On a resolvent approach for perturbed semigroups and application to $L^1$-neutron transport theory

Abstract

We give new sufficient and practical conditions in terms of the generators ensuring the stability of the critical or the essential type of a perturbed $C_0$-semigroup in general Banach spaces. We apply our theoretical results in order to investigate the control and in particular the time asymptotic behavior of solutions to a broad class of transport equations in $L^1$-spaces and higher dimension. Our results improve, complete and enrich several earlier works.

1 Introduction and main results

Let ${\mathcal {X}}$ be a complex Banach space and let $\mathcal T:D({\mathcal {T}})\subseteq {\mathcal {X}} \rightarrow {\mathcal {X}}$ be the infinitesimal generator of a $C_0$-semigroup $({\mathcal {U}}(t))_{t\ge 0}$ acting on ${\mathcal {X}}$. We denote by ${\mathcal {L}}({\mathcal {X}})$ the algebra of all bounded linear operators acting on ${\mathcal {X}}$. Let us consider the Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll}\!\displaystyle {d\psi \over dt} &{}\!\!\!\!\!\!\,\,\,\,\,={\mathcal {T}} \psi +{\mathcal {B}} \psi \\ \\ \!\psi (0)&{}\!\!\!\!\!\,\,\,\,=\psi _0\end{array}\right. \end{aligned}$$

(1.1)

where ${\mathcal {B}}\in {\mathcal {L}}({\mathcal {X}})$ and $\psi _0\in \mathcal X$. By the classical perturbation theory [7], the perturbed operator ${\mathcal {A}}:={\mathcal {T}}+ {\mathcal {B}}$ generates also a strongly continuous semigroup $({\mathcal {V}}(t))_{t\ge 0}$ given for every $t\ge 0$ by the Dyson–Phillips expansion

$$\begin{aligned} {\mathcal {V}}(t)=\sum _{j=0}^{m-1} {\mathcal {U}}_j(t)+ {\mathcal {R}}_m(t), \end{aligned}$$

(1.2)

where

$$\begin{aligned} {\mathcal {U}}_0(t)={\mathcal {U}}(t),\ \ {\mathcal {U}}_j(t)=\int _0^t{\mathcal {U}}(t-s){\mathcal {B}}{\mathcal {U}}_{j-1}(s)\,ds ,\ \ (j\ge 1) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {R}}_m(t)=\sum _{j= m}^{+\infty }{\mathcal {U}}_j(t),\ (m\ge 1). \end{aligned}$$

Consequently, for every $\psi _0\in D({\mathcal {A}})$, the Cauchy problem (1.1) admits the unique solution $\psi (t)=\mathcal V(t)\psi _0$, $t\ge 0$.

In order to investigate the asymptotic behavior of $\psi (t)$ (for large time), two fundamental approaches are at our disposal. The first approach, called the resolvent approach, is based on the asymptotic spectrum of the generator ${\mathcal {A}}$:

$$\begin{aligned} \sigma _{ass}({{\mathcal {A}}}):=\{\lambda \in {\mathbb {C}}:\ \hbox {Re}\lambda >\omega ({\mathcal {U}})\}, \end{aligned}$$

where $\omega ({\mathcal {U}})$ stands for the type of $(\mathcal U(t))_{t\ge 0}.$

This approach was initiated by Vidav [23] in a particular case and developed afterwords by Mokhtar-Kharroubi [16]. It is essentially based on the following hypothesis:

$$\begin{aligned} ({{\mathcal {H}}}({\mathcal {T}},{\mathcal {B}}))\left\{ \begin{array}{l}\hbox {There exists an integer}\ j\ \hbox {and}\ \omega >\omega ({\mathcal {U}})\ \hbox {such that} \\ (\hbox {i})\ [{\mathcal {B}} (\lambda -{\mathcal {T}})^{-1}]^j\ \hbox {is compact for every}\ \lambda \in {\mathcal {R}}_\omega ;\\ (\hbox {ii})\ \displaystyle \lim _{|\hbox {Im}\,\lambda |\rightarrow +\infty }\left\| [{\mathcal {B}}(\lambda -{\mathcal {T}})^{-1}]^j\right\| =0\ \hbox {uniformly on}\ {\mathcal {R}}_\omega , \end{array}\right. \end{aligned}$$

where $(\cdot -{\mathcal {T}})^{-1}$ denotes the resolvent operator of ${\mathcal {T}}$ and $\mathcal R_\omega =\{\lambda \in {\mathbb {C}}:\ \hbox {Re}\lambda \ge \omega \}$. It has been shown in Ref. [16] that under the hypothesis $({{\mathcal {H}}}({\mathcal {T}},{\mathcal {B}}))$ the part of the spectrum of the perturbed generator ${\mathcal {A}}={\mathcal {T}}+{\mathcal {B}}$ lying in the half-plane ${\mathcal {R}}_\omega$ consists of at most of a finite number of isolated eigenvalues with finite algebraic multiplicity $\{\lambda _1,\ldots \lambda _n\}$ . Let $\beta _1=\sup \{$Re$\lambda :\ \lambda \in \sigma ({\mathcal {T}}+{\mathcal {K}})$ and Re$\lambda <\omega \}$ and $\beta _2=\min \{$Re$\lambda _j,\ 1\le j\le n\}$. Let $P_j$ and $D_j$ denote the spectral projection and the nilpotent operator associated with $\lambda _j$, $1\le j\le n$, respectively. The solution of the Cauchy problem (1.1) fulfils

$$\begin{aligned} \big \Vert \psi (t)-\sum _{j=1}^ne^{\lambda _j t}e^{D_jt}P_j\psi _0\big \Vert =o(e^{\beta ^*t})\ \hbox {for every}\ \psi _0\in D({\mathcal {A}}^2), \end{aligned}$$

where $\beta _1<\beta ^*<\beta _2$. The main disadvantage of this approach lies in the fact that the quantity $\psi (t)-\sum _{j=1}^ne^{\lambda _j t}e^{D_jt}P_j\psi _0$ can be evaluated only for initial data $\psi _0$ in $D({\mathcal {A}}^2)$. This condition was weakened to $\psi _0\in D(A)$ by D. Song [21] for special Banach spaces and by Song and Greenberg [22] in some special cases. Recently, Latrach and the first author [9] weakened the condition $\psi _0\in D({\mathcal {A}}^2)$ to $\psi _0\in D({\mathcal {A}})$ by considering instead of condition $({{\mathcal {H}}}({\mathcal {T}},{\mathcal {B}}))$, the condition

$$\begin{aligned} (\tilde{{\mathcal {H}}}({\mathcal {T}},{\mathcal {B}}))\left\{ \begin{array}{l}\hbox {There exist an integer}\ j\ \hbox {and two reals}\ \ \omega>\omega ({\mathcal {U}})\ \hbox {and}\ C>0\ \hbox {such that} \\ (\hbox {i})\ [{\mathcal {B}} (\lambda -{\mathcal {T}})^{-1}]^j\ \hbox {is compact for every}\ \lambda \in {\mathcal {R}}_\omega ;\\ (\hbox {ii})\ |\hbox {Im}\,\lambda |\,\left\| [\mathcal B(\lambda -{\mathcal {T}})^{-1}]^j\right\| \le C\ \hbox {uniformly on}\ {\mathcal {R}}_\omega . \end{array}\right. \end{aligned}$$

