi {\mid }_{x_i=a_i},i=1,\cdots ,N\}.$$

Now, we recall that each function $\phi$ in $L^2(\mathsf{\Omega }\times V,dxdv)$ can be written in the form

$$\phi (x,v)=\sum _{\begin{array}{c}k\in {\mathbb{Z}}^N\end{array}}{\displaystyle \frac{1}{\sqrt{2a}}}{\phi }_k\left(v\right)e^{i\pi {\displaystyle \frac{k}{a}}.x}$$

where ${\phi }_k(·)$ is the Fourier coefficient of $\phi$ of order k, that is,

$${\displaystyle {\phi }_k\left(v\right)={\int }_{\mathsf{\Omega }}\frac{1}{\sqrt{2a}}{e}^{-i\pi \frac{k}{a}.x}\phi (x,v)dx,\phantom{aa}v\in V,\phantom{a}k\in {\mathbb{Z}}^N.}$$

We recall that $\left|\mathsf{\Omega }\right|=2{\prod }_{\begin{array}{c}i=1\end{array}}^N{a}_i,a_i>0,i=1,\cdots ,N$, is the volume of the paved set $\mathsf{\Omega }$. For convention, we put

$$\sqrt{2a}=\sqrt{\left|\mathsf{\Omega }\right|}=\prod _{\begin{array}{c}i=1\end{array}}^N\sqrt{2a_i}\mathrm{and}{\displaystyle \frac{k}{a}}=({\displaystyle \frac{k_1}{a_1}},···,{\displaystyle \frac{k_N}{a_N}}),(k_i\in \mathbb{Z},i=1,···,N).$$

From the Parseval formula, we have

$${∥\phi ∥}_{L^2(\mathsf{\Omega }\times V,dxdv)}^2=\sum _{\begin{array}{c}k\in {\mathbb{Z}}^N\end{array}}{∥{\phi }_k∥}_{L^2(V,dv)}^2.$$

So, the domain of the operator T may be written in the form

$$\mathcal{D}\left(T\right):=\{\phi (x,v)\in L^2(\mathsf{\Omega }\times V,dxdv):\phi (x,v)=\sum _{\begin{array}{c}k\in {\mathbb{Z}}^N\end{array}}{\displaystyle \frac{1}{\sqrt{2a}}}{\phi }_k\left(v\right)e^{i\pi {\displaystyle \frac{k}{a}}.x},\phantom{aaaaaaaaaaa}$$

$$\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\sum _{\begin{array}{c}k\in {\mathbb{Z}}^N\end{array}}{∥{\displaystyle \frac{k}{a}}.v{\phi }_k\left(v\right)∥}_{L^2(V,dv)}^2<\infty \}.$$

Therefore, for all $\phi \in \mathcal{D}\left(T\right)$, we have

$$T\phi (x,v):=\sum _{\begin{array}{c}k\in {\mathbb{Z}}^N\end{array}}{\displaystyle \frac{1}{\sqrt{2a}}}(-i\pi {\displaystyle \frac{k}{a}}.v-\sigma \left(v\right)){\phi }_k\left(v\right)e^{i\pi {\displaystyle \frac{k}{a}}.x}.$$

It is well known that T generates a strongly continuous semigroup ${\left(W\left(t\right)\right)}_{t\ge 0}$, which acts as follows

$$W\left(t\right)\phi (x,v):=\sum _{\begin{array}{c}k\in {\mathbb{Z}}^N\end{array}}{\displaystyle \frac{1}{\sqrt{2a}}}{\phi }_k\left(v\right)e^{i\pi {\displaystyle \frac{k}{a}}.(x-vt)}{e}^{-\sigma \left(v\right)t},(x,v)\in \mathsf{\Omega }\times V$$

for all $t\ge 0$ and $\phi \in L^2(\mathsf{\Omega }\times V,dxdv)$.

The operators $K_c$ and $K_e$ are bounded, i.e.,

$$K_c\in \mathcal{L}\left(L^p(\mathsf{\Omega }\times V,dxdv)\right),\phantom{aaaa}{K}_e\in \mathcal{L}\left(L^p(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1},dx\rho d\omega )\right).$$

Remark 1.

As T generates a $C_0$-semigroup on $L^p(\mathsf{\Omega }\times V,dxdv)$ and the operators $K_e$ and $K_c+K_e$ are bounded, from the classical perturbation theory (see, for example, [8], Chapter III, Section 1, p. 157), the operators $T+K_e$ and $T+K_c+K_e$ generate strongly continuous semigroups ${\left(U\left(t\right)\right)}_{t\ge 0}$ and ${\left(V\left(t\right)\right)}_{t\ge 0}$, respectively. In the remainder of this work, we will sometimes use notations $U\left(t\right)$ and $V\left(t\right)$ or $e^{t(T+K_e)}$ and $e^{t(T+K_c+K_e)}$.