Following [25], the essential spectrum of $\mathcal O\in {\mathcal {L}}({\mathcal {X}})$ is defined by:

$$\begin{aligned} \sigma _{ess}({\mathcal {O}}):=\{\lambda \in \sigma ({\mathcal {O}})\ \hbox {but}\ \lambda \ \hbox {is not an eigenvalue of finite algebraic multiplicity}\}, \end{aligned}$$

and the essential spectral radius of ${\mathcal {O}}$ is

$$\begin{aligned} r_{ess}({\mathcal {O}}):=\sup \{|\lambda |:\ \lambda \in \sigma _{ess}({\mathcal {O}})\}. \end{aligned}$$

Let $({\mathcal {W}}(t))_{t\ge 0}$ be $C_0$-semigroup. According to [25, Lemma 2.1], there exists a real number $\omega _{ess}({\mathcal {W}})\in [-\infty ,\omega ({\mathcal {W}})]$, called the essential type of $({\mathcal {W}}(t))_{t\ge 0}$, such that

$$\begin{aligned} r_{ess}({\mathcal {W}}(t))=e^{\omega _{ess}({\mathcal {W}})\,t}\ \hbox {for every}\ t\ge 0. \end{aligned}$$

The second approach, called the semigroup approach, is based on the asymptotic spectrum of ${\mathcal {V}} (t)$:

$$\begin{aligned} \sigma _{ass}({{\mathcal {V}}(t)}):=\sigma ({\mathcal {V}}(t))\cap \{\lambda \in {\mathbb {C}}:\ |\lambda |>e^{\omega ({\mathcal {U}})t}\}. \end{aligned}$$

This approach makes use of the fact that if some remainder of the Dyson–Phillips expansion ${\mathcal {R}}_m(t)$ is compact (or strictly singular) for $t\ge 0$, then  [16, 26]  $(\mathcal U(t))_{t\ge 0}$ and $(\mathcal V(t))_{t\ge 0}$  have the same essential type. Therefore, the part of the spectrum of the perturbed semigroup ${\mathcal {V}}(t)$ outside the circle $|\mu |=e^{t\omega ({\mathcal {U}})}$ consists only of at most isolated eigenvalues with finite algebraic multiplicities. If these eigenvalues exist, the semigroup $(\mathcal V(t))_{t\ge 0}$ may be decomposed in two parts: the first containing the time development of finitely many eigenmodes, the second being of faster decay. Applying now the spectral mapping theorem for the point spectrum [6, 3.7 Spectral Mapping Theorem for Point and Residual Spectrum], we deduce that for any fixed $\omega >\omega ({\mathcal {U}})$, the set $\sigma ({\mathcal {T}}+\mathcal B)\cap \{\lambda \in {\mathbb {C}}:\ \hbox {Re}\lambda \ge \omega \}$ consists of finitely many eigenvalues $\{\lambda _1,\ldots \lambda _n\}$. The solution of the Cauchy problem (1.1) satisfies

$$\begin{aligned} \big \Vert \psi (t)-\sum _{j=1}^ne^{\lambda _j t}e^{D_jt}P_j\psi _0\big \Vert =o(e^{\beta ^*t})\ \hbox {for every}\ \psi _0\in D({\mathcal {A}}), \end{aligned}$$

where $P_j$, $D_j$ and $\beta ^*$ have the same meaning as above. For sake of completeness, we note that a similar result concerning the semigroup approach was given in [19] (see also [2, Chapter 19]).

The major drawback of this approach is that it is not applicable unless the unperturbed semigroup is explicit.

Even though these two approaches have a significant merit in their own right, it is of interest to find a balance between them to avoid the aforementioned drawbacks. On that account, the two approaches were linked firstly in the Hilbert spaces setting by Sbihi and subsequently in Banach spaces by Latrach and the authors. Specifically, if ${\mathcal {X}}$ is a Hilbert space, ${\mathcal {T}}$ is dissipative and there exists $\alpha > \omega ({\mathcal {U}})$ such that

$$\begin{aligned} (\alpha +i\beta -{\mathcal {T}})^{-1}{\mathcal {B}} (\alpha +i\beta -{\mathcal {T}})^{-1}\ \hbox {is compact for all}\ \beta \in {\mathbb {R}}, \end{aligned}$$

and

$$\begin{aligned} \displaystyle \lim _{\beta \rightarrow \infty }\Vert {\mathcal {B}}^*(\alpha +i\beta -{\mathcal {T}})^{-1}{\mathcal {B}}\Vert +\Vert \mathcal B(\alpha +i\beta -{\mathcal {T}})^{-1}{\mathcal {B}}^*\Vert =0, \end{aligned}$$

(1.3)

where ${\mathcal {B}}^*$ denotes the dual operator of ${\mathcal {B}}$, then [20, Theorem 2.3 and Lemma 2.8] ${\mathcal {R}}_1(t)$ is compact on ${\mathcal {X}}$ for all $t\ge 0$. This implies that, for each $t \ge 0$, ${\mathcal {U}}(t)$ and ${\mathcal {V}}(t)$ have the same essential spectrum. Further for a general Banach space ${\mathcal {X}}$, if the condition $({\tilde{{\mathcal {H}}}}({\mathcal {T}},{\mathcal {B}}))$ holds, then [8, Theorem 1.1] the remainder term $\mathcal R_{2j+3}$ is compact for each $t\ge 0$, and consequently, $({\mathcal {V}}(t))_{t\ge 0}$ and $({\mathcal {U}}(t))_{t\ge 0}$ have the same essential type.

In many applications, including multi-dimensional transport equations, the item (ii) in $({\tilde{{\mathcal {H}}}}(\mathcal T,{\mathcal {B}}))$, seems to be not always verified because 0 gives a singularity point in an integral representation of $KR(\lambda , T)K$, where T and K denotes respectively the streaming operator and the collision operator (see, Proposition 4.2 below and its proof). In this work, we present and describe an efficient method to avoid singularities. The idea of the method is to construct a sequence of operators $(\vartheta _0^\varepsilon (\mathcal T,{\mathcal {B}})(\lambda ))_{\varepsilon >0}$ which converges in ${\mathcal {L}}({\mathcal {X}})$ to $(\lambda -{\mathcal {T}})^{-1}$ as $\varepsilon$ goes to zero, uniformly on ${\mathcal {R}}_\omega$ (see, Lemma 3.4 below), and satisfies (see, Proposition 4.2 below): for every $\varepsilon >0$, there exists $C_\varepsilon >0$ such that

$$\begin{aligned} |\hbox {Im}\,\lambda |\,\left\| {\mathcal {B}}\vartheta _0^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda ){\mathcal {B}}\right\| \le C_\varepsilon \ \hbox {uniformly on}\ {\mathcal {R}}_\omega , \end{aligned}$$

or more generally, we construct a sequence of operators $(\vartheta _j^\varepsilon ({\mathcal {T}},\mathcal B)(\lambda ))_{\varepsilon >0}$ which converges in $\mathcal L({\mathcal {X}})$ to $\vartheta _j({\mathcal {T}},\mathcal B)(\lambda ):=(\lambda -{\mathcal {T}})^{-1}[{\mathcal {B}}(\lambda -\mathcal T)^{-1}]^j$ as $\varepsilon$ goes to zero, uniformly on $\mathcal R_\omega$, and satisfies the hypothesis

$$\begin{aligned} ({\mathcal {A}}1(j))\left\{ \begin{array}{l}\hbox {For every fixed}\ \varepsilon>0, \hbox {there exists}\ \nu \ge \max (\omega ({\mathcal {U}}),0)\ \hbox {satisfying}\\ \displaystyle \ \sup _{\xi > \nu } \int _{-\infty }^{+\infty } \Vert \vartheta ^\varepsilon _j(\mathcal T,{\mathcal {B}})(\xi +i\tau )\Vert \,d\tau <\infty . \ \end{array} \right. \end{aligned}$$

For further purposes, we recall the following:

Theorem 1.1

• (i) [4] Let $j\ge 1$, then, ${\mathcal {U}}_j(t)$ is compact for all $t \ge 0$ if and only if, the map $t \mapsto {\mathcal {U}}_j(t)$ is norm continuous for $t \ge 0$, and $(\lambda -{\mathcal {T}})^{-1}[{\mathcal {B}} (\lambda -{\mathcal {T}})^{-1}]^j$ is compact for all $s\in {\mathbb {R}}$.

• (ii) [5] If the mapping $t\mapsto {\mathcal {R}}_j(t)$ is norm right continuous for every $t>0$ and some $j\in {\mathbb {N}}$, then one has $\omega _{crit}({\mathcal {V}})= \omega _{crit}(\mathcal U)$.

(for the definition of the critical type $\omega _{crit}({\mathcal {W}})$ of a given $C_0$-semigroup $(\mathcal W(t))_{t\ge 0}$, see Sect. ). A natural relevant problem occurs here; namely, for ${\mathcal {B}}\in {\mathcal {L}}(\mathcal X)$ and $j\in {\mathbb {N}}$ with $j\ge 1$, does exist necessary and sufficient conditions involving ${\mathcal {T}}$ and ${\mathcal {B}}$ to get norm continuity of the j th order term ${\mathcal {U}}_j(t)$ for $t>0$, of the Dyson–Phillips expansion defining the $C_0$-semigroup $({\mathcal {V}}(t))_{t\ge 0}$ generated by ${\mathcal {T}}+{\mathcal {B}}.$ This problem has been addressed by several investigators in the Hilbert spaces setting (cf. [3, 11, 18, 20] and [1]).

In [20], Sbihi gave a practical sufficient condition to get norm continuity of ${\mathcal {R}}_1(t)$ for $t\ge 0$, in the case where X is a Hilbert space. His result states as follows:

Lemma 1.1

[20, Corollary 2.9] Assume that X is a Hilbert space, ${\mathcal {T}}$ is dissipative and for some $\alpha >0$, the condition (1.3) holds true, then the map $t \mapsto {\mathcal {U}}_1(t)$ is norm continuous for $t \ge 0$.

More recently, in [1], we showed the following:

Theorem 1.2

If X is a Hilbert space then ${\mathcal {V}}(t)-{\mathcal {U}}(t)$ is norm continuous for $t\ge 0$ if and only if,

$$\begin{aligned} \sup _{x\in H, \Vert x\Vert =1} \int _{| s | \ge a}\Vert R(\alpha + is,A )KR(\alpha + is,A )x \Vert ^{2}ds \rightarrow 0\ \hbox {as}\ a \rightarrow +\infty \end{aligned}$$

and

$$\begin{aligned} \sup _{x\in H, \Vert x\Vert =1} \int _{| s | \ge a}\Vert R(\alpha + is ,A^*)K^*R(\alpha + is ,A^*)x \Vert ^{2}ds \rightarrow 0\ \hbox {as}\ a \rightarrow +\infty . \end{aligned}$$

This problem becomes distinctly more complicated when going from Hilbert spaces to Banach spaces. Notice that according to Theorem 1.1, the norm continuity of one of the terms $(\mathcal U_j(t))_{t\ge 0}$, $j\ge 1$, implies that the critical type $\omega _{crit}({\mathcal {V}})$ of the perturbed semigroup $(\mathcal U(t))_{t\ge 0}$ equals to the critical type $\omega _{crit}({\mathcal {U}})$ of the unperturbed semigroup $(\mathcal V(t))_{t\ge 0}$. This enables us to study the control for $t\ge 0$ or in particular the time asymptotic behavior of solutions to several evolution equations. Indeed, according to the partial spectral mapping theorem (see, Theorem 2.1 below), for $\beta >\omega ({\mathcal {U}})$, if $\sigma ({\mathcal {T}}+{\mathcal {B}})\cap \{\lambda \in {\mathbb {C}}: Re\lambda =\beta \}=\emptyset$ and P is the spectral projection associated to the closed part of the spectrum $\sigma ({\mathcal {T}}+{\mathcal {B}})\cap \{\lambda \in {\mathbb {C}}: Re\lambda >\beta \}$, then there exists $M>0$ satisfying

$$\begin{aligned} \Vert {\mathcal {V}}(t)(I-P)\Vert \le M e^{\beta t}\ \hbox {for all}\ t\ge 0. \end{aligned}$$

In this paper we provide sufficient conditions in terms of the resolvent operator $(\cdot -{\mathcal {T}})^{-1}$ of the generator ${\mathcal {T}}$ of the semigroup $({\mathcal {U}}(t))_{t\ge 0}$ and the perturbation operator ${\mathcal {B}}$, ensuring the norm continuity of the mapping $t\mapsto {\mathcal {R}}_j(t)$ for $t>0$ (see Theorem 3.1 and Corollary 3.4 below). Furthermore, we show that, if condition $({\mathcal {A}}1(j))$ and assertion (i) of hypothesis $({{\mathcal {H}}}({\mathcal {T}},{\mathcal {B}}))$ hold true simultaneously, then the $j^{th}$-order term ${\mathcal {R}}_j(t)$ of the Dyson–Phillips expansion (4.4) is compact on ${\mathcal {X}}$ for every $t\ge 0$ (see Theorem 3.2 and Corollary 3.6 below). These theoretical results apply directly to discuss the control for every $t\ge 0$ and consequently the time asymptotic behavior (for large times) of solutions to a broad class of multi-dimensional neutron transport equations on $L^1$-spaces.