We recall that

$$e^{t(T+K_e+K_c)}=\sum _{i=0}^{\infty }{V}_i\left(t\right)=\sum _{i=0}^{n-1}{V}_i\left(t\right)+R_n\left(t\right)$$

where

$${\displaystyle V_0\left(t\right)=e^{t(T+K_e)}\phantom{aa}\mathrm{and}\phantom{aa}{V}_{i+1}\left(t\right)={\int }_0^{t}U\left(s\right)K_c{V}_i(t-s)ds.}$$

with the nth order remainder term $R_n\left(t\right)$ is given by

$${\displaystyle R_n\left(t\right)={\int }_{s_1+s_2+\cdots +s_n\le t,s_i\ge 0}U\left(s_1\right)K_c\cdots K_{c}U\left(s_n\right)K_{c}V\left(t-\sum _{i=1}^n{s}_i\right)d{s}_1\cdots d{s}_n.}$$

Definition 1.

Let $F,G:[0,\infty )\to \mathcal{L}\left(X\right)$ be strongly continuous mappings. We define their convolution by

$${\displaystyle F\ast G\left(t\right)={\int }_0^{t}F\left(s\right)G(t-s)ds.}$$

The convolution of F j-times will be denoted by ${\left[F\right]}^j=F\ast \cdots \ast F$ (j times).

According to Definition 1, the remainder term of order n, $R_n\left(t\right)$, can be written in the following form

$$R_n\left(t\right)=\sum _{i=n}^{\infty }{V}_i\left(t\right)={\left[e^{t(T+K_e)}{K}_c\right]}^n\ast e^{t(T+K_e+K_c)}.$$

The following result will be required in the sequel (see, for example, [6], Theorem $2.10$, p. 24).

Proposition 1.

Assume that the remainder term $R_n\left(t\right)(n\ge 1)$ is compact for all $t\ge 0$ on $L^p(\mathsf{\Omega }\times V,dxdv)$ with $p\in (1,+\infty )$. Then, the $C_0$-semigroups ${\left(V\left(t\right)\right)}_{t\ge 0}$ and ${\left(U\left(t\right)\right)}_{t\ge 0}$ have the same essential type, that is, ${\omega }_{ess}\left(V\left(t\right)\right)={\omega }_{ess}\left(U\left(t\right)\right)$ for all $t\ge 0$.

We also recall the following interpolation result for positive compact operators (cf. [13], Theorem $3.10$, p. 57).

Proposition 2.

Let $(\mathsf{\Lambda },\eta )$ be a positive measure space and let B be a bounded linear operator, such that

$$B\in \bigcap _{p\ge 1}\mathcal{L}\left(L^p(\mathsf{\Lambda },d\eta )\right).$$

If there exists $1<p_0<\infty$, such that B is compact in $L^{p_0}(\mathsf{\Lambda },d\eta )$, then B is compact in all $L^p(\mathsf{\Lambda },d\eta )$, $1<p<\infty$.

Recall that in neutron transport theory, the collision operator $K_c$ has the form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{K}_c:L^p(\mathsf{\Omega }\times V,dxdv)\to L^p(\mathsf{\Omega }\times V,dxdv),\hfill \\ {\displaystyle \phi \mapsto {\int }_V{\kappa }_c(x,v,v^{\prime })\phi (x,v^{\prime })d{v}^{\prime },}\hfill \end{array}\right.\end{array}$$

(6)

where ${\kappa }_c(·,·,·)$ is a nonnegative measurable function on $\mathsf{\Omega }\times V\times V$.

Note that $K_c$ is local with respect to the variable space $x\in \mathsf{\Omega }$. So, it may be viewed as an operator valued mapping from $\mathsf{\Omega }$ into $\mathcal{L}\left(L^p(V,dv)\right)$, that is,

$$\begin{array}{c}\hfill \mathsf{\Omega }\ni x\mapsto K_c\left(x\right)\in \mathcal{L}(L^p(V,dv),\end{array}$$

where

$$\begin{array}{c}\hfill {\displaystyle K_c\left(x\right):\phi \in L^p\left(V\right)\mapsto K_c\left(x\right)\phi ={\int }_V{\kappa }_c(x,v,v^{\prime })\phi \left(v^{\prime }\right)d{v}^{\prime }.}\end{array}$$

Now, we are ready to state the definition of regular collisions operators [2].

Definition 2.

Let $p\in (1,+\infty )$. A collision operator $K_c$ is said to be regular on

$L^p(\mathsf{\Omega }\times V,dxdv)$ if:

• $\{K_c\left(x\right):x\in \mathsf{\Omega }\}$ is a set of collectively compact operators on $L^p(V,dv),$ i.e.,

  $$\left\{K_c\left(x\right)\phi ;x\in \mathsf{\Omega },∥\phi ∥\le 1\right\}is relatively compact in L^p(V,dv).$$
• For $\psi \in L^q(V,dv),$

  $$\left\{K_c^{\prime }\left(x\right)\psi ;x\in \mathsf{\Omega },∥\psi ∥\le 1\right\}is relatively compact in{L}^q(V,dv).$$

Here, $K_c^{\prime }\left(x\right)$ denotes the dual operator of $K_c\left(x\right)$ and ${\displaystyle q=\frac{p}{p-1}}$.