2 Critical spectrum

Let us recall the concept of critical spectrum introduced by R. Nagel and J. Poland [17]. Let $({\mathcal {W}}(t))_{t\ge 0}$ be a $C_0$-semigroup on a given Banach space ${\mathcal {Y}}$. We consider the Banach space $\tilde{{\mathcal {Y}}}:=\ell ^\infty ({\mathcal {Y}})$ of all bounded sequences in ${\mathcal {Y}}$ endowed with the norm

$$\begin{aligned} \Vert {\tilde{y}}\Vert :=\sup _{n\in {\mathbb {N}}}\Vert y_n\Vert \ \hbox {for}\ {\tilde{y}}=(y_n)_{n\in {\mathbb {N}}}\in \tilde{{\mathcal {Y}}}. \end{aligned}$$

We extend the semigroup $({\mathcal {W}}(t))_{t\ge 0}$ to $\tilde{{\mathcal {Y}}}$ and obtain a new semigroup $(\tilde{\mathcal W}(t))_{t\ge 0}$ defined by

$$\begin{aligned} \tilde{{\mathcal {W}}}(t){\tilde{y}}:=({\mathcal {W}}(t)y_n)_{n\in {\mathbb {N}}}\ \hbox {for}\ {\tilde{y}}=(y_n)_{n\in {\mathbb {N}}}\in \tilde{{\mathcal {Y}}}. \end{aligned}$$

Let ${\tilde{Y}}_{\mathcal {W}}$ be the subspace of strong continuity of $(\tilde{{\mathcal {W}}}(t))_{t\ge 0}$

$$\begin{aligned} {\tilde{Y}}_{\mathcal {W}}:=\left\{ {\tilde{y}}\in {\tilde{Y}}:\ \lim _{t\downarrow 0} \left\| \tilde{\mathcal W}(t){\tilde{y}}-{\tilde{y}}\right\| =0\right\} . \end{aligned}$$

This subspace is closed and $(\tilde{{\mathcal {W}}}(t))_{t\ge 0}$-invariant. On the quotient space ${\hat{Y}}:={\tilde{Y}}/ {\tilde{Y}}_{\mathcal {W}}$, the semigroup $(\tilde{\mathcal W}(t))_{t\ge 0}$ induces the quotient semigroup $(\hat{\mathcal W}(t))_{t\ge 0}$ given by

$$\begin{aligned} \hat{{\mathcal {W}}}(t){\hat{y}}:=\tilde{{\mathcal {W}}}(t){\tilde{y}}+{\tilde{Y}}_{\mathcal {W}}\ \hbox {for}\ {\hat{y}}={\tilde{y}}+{\tilde{Y}}_{\mathcal {W}} \ \hbox {and}\ t\ge 0. \end{aligned}$$

The critical spectrum of ${{\mathcal {W}}}(t)$ is then defined as

$$\begin{aligned} \sigma _{crit}({{\mathcal {W}}}(t)):=\sigma (\hat{{\mathcal {W}}}(t)) \end{aligned}$$

and the critical spectral radius is defined as

$$\begin{aligned} r_{crit}({{\mathcal {W}}}(t)):=r(\hat{{\mathcal {W}}}(t)) \end{aligned}$$

while the critical type or the critical growth bound of $({{\mathcal {W}}}(t))_{t\ge 0}$ is defined as

$$\begin{aligned} \omega _{crit}({\mathcal {W}}):=\omega (\hat{{\mathcal {W}}}). \end{aligned}$$

The critical spectrum enjoys nice properties (see, [5, 17] for general theory); in particular, we have

Theorem 2.1

[17, Proposition 4.3] Let $({{\mathcal {W}}}(t))_{t\ge 0}$ be a $C_0$-semigroup on a Banach space ${\mathcal {Y}}$ with infinitesimal generator $\Xi$. Then

$$\begin{aligned} \omega ({\mathcal {W}})=\max \{s(\Xi ), \omega _{crit}({\mathcal {W}})\}. \end{aligned}$$

Moreover, the following partial spectral mapping theorem holds

$$\begin{aligned} \sigma ({{\mathcal {W}}}(t))\cap {{\mathcal {Q}}}_{crit}({{\mathcal {W}}}(t)) = e^{t\sigma (\Xi )} \cap {{\mathcal {Q}}}_{crit}({{\mathcal {W}}}(t))\ \hbox {for each}\ t\ge 0, \end{aligned}$$

where ${{\mathcal {Q}}}_{crit}({{\mathcal {W}}}(t)):=\{\lambda \in {\mathbb {C}}: |\lambda |>r_{crit}({{\mathcal {W}}}(t))\}$.

3 Resolvent approach for perturbed semigroups

3.1 Norm continuity of ${\mathcal {U}}_j(\cdot )$

For every $\varepsilon >0$ small enough, set

$$\begin{aligned} {\mathcal {U}}_{0}^\varepsilon (t):={\mathcal {U}}(t)\, \chi _{(\varepsilon ,+\infty )}(t),\ t\ge 0. \end{aligned}$$

(3.1)

Remark 3.1

• (1) For every $\varepsilon >0$ and $x\in {\mathcal {X}}$, ${\mathcal {U}}_0^\varepsilon (\cdot )x=0$ on $(0,\varepsilon )$ and since $({\mathcal {U}}(t))_{t\ge 0}$ is a strongly continuous semigroup on ${\mathcal {X}}$, the map ${\mathcal {U}}_0^\varepsilon (\cdot )x$ is continuous on $(\varepsilon ,+\infty )$ (and therefore, the map $\mathcal U_0^\varepsilon (\cdot )x$ is measurable on $(0,+\infty )$) and for every $\omega >\omega ({\mathcal {U}})$, there exists $M\ge 1$ satisfying

  $$\begin{aligned} \Vert {\mathcal {U}}(t)\Vert \le Me^{\omega t}\ \hbox {for every}\ t>0, \end{aligned}$$

  and therefore,

  $$\begin{aligned} \Vert {\mathcal {U}}_0^\varepsilon (t)\Vert \le Me^{\omega t}\ \hbox {for every}\ t>0. \end{aligned}$$

  (3.2)

• (2) It is well known that (see, [7] for example) for every $\omega >\omega ({\mathcal {U}})$, there exists $M\ge 1$ satisfying

  $$\begin{aligned} \Vert {\mathcal {U}}_j(t)\Vert \le M_j\,t^j \,e^{\omega t}\ \hbox {for every}\ t>0\ \hbox {and}\ j\ge 0 \end{aligned}$$

  (3.3)

  where $M_j:= ({M^{j+1}\ \Vert {\mathcal {B}}\Vert ^j})/{j!}$, $j\ge 0$.

Let $\varepsilon >0$. Formally, we set

$$\begin{aligned} {\mathcal {U}}_j^\varepsilon (t)x=\int _0^t{\mathcal {U}}_{j-1}^\varepsilon (t-s) \,{\mathcal {B}}\,{\mathcal {U}}^\varepsilon _0(s)x\,ds,\ t\ge 0,\ x\in {\mathcal {X}}\ \hbox {and}\ j\ge 1. \end{aligned}$$

In the following lemma, we prove that the operators $\mathcal U_j^\varepsilon (t)$ are well defined for almost all $t\ge 0$ and all $j\in {\mathbb {N}}$.

Lemma 3.1

For every $j\ge 0$, $\varepsilon >0$ and $x\in {\mathcal {X}}$, ${\mathcal {U}}_j^\varepsilon (\cdot )x=0$ on $(0,\varepsilon )$, the map ${\mathcal {U}}_j^\varepsilon (\cdot )x$ is continuous on $(\varepsilon ,+\infty )$, and therefore, it is measurable on $(0,+\infty )$. Furthermore, for every $\omega >\omega ({\mathcal {U}})$, there exists $M\ge 1$ satisfying

$$\begin{aligned} \Vert {\mathcal {U}}_j^\varepsilon (t)\Vert \le M_j\,t^j \,e^{\omega t}\ \hbox {for every}\ t>0,\ \varepsilon >0\ \hbox {and}\ j\ge 0 \end{aligned}$$

(3.4)

where $M_j:= ({M^{j+1}\ \Vert {\mathcal {B}}\Vert ^j})/{j!}$, $j\ge 0$.

Proof

We proceed by mathematical induction on $j\in \mathbb N$. For $j=0$, the result follows from the first item of Remark 3.1. Let $j\ge 0$ and assume that the result holds for j. Then, for $\varepsilon >0$ and $x\in {\mathcal {X}}$, we have

$$\begin{aligned} {\mathcal {U}}_{j+1}^\varepsilon (t)x=\int _0^t{\mathcal {U}}_{j}^\varepsilon (t-s) \,{\mathcal {B}}\,{\mathcal {U}}^\varepsilon _0(s)x\,ds,\ t\ge 0. \end{aligned}$$

(3.5)

Since ${\mathcal {U}}_{0}^\varepsilon (\cdot )x=0$ on $(0,\varepsilon )$, then, ${\mathcal {U}}_{j+1}^\varepsilon (\cdot )x=0$ on $(0,\varepsilon )$. Linking (3.2), (3.4) and (3.5), we get

$$\begin{aligned} \Vert {\mathcal {U}}_{j+1}^\varepsilon (t)\Vert \le M_{j+1}\,t^{j+1} \,e^{\omega t},\ t> 0. \end{aligned}$$

To achieve the proof, it remains to show that the map $\mathcal U_{j+1}^\varepsilon (\cdot )x$ is continuous on $(\varepsilon ,+\infty )$. For $\varepsilon<t_0<t$, we have

$$\begin{aligned}&{\mathcal {U}}_{j+1}^\varepsilon (t)x-{\mathcal {U}}_{j+1}^\varepsilon (t_0)x\\&\quad =\displaystyle \int _0^t{\mathcal {U}}_{j}^\varepsilon (t-s) \,{\mathcal {B}}\,{\mathcal {U}}^\varepsilon _0(s)x\,ds-\int _0^{t_0}{\mathcal {U}}_{j}^\varepsilon (t_0-s) \,{\mathcal {B}}\,{\mathcal {U}}^\varepsilon _0(s)x\,ds\\&\quad =\displaystyle \int _0^{t_0}[{\mathcal {U}}_{j}^\varepsilon (t-s)-\mathcal U_{j}^\varepsilon (t_0-s)] \,{\mathcal {B}}\,\mathcal U^\varepsilon _0(s)x\,ds+ \int _{t_0}^t{\mathcal {U}}_{j}^\varepsilon (t-s) \,{\mathcal {B}}\,\mathcal U^\varepsilon _0(s)x\,ds \end{aligned}$$

and hence,

$$\begin{aligned}&\Vert {\mathcal {U}}_{j+1}^\varepsilon (t)x-{\mathcal {U}}_{j+1}^\varepsilon (t_0)x\Vert \\&\quad \le \left\| \displaystyle \int _0^{t_0}[{\mathcal {U}}_{j}^\varepsilon (t-s)-{\mathcal {U}}_{j}^\varepsilon (t_0-s)] \,{\mathcal {B}}\,{\mathcal {U}}^\varepsilon _0(s)x\,ds\right\| + \int _{t_0}^t\Vert {\mathcal {U}}_{j}^\varepsilon (t-s) \,{\mathcal {B}}\,\mathcal U^\varepsilon _0(s)x\Vert \,ds \end{aligned}$$

and similarly for $\varepsilon<t<t_0$, we have

$$\begin{aligned}&\Vert {\mathcal {U}}_{j+1}^\varepsilon (t_0)x-{\mathcal {U}}_{j+1}^\varepsilon (t)x\Vert \\&\quad \le \left\| \displaystyle \int _0^{t}[{\mathcal {U}}_{j}^\varepsilon (t_0-s)-{\mathcal {U}}_{j}^\varepsilon (t-s)] \,{\mathcal {B}}\,{\mathcal {U}}^\varepsilon _0(s)x\,ds\right\| + \int _{t}^{t_0}\Vert {\mathcal {U}}_{j}^\varepsilon (t_0-s) \,\mathcal B\,{\mathcal {U}}^\varepsilon _0(s)x\Vert \,ds. \end{aligned}$$

Now the continuity of ${\mathcal {U}}_{j+1}^\varepsilon (\cdot )x$ at $t_0\in (\varepsilon ,+\infty )$ follows from the fact that $\Vert {\mathcal {U}}_{j}^\varepsilon (t') \,{\mathcal {B}}\,\mathcal U^\varepsilon _0(s)x\Vert$ is uniformly bounded for $t'$ and s belonging to bounded subsets of $(0,+\infty )$ [see (3.4)] together with the boundedness of ${\mathcal {B}}$ and the Lebesgue dominated convergence theorem. This ends the proof. $\square$

Lemma 3.2

For every $j\in {\mathbb {N}}$, the sequence $(\mathcal U^\varepsilon _j(t))_{\varepsilon >0}$ converges in $\mathcal L({\mathcal {X}})$ to ${\mathcal {U}}_j(t)$ as $\varepsilon$ goes to 0, uniformly on bounded and closed intervals of $(0,+\infty )$.