This class of operators enjoys the useful approximation property (cf [2], Proposition 1).

Lemma 1.

The class of regular collision operators is the closure in the operator norm of the class of collision operator with kernels of the form

$${\displaystyle {\kappa }_c(x,v,v^{\prime })=\sum _{i\in I}{\alpha }_i\left(x\right)f_i\left(v\right)g_i\left(v^{\prime }\right)}$$

(7)

where I is finite, $\alpha (·)\in L^{\infty }(\mathsf{\Omega };dx)$, $f_i(·)\in L_p(V;dv)$ and $g_i(·)\in L_q(V;dv)$ with $q={\displaystyle \frac{p}{p-1}}$.

According to Lemma 1, for our analysis, it is sufficient to consider collision operators $K_c$ with kernels of the form (7).

The elastic collision operator $K_e$ is defined by

$$\left\{\begin{array}{c}{K}_e:L^p(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1},dxd\rho d\omega )⟶L^p(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1},dxd\rho d\omega )\hfill \\ {\displaystyle \psi \mapsto {\int }_{{\mathbb{S}}^{N-1}}{k}_e(x,\rho ,\omega ,{\omega }^{\prime })\psi (x,\rho ,{\omega }^{\prime })d{\omega }^{\prime }.}\hfill \end{array}\right.$$

(8)

We introduce the following hypothesis:

$\left(\mathsf{A}1\right)$ for all $\psi \in L^p({\mathbb{S}}^{N-1},d\omega )$, the set

$${\displaystyle \left\{{\int }_{{\mathbb{S}}^{N-1}}{k}_e(x,\rho ,\omega ,{\omega }^{\prime })\psi \left({\omega }^{\prime }\right)d{\omega }^{\prime }:(x,\rho )\in \mathsf{\Omega }\times I,{∥\psi ∥}_{L^p({\mathbb{S}}^{N-1},d\omega )}\le 1\right\}}$$

is relatively compact in $L^p({\mathbb{S}}^{N-1},d\omega )$ and, for every, ${\psi }^{\prime }\in L^q({\mathbb{S}}^{N-1},d\omega )$,

$${\displaystyle \left\{{\int }_{{\mathbb{S}}^{N-1}}{k}_e(x,\rho ,\omega ,{\omega }^{\prime }){\psi }^{\prime }\left(\omega \right)d\omega :(x,\rho )\in \mathsf{\Omega }\times I\right\}}$$

is relatively compact in $L^q({\mathbb{S}}^{N-1},d\omega )$.

We note that hypothesis $\left(\mathsf{A}1\right)$ was introduced by M. Sbihi in [4] (see also [3]).

Under condition $\left(\mathsf{A}1\right)$, we have

$${∥K_e∥}_{\mathcal{L}\left(L^p(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1},dxd\rho d\omega )\right)}=\underset{(x,\rho )\in \mathsf{\Omega }\times I}{ess\; sup}{∥K_e(x,\rho )∥}_{\mathcal{L}\left(L^p({\mathbb{S}}^{N-1},d\omega )\right)}$$

(9)

where $K_e(x,\rho )$ is the operator defined by

$${\displaystyle L^p\left({\mathbb{S}}^{N-1}\right)\ni \psi ⟼{\int }_{{\mathbb{S}}^{N-1}}{k}_e(x,\rho ,\omega ,{\omega }^{\prime })\psi \left({\omega }^{\prime }\right)d{\omega }^{\prime }.}$$

We also recall the following result, owing to M. Sbihi [3].

Lemma 2.

An elastic collision operator satisfying $\left(\mathsf{A}1\right)$ can be approximated in the norm operator topology by collision operators with kernels of the form

$$k_e^i(x,\rho ,\omega ,{\omega }^{\prime })=\sum _{i\in J}{\alpha }_i(x,\rho )f_i\left(\omega \right)g_i\left({\omega }^{\prime }\right)$$

where ${\alpha }_i\in L^{\infty }(\mathsf{\Omega }\times I)$, $f_i\in L^p\left({\mathbb{S}}^{N-1}\right)$ and $g_i\in L^q\left({\mathbb{S}}^{N-1}\right)$, $i\in J$ with q denotes the conjugate exponent of p, while J a finite set.

3. Time Asymptotic Behavior of the Solution

In this section, we are concerned with the compactness properties of the second-order remainder term of the Dyson–Philips expansion, $R_2(·)$, on $L^p(\mathsf{\Omega }\times V,dxdv)$ spaces with $p\in (1,+\infty )$. This will allow us to show that the essential type of the semigroups generated by the operators $T+K_c+K_e$ is less or equal to that generated by $T+K_c$. This permits us to derive the time asymptotic behavior of the solution to Problems (1) and (2) via standard arguments.

Racall that $R_2(·)$ is written in the form

$${\displaystyle R_2\left(t\right)={\int }_0^{t}ds{\int }_0^{s}U(t-s)K_{c}U(t-\tau )K_{c}V\left(\tau \right)d\tau \forall t>0.}$$

The aim of the following result is to establish the compactness of the operator $R_2(·)$.