Proof

We proceed by induction on $j\in {\mathbb {N}}$. For $[a,b]\subset (0,+\infty )$, we have

$$\begin{aligned} \Vert {\mathcal {U}}^\varepsilon _0(t)-{\mathcal {U}}_0(t)\Vert =\Vert {\mathcal {U}}(t)\chi _{(0,\varepsilon )}(t)\Vert =0\ \hbox {for every} \ t\in [a,b]\ \hbox {and}\ 0<\varepsilon <a. \end{aligned}$$

This gives the result for $j=0$. Let $j\in {\mathbb {N}}$ and assume that the sequence $({\mathcal {U}}^\varepsilon _j(t))_{\varepsilon >0}$ converges in ${\mathcal {L}}(X)$ to ${\mathcal {U}}_j(t)$ as $\varepsilon$ goes to 0, uniformly on bounded and closed intervals of $(0,+\infty )$. We will show the result for $j+1$. Notice, for every $\varepsilon >0$ and $t\in (0,+\infty )$, we have

$$\begin{aligned}\begin{array}{ll}\Vert \mathcal U_{j+1}^\varepsilon (t)- \mathcal U_{j+1}(t)\Vert &{}\!\le \displaystyle \sup _{\begin{array}{c}x\in \mathcal X\\ \Vert x\Vert =1\end{array}}\Big \Vert \displaystyle \int _0^t \mathcal U_j^\varepsilon (t-s)\mathcal B \mathcal U_0^\varepsilon (s)x\,ds,\\ {} &{}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ -\displaystyle \int _0^t \mathcal U_j(t-s)\mathcal B\mathcal U_0(s)x\,ds\Big \Vert \\ &{}\!\le \displaystyle \sup _{\begin{array}{c}x\in \mathcal X\\ \Vert x\Vert =1\end{array}}\Big \Vert \displaystyle \int _0^t[\mathcal U_j^\varepsilon (t-s)-\mathcal U_j(t-s)]\mathcal B\mathcal U_0^\varepsilon (s)x\,ds\Big \Vert \\ {} &{}\,\,\, +\displaystyle \sup _{\begin{array}{c} x\in \mathcal X\\ \Vert x\Vert =1\end{array}} \Big \Vert \displaystyle \int _0^t\mathcal U_j(t-s)\mathcal B[\mathcal U_0^\varepsilon (s)-\mathcal U_0(s)]x\,ds\Big \Vert \\ &{}\!\le \displaystyle \int _0^t\Vert \mathcal U_j^\varepsilon (t-s)-\mathcal U_j(t-s)\Vert \,\Vert \mathcal B\Vert \,\Vert \mathcal U_0^\varepsilon (s)\Vert \,ds\\ {} &{}\,\,\, + \displaystyle \int _0^t\Vert \mathcal U_j(t-s)\Vert \,\Vert \mathcal B\Vert \,\Vert \mathcal U_0^\varepsilon (s)-\mathcal U_0(s)\Vert \,ds.\end{array}\end{aligned}$$

Now, taking into account (3.3), (3.4) and the boundedness of ${\mathcal {B}}$, the result follows from the Lebesgue dominated convergence theorem. $\square$

Set

$$\begin{aligned}\begin{array}{ll} \vartheta _j({\mathcal {T}},{\mathcal {B}})(\lambda ) =(\lambda -T)^{-1}[{\mathcal {B}}(\lambda -T)^{-1}]^j\ \hbox {for Re}\lambda >\omega ({\mathcal {U}})\ \hbox {and}\ j\ge 1. \end{array} \end{aligned}$$

For $\varepsilon >0$, set by induction,

$$\begin{aligned} {\mathcal {L}}({\mathcal {X}})\ni \vartheta _0^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda ):=\int _0^{+\infty }e^{-\lambda t}{\mathcal {U}}_0^\varepsilon (t)\,dt\ \hbox {for Re}\lambda >\omega ({\mathcal {U}}) \end{aligned}$$

(3.6)

in the strong sense, and

$$\begin{aligned} {\mathcal {L}}({\mathcal {X}})\ni \vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda ):=\vartheta _{j-1}^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda )\,{\mathcal {B}}\,\vartheta _{0}^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda ),\ \hbox {for Re}\lambda >\omega ({\mathcal {U}})\ \hbox {and}\ j\ge 1. \end{aligned}$$

For $\omega >\omega ({\mathcal {U}})$, set

$$\begin{aligned} {\mathcal {R}}_\omega := \{\lambda \in {\mathbb {C}}: \hbox {Re}\lambda \ge \omega \}. \end{aligned}$$

Remark 3.2

In the strong sense, we have [7]

$$\begin{aligned} \vartheta _j({\mathcal {T}},{\mathcal {B}})(\lambda )=\int _0^{+\infty }e^{-\lambda t}\ {\mathcal {U}}_j(t)\,dt\ \hbox {for Re}\lambda >\omega ({\mathcal {U}})\ \hbox {and}\ j\ge 0. \end{aligned}$$

(3.7)

Lemma 3.3

For every $j\ge 0$ and $\varepsilon >0$, we have, in the strong sense,

$$\begin{aligned} \int _0^{+\infty }e^{-\lambda t}\,{\mathcal {U}}_j^\varepsilon (t)\,dt=\vartheta ^\varepsilon _j({\mathcal {T}},{\mathcal {B}})(\lambda )\ \hbox {for Re}\lambda >\omega ({\mathcal {U}}) \end{aligned}$$

(3.8)

and consequently for $\omega >\omega ({\mathcal {U}})$, there exists $M_j\ge 1$ satisfying

$$\begin{aligned} \begin{array}{c}\Vert \vartheta ^\varepsilon _j(\mathcal T,\mathcal B)(\lambda )\Vert \le M_j\displaystyle \Big (\int _0^{+\infty }t^je^{(\omega -\omega ')\,t}\,dt\Big )\ \hbox { for Re}\lambda>\omega '>\omega .\end{array}\end{aligned}$$

(3.9)