Theorem 1.

Let $K_c$ be a regular collision operator and assume that $K_e$ satisfies $\left(\mathsf{A}1\right)$. If, for all bounded subsets D contained in $I\times {\mathbb{S}}^{N-1}$, we have

$${\displaystyle \underset{\left|x\right|\to \infty }{lim}{\int }_D{e}^{-ix.\rho \omega }{\rho }^{N-1}d\rho d\omega =0,}$$

(10)

then, $R_2\left(t\right)$ is compact for all $t\ge 0$.

Remark 2.

It should be noticed that the hypothesis (10) was first used by M. Mokhtar-Kharroubi to prove Theorem 4.7 in [6] in the context of transport equations with vacuum boundary conditions.

Proof of Theorem 1.

As $R_2\left(t\right)=e^{t(T+K_e)}\ast K_c{e}^{t(T+K_e)}{K}_c\ast e^{t(T+K_e+K_c)}$ (see Lemma 2.1, p. 14 in [6]), it suffices to show that the compactness of

$$\begin{array}{c}\hfill K_c{e}^{t(T+K_e)}{K}_c=K_{c}U\left(t\right)K_c.\end{array}$$

for all $t>0$. As $K_c$ is regular, from Lemma 1, it is enough to establish the proof for a collision operator $K_c$ of the form

$${\displaystyle K_c\phi (x,v^{\prime }):=\alpha \left(x\right)f\left(v\right){\int }_{V}g\left(v^{\prime }\right)\phi (x,v^{\prime })d{v}^{\prime }}$$

(11)

where $\alpha \in L^{\infty }\left(\mathsf{\Omega }\right)$ and the functions $f(·)$ and $g(·)$ are continuous with compact support.

According to Proposition 2, it is enough to prove the result for $p=2.$ Note that the operator $K_{c}W\left(t\right)K_c=P_{1}O\left(t\right)\phantom{a}(t\ge 0)$ is defined by

$$P_1:L^2\left(\mathsf{\Omega }\right)\ni \psi \mapsto \alpha \left(x\right)f\left(v\right)\psi \left(x\right)\in L^2(\mathsf{\Omega }\times V),\phantom{aaaaaaaa}$$

$${\displaystyle O\left(t\right):L^2(\mathsf{\Omega }\times V)\ni \phi \mapsto \sum _{k\in {\mathbb{Z}}^N}\frac{1}{\sqrt{2a}}\left({\int }_{V}g\left(v\right)f\left(v\right)e^{-\sigma \left(v\right)t}{e}^{-i\pi .\frac{k}{a}.vt}dv\right)}$$

$${\displaystyle \phantom{aaaaaaaaaaaaaaaaaaa}\times {\int }_{V}g\left(v^{\prime }\right)N_k\left[\phi \right]\left(v^{\prime }\right)d{v}^{\prime }{e}^{i\pi .\frac{k}{a}.x}\in L^2\left(\mathsf{\Omega }\right)}$$

where

$${\displaystyle N_k\left[\phi \right]\left(v\right):=\frac{1}{\sqrt{2a}}{\int }_{\mathsf{\Omega }}\alpha \left(x\right)\phi (x,v)e^{-i\pi .\frac{k}{a}.x}dx,\phantom{aa}v\in V,k\in {\mathbb{Z}}^N.}$$

(12)

Further, we can write ${\left(U\left(t\right)\right)}_{t\ge 0}$ as a Dyson–Phillips expansion relative to ${\left(W\left(t\right)\right)}_{t\ge 0}$, that is

$$\begin{array}{c}\hfill U\left(t\right)=\sum _{j=0}^{\infty }{U}_j\left(t\right)\end{array}$$

where

$${\displaystyle U_0\left(t\right)=W\left(t\right)\mathrm{and}{U}_{j+1}\left(t\right)={\int }_0^{t}W\left(s\right)K_e{U}_j(t-s)ds.}$$

Moreover, the sequence ${\left({\sum }_{j=0}^n{K}_c{U}_j\left(t\right)K_c\right)}_{n\in \mathbb{N}}$ converges in $\mathcal{L}\left(L^2(\mathsf{\Omega }\times V,dxdv)\right)$ uniformly for bounded times to $K_c{e}^{t(T+K_e)}{K}_c$; thus, it suffices to prove that $K_c{U}_j\left(t\right)K_c$ is compact for all $j\in \mathbb{N}$.

For $j=0$, we have to show the compactness of $K_{c}W\left(t\right)K_c$. From the Riesz–Fréchet–Kolmogorov theorem (see, for example, [14], Theorem 4.26), it suffices to show that, for all bounded subsets $\mathfrak{B}$ of $L^2(\mathsf{\Omega }\times V,dxdv),$ we have

$$\underset{h\to 0}{lim}\underset{\begin{array}{c}\phi \in \mathfrak{B}\end{array}}{sup}{∥{\tau }_{h}O\left(t\right)\phi -O\left(t\right)\phi ∥}_{L^2(\mathsf{\Omega }\times V)}=0$$

(13)

where ${\tau }_h$ is the translation operator in x variable, defined by

$$\begin{array}{c}\hfill {\tau }_h\psi (x,v)=\psi (x-h,v),\forall \psi \in L^2(\mathsf{\Omega }\times V,dxdv).\end{array}$$

Equation (13) was established in ([15], Theorem 2.1).

Now, let $j\ge 1$. It is clear that

$$\begin{array}{cc}\hfill K_c{U}_j\left(t\right)K_c& =K_c{\left[W\left(t\right)K_e\right]}^j\ast W\left(t\right)K_c\hfill \\ \hfill & =K_{c}W\left(t\right)K_e\ast {\left[W\left(t\right)K_e\right]}^{j-1}\ast W\left(t\right)K_c.\hfill \end{array}$$

It is sufficient to prove that $K_{c}W\left(t\right)K_e$ is compact. Taking into account Proposition 2, it is enough to show the result for $p=2.$

Note that, according to Lemma 2 and Equation (11), $K_{c}W\left(t\right)K_e$ can be written as follows

$$K_{c}W\left(t\right)K_e={\mathsf{\Theta }}_{1}S\left(t\right)$$

where

$${\mathsf{\Theta }}_1:L^2\left(\mathsf{\Omega }\right)\ni \psi \mapsto {\alpha }_1\left(x\right)f_1\left(v\right)\psi \left(x\right)\in L^2(\mathsf{\Omega }\times V),\phantom{aaaaaaaa}$$

$${\displaystyle S\left(t\right):L^2(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1})\ni \phi \mapsto \sum _{k\in {\mathbb{Z}}^N}\frac{1}{\sqrt{2a}}({\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}}$$

$${\displaystyle e^{-i\pi .\frac{k}{a}.\rho \omega x}{\rho }^{N-1}d\rho d\omega )\times {\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_2\left({\omega }^{\prime }\right)M_k\left[\phi \right](\rho ,{\omega }^{\prime }){\rho }^{N-1}d\rho d{\omega }^{\prime }{e}^{i\pi .\frac{k}{a}.x}\in L^2\left(\mathsf{\Omega }\right)}$$

where

$${\displaystyle M_k\left[\phi \right](\rho ,\omega ):=\frac{1}{\sqrt{2a}}{\int }_{\mathsf{\Omega }}{\alpha }_2(x,\rho )\phi (x,\rho \omega )e^{-i\pi .\frac{k}{a}.x}dx,\phantom{aa}\omega \in {\mathbb{S}}^{N-1},\rho \in I,k\in {\mathbb{Z}}^N.}$$

(14)

It is sufficient to show that $S\left(t\right)$ is compact. According to Riesz–Fréchet–Kolmogorov’s theorem, we only have to prove that, if $\mathfrak{B}$ is a bounded subset of $L^2(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1},dxd\rho d{\omega }^{\prime })$, then

$$\underset{h\to 0}{lim}\underset{\begin{array}{c}\phi \in \mathfrak{B}\end{array}}{sup}{∥{\tau }_{h}S\left(t\right)\phi -S\left(t\right)\phi ∥}_{L^2\left(\mathsf{\Omega }\right)}=0.$$

(15)

Let $\phi \in \mathfrak{B}$. Putting

$$\begin{array}{c}\hfill {\displaystyle {\psi }_k:={\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_2\left(\omega \right)M_k\left[\phi \right](\rho ,\omega ){\rho }^{N-1}d\rho d\omega \phantom{aa}(k\in {\mathbb{Z}}^N).}\end{array}$$