Accordingly, by applying the Lebesgue dominated convergence theorem, we obtain

Corollary 3.1

For every $j\ge 1$, $\varepsilon >0$, $\omega >\omega ({\mathcal {U}})$ and $x\in X$, the function

$$\begin{aligned} R_{\omega }\ni \lambda \longmapsto \vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\cdot )x\in {\mathcal {X}} \end{aligned}$$

is holomorphic on the interior of $R_{\omega }.$

Proof of Lemma 3.3

We proceed by induction. For $j=0$, the result follows from (3.6). Let $j\ge 0$ and assume that (3.8) holds for j. For Re$\lambda >\omega (\mathcal U)$ and $x\in X$, we have by Fubini’s theorem

$$\begin{aligned} \begin{array}{ll} \displaystyle \int _0^{+\infty }e^{-\lambda t}\,{\mathcal {U}}_{j+1}^\varepsilon (t)x\,dt &{}=\displaystyle \int _0^{+\infty }dt\,e^{-\lambda t}\int _0^{+\infty }ds\,{\mathcal {U}}_{j}^\varepsilon (t-s) \,\mathcal B\,{\mathcal {U}}_0^\varepsilon (s)x \\ {} &{}= \displaystyle \int _0^{+\infty }ds\int _0^{+\infty }dt\,\,e^{-\lambda t}\,{\mathcal {U}}_{j}^\varepsilon (t-s) \,{\mathcal {B}}\,{\mathcal {U}}_0^\varepsilon (s)x \end{array} \end{aligned}$$

where ${\mathcal {U}}_{j}^\varepsilon (.)$ is identified to its natural extension (${\mathcal {U}}_{j}^\varepsilon (u)=0$ for $u<0$) on $\mathbb R$. Now using the change of unknowns $t\mapsto u:=t-s$, we get

$$\begin{aligned} \displaystyle \int _0^{+\infty }e^{-\lambda t}\,{\mathcal {U}}_{j+1}^\varepsilon (t)x\,dt= & {} \displaystyle \int _0^{+\infty }ds\,\,e^{-\lambda s}\int _0^{+\infty }du\,\,e^{-\lambda u}\,{\mathcal {U}}_{j}^\varepsilon (u) \,{\mathcal {B}}\,{\mathcal {U}}_0^\varepsilon (s)x\\= & {} \displaystyle \int _0^{+\infty }ds\,\,e^{-\lambda s}\,\vartheta ^\varepsilon _j({\mathcal {T}},{\mathcal {B}})(\lambda ) \,{\mathcal {B}}\,{\mathcal {U}}_0^\varepsilon (s)x. \end{aligned}$$

And, by linearity and continuity,

$$\begin{aligned} \displaystyle \int _0^{+\infty }e^{-\lambda t}\,{\mathcal {U}}_{j+1}^\varepsilon (t)x\,dt= & {} \vartheta ^\varepsilon _j({\mathcal {T}},{\mathcal {B}})(\lambda ) \,{\mathcal {B}}\,\vartheta ^\varepsilon _0({\mathcal {T}},{\mathcal {B}})(\lambda )x\\ {}= & {} \vartheta ^\varepsilon _{j+1}({\mathcal {T}},{\mathcal {B}})(\lambda )x. \end{aligned}$$

To achieve the proof, we use formula (3.4). $\square$

Lemma 3.4

Let $j\in {\mathbb {N}}$ and $\omega >\omega ({\mathcal {U}})$, then uniformly on ${\mathcal {R}}_\omega$, the sequence $(\vartheta _j^\varepsilon ({\mathcal {T}},\mathcal B)(\lambda ))_{\varepsilon >0}$ converges in ${\mathcal {L}}(\mathcal X)$ to $\vartheta _j({\mathcal {T}},{\mathcal {B}})(\lambda )$ as $\varepsilon$ goes to zero.

Proof

For every $\lambda \in {\mathcal {R}}_\omega$, from (3.8) and (3.7), we have

$$\begin{aligned} \Vert \vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda )-\vartheta _j({\mathcal {T}},{\mathcal {B}})(\lambda )\Vert= & {} \displaystyle \sup _{x\in {\mathcal {X}}\mathop \Vert x\Vert =1}\left\| \int _0^{+\infty } e^{-\lambda t}\,[{\mathcal {U}}_j^\varepsilon (t)x-{\mathcal {U}}_j(t)x]\,dt\right\| \\\le & {} \displaystyle \int _0^{+\infty } e^{-\omega t}\,\Vert {\mathcal {U}}_j^\varepsilon (t)-{\mathcal {U}}_j(t)\Vert \,dt. \end{aligned}$$

Let $\omega '\in (\omega ({\mathcal {U}}),\omega )$. From (3.3) and (3.4), there exists $M'\ge 0$ satisfying

$$\begin{aligned} \max (\Vert {\mathcal {U}}^\varepsilon _j(t)\Vert ,\Vert {\mathcal {U}}_j(t)\Vert )\le M'_j\,t^j\,e^{\omega ' t}\ \hbox {for every}\ t\ge 0 \end{aligned}$$

where $M'_j:=((M')^{j+1}\,\Vert {\mathcal {B}}\Vert ^j)/j!$. We have

$$\begin{aligned} 0\le e^{-\omega t}\,\Vert {\mathcal {U}}_j^\varepsilon (t)-{\mathcal {U}}_j(t)\Vert \le 2M'_j\,t^j\,e^{(\omega '-\omega ) t}\ \hbox {for every}\ t\ge 0 \end{aligned}$$

and the map: $(0,+\infty )\ni t\longmapsto 2M'_j\,t^j\,e^{(\omega '-\omega ) t}\in {\mathbb {R}}_+$, belongs to $L^1(0,+\infty )$. Now we achieve the proof by using Lemma 3.2 and applying the Lebesgue dominated convergence theorem. $\square$

The following lemma will be crucial in the proof of the two results below.

Lemma 3.5

For every $j\ge 1$, $\varepsilon >0$ and $\omega >\omega ({\mathcal {U}})$, the function

$$\begin{aligned} R_{\omega }\ni \lambda \longmapsto \vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda )\in {\mathcal {L}}({\mathcal {X}}) \end{aligned}$$

is holomorphic on the interior of $R_{\omega }$, and,

$$\begin{aligned} \displaystyle {d\vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})\over d\lambda }(\lambda )x=\Upsilon (\lambda )x:=-\displaystyle \int _0^{+\infty }te^{-\lambda t}\,{\mathcal {U}}^\varepsilon _j(t)x\,dt\, \hbox { for every}\ \lambda \in \hbox {int}({\mathcal {R}}_\omega )\ \hbox {and}\ x\in {\mathcal {X}}, \end{aligned}$$

where int(${\mathcal {R}}_\omega$) denotes the interior of $\mathcal R_\omega$.