Simple calculations show that

$$\begin{array}{cc}\hfill |{\psi }_k|& {\displaystyle \le {\int }_I{\int }_{{\mathbb{S}}^{N-1}}\left|{\displaystyle \left(g_2\left(\omega \right)\frac{1}{\sqrt{2a}}{\int }_{\mathsf{\Omega }}{\alpha }_2(x,\rho )\phi (x,\rho \omega )e^{-i\pi .\frac{k}{a}.x}dx\right)}\right|{\rho }^{N-1}d\rho d\omega }\hfill \\ \hfill \phantom{aaa}& {\displaystyle \le {\left({\int }_I{\int }_{{\mathbb{S}}^{N-1}}{\left|g_2\left(\omega \right)\right|}^2{\rho }^{N-1}d\rho d\omega \right)}^{{\displaystyle \frac{1}{2}}}\frac{1}{\sqrt{2a}}({\int }_I{\int }_{{\mathbb{S}}^{N-1}}}\hfill \\ \hfill & \phantom{aaaaaaaaaaaaaaaaaaaaaaa}{{\displaystyle \left|{\int }_{\mathsf{\Omega }}{\alpha }_2(x,\rho )\phi (x,\rho \omega )dx\right|}}^2{\rho }^{N-1}d\rho d\omega {)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & {\displaystyle \le {\left(\frac{{\rho }_{max}^N}{N}\right)}^{1/2}\frac{{∥{\alpha }_2∥}_{L^{\infty }(\mathsf{\Omega }\times I)}}{\sqrt{2a}}{∥g_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}}\hfill \\ \hfill & {\displaystyle \phantom{aaaaaaaaaaaaaaaaaa}{\left({\int }_I{\int }_{{\mathbb{S}}^{N-1}}{\left({\int }_{\mathsf{\Omega }}\left|\phi (x,\rho \omega )\right|dx\right)}^2{\rho }^{N-1}d\rho d\omega \right)}^{1/2}}\hfill \\ \hfill & {\displaystyle \le {\left(\frac{{\rho }_{max}^N}{N}\right)}^{1/2}\frac{{∥{\alpha }_2∥}_{L^{\infty }(\mathsf{\Omega }\times I)}}{\sqrt{2a}}{∥g_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}}\hfill \\ \hfill & {\displaystyle {\left({\int }_I{\int }_{{\mathbb{S}}^{N-1}}\left({\int }_{\mathsf{\Omega }}{I}_{\mathsf{\Omega }}\left(x\right)dx\right)\left({\int }_{\mathsf{\Omega }}{\left|\phi (x,\rho \omega )\right|}^{2}dx\right){\rho }^{N-1}d\rho d\omega \right)}^{1/2}}\hfill \\ \hfill & {\displaystyle ={\left(\frac{{\rho }_{max}^N}{N}\right)}^{1/2}\frac{{∥{\alpha }_2∥}_{L^{\infty }(\mathsf{\Omega }\times I)}}{\sqrt{\left|\mathsf{\Omega }\right|}}{∥g_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}\sqrt{\left|\mathsf{\Omega }\right|}{∥\phi ∥}_{L^2(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1})}.}\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\displaystyle |{\psi }_k{|}^2\le \frac{{\rho }_{max}^N}{N}{∥{\alpha }_2∥}_{L^{\infty }(\mathsf{\Omega }\times I)}^2{∥g_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}^2{∥\phi ∥}_{L^2(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1})}^2.}\end{array}$$

(16)

Using Parseval’s identity, we obtain

$${∥{\tau }_{h}S\left(t\right)\phi -S\left(t\right)\phi ∥}_{L^2\left(\mathsf{\Omega }\right)}^2=\sum _{k\in {\mathbb{Z}}^N}|e^{i\pi {\displaystyle \frac{k}{a}}.h}-1{|}^2{\left|{\psi }_k\right|}^2\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$$

$$\phantom{aaaaaaaaaaaaaaaaaaaa}{\left|{\displaystyle {\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2.$$

Let $N_0\in \mathbb{N}$ and assume that $\left|{\displaystyle \frac{k}{a}}\right|\le N_0$. From the Cauchy–Schwarz inequality, we have

$$\sum _{\left|{\displaystyle \frac{k}{a}}\right|\le N_0}{|e^{i\pi {\displaystyle \frac{k}{a}}.h}-1|}^2|{\psi }_k{|}^2{\left|{\displaystyle {\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2\phantom{aaaaaaaaaaaa}$$

$$\le 4\underset{\left|{\displaystyle \frac{k}{a}}\right|\le N_0}{sup}{|sin({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{k}{a}}.h)|}^2\sum _{k\in {\mathbb{Z}}^N}{\left|{\psi }_k\right|}^2\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$$

$$\phantom{aaaaaaaaaaaaaaaa}{{\displaystyle |{\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }|}^2\phantom{aaa}$$

$${\displaystyle \le 4\underset{\left|\frac{k}{a}\right|\le N_0}{sup}{|sin(\frac{\pi }{2}\frac{k}{a}.h)|}^2\sum _{k\in {\mathbb{Z}}^N}{\left|{\psi }_k\right|}^2{\left\{{\int }_I{\int }_{{\mathbb{S}}^{N-1}}|g_1\left(\rho \omega \right)||f_2\left(\omega \right)|{\rho }^{N-1}d\rho d\omega \right\}}^2\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}}$$

$${\displaystyle \le 4\underset{\left|\frac{k}{a}\right|\le N_0}{sup}{|sin(\frac{\pi }{2}\frac{k}{a}.h)|}^2\sum _{k\in {\mathbb{Z}}^N}{\left|{\psi }_k\right|}^2\left({\int }_I{\int }_{{\mathbb{S}}^{N-1}}{\left|g_1\left(\rho \omega \right)\right|}^2{\rho }^{N-1}d\rho d\omega \right)\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}}$$

$${\displaystyle \phantom{aaaaaaaaaaaaaaaaaaaaaaaaaa}\left({\int }_I{\int }_{{\mathbb{S}}^{N-1}}{\left|f_2\left(\omega \right)\right|}^2{\rho }^{N-1}d\rho d\omega \right)\phantom{aa}}$$

$${\displaystyle \le 4\underset{\left|\frac{k}{a}\right|\le N_0}{sup}{|sin(\frac{\pi }{2}\frac{k}{a}.h)|}^2\sum _{k\in {\mathbb{Z}}^N}{\left|{\psi }_k\right|}^{2}d\omega )\left({\int }_I{\rho }^{N-1}d\rho \right)\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}}$$

$${\displaystyle \phantom{aaaaaaaaaaaaaaaa}({\int }_I{\int }_{{\mathbb{S}}^{N-1}}{\left|g_1\left(\rho \omega \right)\right|}^2{\rho }^{N-1}d\rho \left({\int }_{{\mathbb{S}}^{N-1}}{\left|f_2\left(\omega \right)\right|}^{2}d\omega \right)\phantom{aa}}$$

$$=4\underset{\left|{\displaystyle \frac{k}{a}}\right|\le N_0}{sup}{|sin({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{k}{a}}.h)|}^2\sum _{k\in {\mathbb{Z}}^N}{\left|{\psi }_k\right|}^2{C}_g{∥f_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}^2\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$$

where

$${\displaystyle C_g:=\frac{{\rho }_{max}^N}{N}\left({\int }_I{\int }_{{\mathbb{S}}^{N-1}}{\left|g_1\left(\rho \omega \right)\right|}^2{\rho }^{N-1}d\rho d\omega \right).}$$

Using Equation (16) again, we can write

$$\sum _{\left|{\displaystyle \frac{k}{a}}\right|\le N_0}{|e^{i\pi \frac{k}{a}.h}-1|}^2{\left|{\psi }_k\right|}^2\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$$

$$\phantom{aaaaaaaaaaaaa}{{\displaystyle |{\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }|}^2$$

$${\displaystyle \le 4\underset{\left|\frac{k}{a}\right|\le N_0}{sup}{\left|sin(\frac{\pi }{2}\frac{k}{a}.h)\right|}^2\frac{{\rho }_{max}^N}{N}{∥{\alpha }_2∥}_{L^{\infty }(\mathsf{\Omega }\times I)}^2{∥g_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}^2\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}}$$

$$\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaa}{∥\phi ∥}_{L^2(\mathsf{\Omega }\times I\times {\mathbb{S}}^{N-1})}^2{C}_g{∥f_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}^2$$

As $\mathfrak{B}$ is bounded, it is clear that when $h\to 0$, $\left|sin({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{k}{a}}.h)\right|\to 0$, and thus one can obtain

$$\underset{h\to 0}{lim}\underset{\phi \in \mathfrak{B}}{sup}\underset{\begin{array}{c}\rho \in I\end{array}}{sup}\sum _{\left|{\displaystyle \frac{k}{a}}\right|\le N_0}\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$$

$${|e^{i\pi {\displaystyle \frac{k}{a}}.h}-1|}^2|{\psi }_k{|}^2{\left|{\displaystyle {\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2=0.$$

(17)

We suppose now, that $\left|{\displaystyle \frac{k}{a}}\right|\ge N_0$. So, we have

$$\sum _{\left|{\displaystyle \frac{k}{a}}\right|\ge N_0}{|e^{i\pi {\displaystyle \frac{k}{a}}.h}-1|}^2|{\psi }_k{|}^2{\left|{\displaystyle {\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2\phantom{aaaaaaaaaaaaa}$$

$$\le 4\sum _{k\in {\mathbb{Z}}^N}{\left|{\psi }_k\right|}^2\underset{\left|{\displaystyle \frac{k}{a}}\right|\ge N_0}{sup}{∥f_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}^2\phantom{aaaaaaaa}\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$$

$$\phantom{aaaaaaaaaaaaaaaaaaa}{\left|{\displaystyle \int {\int }_{\mathrm{supp}\left(g_1\right)}{g}_1\left(\rho \omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2.$$

Furthermore, using Equation (16), one can write

$$\begin{array}{cc}\hfill \sum _{\left|{\displaystyle \frac{k}{a}}\right|\ge N_0}{\left|e^{i\pi {\displaystyle \frac{k}{a}}.h}-1\right|}^2& |{\psi }_k{|}^2{\left|{\displaystyle {\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2\hfill \\ \hfill & {\displaystyle \le 4\frac{{\rho }_{max}^N}{N}{∥g_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}^2{∥{\alpha }_2∥}_{L^{\infty }(\mathsf{\Omega }\times I)}^2{∥\phi ∥}^2{∥f_2∥}_{L^2\left({\mathbb{S}}^{N-1}\right)}^2}\hfill \\ \hfill & |{\displaystyle \int {\int }_{\mathrm{supp}\left(g_1\right)}{g}_1\left(\rho \omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }{|}^2\phantom{aaaaaaa}\hfill \end{array}$$

As $\mathrm{Supp}\left(g_1\right)$ is compact and the measure $g_1\left(\rho \omega \right)e^{-\sigma \left(\rho \omega \right)t}{\rho }^{N-1}d\rho d\omega$ is absolutely continuous with respect to ${\rho }^{N-1}d\rho d\omega$, the use of hypothesis (10) implies that

$$\underset{N_0\to \infty }{lim}\underset{\left|{\displaystyle \frac{k}{a}}\right|\ge N_0}{sup}\sum _{\left|{\displaystyle \frac{k}{a}}\right|\ge N_0}{\left|{\displaystyle \int {\int }_{\mathrm{supp}\left(g_1\right)}{g}_1\left(\rho \omega \right)e^{-\sigma t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2=0.$$

(18)

As $\mathfrak{B}$ is bounded, using Equation (18), we obtain

$$\begin{array}{cc}\hfill \underset{N_0\to \infty }{lim}\underset{h\in {\mathbb{R}}^N}{sup}& \underset{\phi \in \mathfrak{B}}{sup}\underset{\begin{array}{c}\rho \in I\end{array}}{sup}\sum _{\left|{\displaystyle \frac{k}{a}}\right|\ge N_0}{|e^{i\pi {\displaystyle \frac{k}{a}}.h}-1|}^2{\left|{\psi }_k\right|}^2\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}\hfill \\ \hfill & \phantom{aaaaaaa}{\left|{\displaystyle {\int }_I{\int }_{{\mathbb{S}}^{N-1}}{g}_1\left(\rho \omega \right)f_2\left(\omega \right)e^{-\sigma \left(\rho \omega \right)t}{e}^{-i\pi \frac{k}{a}.\rho \omega t}{\rho }^{N-1}d\rho d\omega }\right|}^2=0.\hfill \end{array}$$

(19)

Next, using Equations (17) and (19), we get

$$\underset{h\to 0}{lim}\underset{\begin{array}{c}\phi \in \mathfrak{B}\end{array}}{sup}{∥{\tau }_{h}S\left(t\right)\phi -S\left(t\right)\phi ∥}_{L^2(\mathsf{\Omega }\times I)}=0$$

which proves the compactness of operator $K_{c}W\left(t\right)K_e$ on $L^2(\mathsf{\Omega }\times V,dxdv)$.

Finally, according to Proposition 2, $R_2\left(t\right)$ is compact on $L^p(\mathsf{\Omega }\times V,dxdv)$ for all $p\in (1,+\infty )$. Now, applying Proposition 1, we infer that ${\omega }_{ess}\left(e^{t(T+K_c+K_e)}\right)={\omega }_{ess}\left(e^{t(T+K_e)}\right),\forall t\ge 0.$ This completes the proof. □

Remark 3.

Note that according to the comments in the last paragraph of the introduction, Theorem 1 implies that

$$\begin{array}{c}\hfill {\omega }_{ess}\left(e^{t(T+K_c+K_e)}\right)={\omega }_{ess}\left(e^{t(T+K_e)}\right)\forall t\ge 0.\end{array}$$

(20)

This allows us to write the solution to the Cauchy problem (3) in the form (4), which allows us to derive its time asymptotic behavior for large times.

Open problems

• Let us point out that the compactness properties of $R_2\left(t\right)$ in space $L^1(\mathsf{\Omega }\times V,dxdv)$ is still an open problem. In fact, in space $L^1(\mathsf{\Omega }\times V,dxdv)$, it is sufficient to prove the weak compactness of $R_2\left(t\right)$. We say that a bounded linear operator B on a Banach space X is strictly power compact if there exists $n\in \mathbb{N}$, such that ${\left(BA\right)}^n$ is compact for all $A\in \mathcal{L}\left(X\right)$ (see [9], p. 154). So, if $R_2\left(t\right)$ is weakly compact, for all $A\in \mathcal{L}\left(L^1(\mathsf{\Omega }\times V,dxdv)\right)$, then ${\left(R_2\left(t\right)A\right)}^2$ will be compact on $L^1(\mathsf{\Omega }\times V,dxdv)$ and, therefore, we will have ${\omega }_{ess}\left(e^{t(T+K_e+K_c)}\right)={\omega }_{ess}\left(e^{t(T+K_e)}\right)\forall t\ge 0$. Unfortunately, we did not succeed in proving the weak compactness of $R_2\left(t\right)$ for $p=1$.
• We note that the compactness of the first-order reminder term, $R_1\left(t\right)$, of the Dyson–Phillips expansion, is also an interesting open problem in $L^p$-spaces with $p\in [1,+\infty )$. For vacuum boundary conditions, this problem was settled by Mokhtar–Kharroubi in [2] for $p\in (1,+\infty )$. However, even for vacuum boundary conditions, it is open for $p=1$. The interest of this result lies in the fact that it implies not only equality (20), but also the invariance of the essential spectrum of ${\left(e^{t(T+K_e)}\right)}_{t\ge 0}$ under perturbation, that is

  $${\sigma }_{ess}\left(e^{t(T+K_e+K_c)}\right)={\sigma }_{ess}\left(e^{t(T+K_e)}\right)\forall t\ge 0$$

  where ${\sigma }_{ess}(·)$ of operator U on Banach space X is defined by

  $${\sigma }_{ess}\left(U\right):=\bigcap _{V\in \mathcal{K}\left(X\right)}\sigma (U+V)$$

  where $\mathcal{K}\left(X\right)$ stands for the set of all compact operators on X.