Proof

For each $t>0$ and $z\in {\mathcal {R}}_\omega$, we put $\phi _t(z):=e^{-zt}$. Let $\lambda \in$ int(${\mathcal {R}}_{\omega '}$) where $\omega '>\omega$. Note that $\phi _t'(\lambda )=-te^{-\lambda t}$. For each $x\in {\mathcal {X}}$ and $h\in {\mathbb {C}}$ such that

$$\begin{aligned} \lambda +h\in \ \hbox {int}({\mathcal {R}}_{\omega '})\ \hbox { and }\ |h^{-1}(\phi _t(h)-1)+t|<1, \end{aligned}$$

(3.10)

we have according to Lemma 3.3,

$$\begin{aligned} \displaystyle&\left\| h^{-1}[\vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda +h)x-\vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda )x]+\int _0^{+\infty } te^{-\lambda t}{\mathcal {U}}^\varepsilon _j(t)x\,dt\right\| \\&\quad =\displaystyle \left\| \int _0^{+\infty } [h^{-1}(\phi _t(\lambda +h)-\phi _t(\lambda ))+te^{-\lambda t}]\, {\mathcal {U}}^\varepsilon _j(t)x\,dt \right\| \\&\quad \le \displaystyle \int _0^{+\infty }|h^{-1}(\phi _t(h)-1)+t|\,e^{-Re\lambda \, t}\,\Vert {\mathcal {U}}^\varepsilon _j(t)x\Vert \,dt. \end{aligned}$$

Hence,

$$\begin{aligned}\begin{array}{ll} &{}\Vert h^{-1}[\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda +h)-\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda )]-\Upsilon (\lambda )\Vert \\ {} &{}\quad =\displaystyle \sup _{x\in \mathcal X,\ \Vert x\Vert \le 1}\Big \Vert h^{-1}[\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda +h)x-\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda )x]+\displaystyle \int _0^{+\infty }\!\! te^{-\lambda t}\mathcal U^\varepsilon _j(t)x\,dt\Big \Vert \\ &{}\quad \le \displaystyle \int _0^{+\infty }|h^{-1}(\phi _t(h)-1)+t|\,e^{-Re\lambda \, t}\,\Vert \mathcal U^\varepsilon _j(t)\Vert \,dt. \end{array}\end{aligned}$$

Note that, by using (3.4), one has

$$\begin{aligned} 0\le |h^{-1}(\phi _t(h)-1)+t|\,e^{-Re\lambda \, t}\,\Vert {\mathcal {U}}^\varepsilon _j(t)\Vert \le M_j\,t^j\, e^{(\omega -Re\lambda )\, t} \le M_j\,t^j\,e^{(\omega -\omega ')\, t} \end{aligned}$$

for each h satisfying (3.10) and

$$\begin{aligned} \displaystyle \int _0^{+\infty }M_j\,t^j\, e^{(\omega -\omega ')\, t}\,dt<\infty . \end{aligned}$$

Therefore, by applying the Lebesgue dominated convergence theorem, we get

$$\begin{aligned}\begin{array}{ll} &{} \!\!\!\!\! \displaystyle \lim _{h\rightarrow 0}\Vert h^{-1}[\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda +h)-\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda )]-\Upsilon (\lambda )\Vert \\ {} &{}\,\,=\displaystyle \lim _{h\rightarrow 0}\Big \Vert h^{-1}[\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda +h)-\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda )] +\displaystyle \int _0^{+\infty }\!\!\!\! te^{-\lambda t}\mathcal U^\varepsilon _j(t)\,dt\Big \Vert \\ {} &{}\,\, \le \displaystyle \lim _{h\rightarrow 0}\int _0^{+\infty }|h^{-1}(\phi _t(h)-1)+t|\,e^{-Re\lambda \, t}\,\Vert \mathcal U^\varepsilon _j(t)\Vert \,dt\\ &{}\,\,=\displaystyle \int _0^{+\infty }\lim _{h\rightarrow 0}|h^{-1}(\phi _t(h)-1)+t|\,e^{-Re\lambda \, t}\,\Vert \mathcal U^\varepsilon _j(t)\Vert \,dt=0. \end{array}\end{aligned}$$

This ends the proof. $\square$

For each fixed $j\ge 0$ and $\varepsilon >0$, let us consider the following hypotheses

$$\begin{aligned} ({\mathcal {A}} 1(j,\varepsilon )) \left\{ \begin{array}{c}\hbox {There exists}\ \nu \ge \max (\omega ({\mathcal {U}}),0)\ \hbox {satisfying}\\ \displaystyle \ \sup _{\xi > \nu } \int _{-\infty }^{+\infty } \Vert \vartheta ^\varepsilon _j({\mathcal {T}},\mathcal B)(\xi +i\tau )\Vert \,d\tau <\infty \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} ({\mathcal {A}} 1(j))\ \hbox {The hypothesis}\ ({\mathcal {A}} 1(j,\varepsilon ))\ \hbox {hols for every (small)}\ \varepsilon >0. \end{aligned}$$

Now, we are in a position to state the following

Theorem 3.1

Let $j\ge 0$ and assume that $({\mathcal {A}} 1(j))$ holds true. Then, ${\mathcal {U}}_{j}(\cdot )$ is norm continuous on $(0,+\infty )$.

Proposition 3.1

Let $j\ge 0$ and $\varepsilon >0$. If the hypothesis $({\mathcal {A}} 1(j,\varepsilon ))$ holds then, ${\mathcal {U}}^\varepsilon _{j}(\cdot )$ is norm continuous on $(\varepsilon ,+\infty )$. Furthermore, in the strong sense and in ${\mathcal {L}}({\mathcal {X}})$, we have for $\xi >\nu$,

$$\begin{aligned} {\mathcal {U}}^\varepsilon _j(t)={ 1\over 2 \pi i}\displaystyle \int _{\xi -i\infty }^{\xi +i\infty } e^{\lambda t}\,\vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda )\ d\lambda ,t> \varepsilon . \end{aligned}$$

(3.11)

To prove this result, we need to recall some relevant definitions and auxiliary results.

Definition 3.1

(See [7, Definition 6.4.1].) Let ${\mathcal {Y}}$ be a Banach space and let $\vartheta (\cdot )$ denote an ${\mathcal {Y}}$-valued function defined on the half-plane $\{\lambda \in {\mathbb {C}}:\ \hbox {Re}\lambda >\alpha \}$ where $\alpha \in {\mathbb {R}}$. We say that $\vartheta (\cdot )$ belongs to the class $H_p(\alpha ,{\mathcal {Y}})$ if the following conditions are satisfied:

(a):

$\vartheta (\cdot )$ is a function on complex numbers to ${\mathcal {Y}}$, which is holomorphic for Re$\lambda >\alpha$;

(b):

$\sup _{\gamma >\alpha }\{\int _{-\infty }^{+\infty }\Vert \vartheta (\gamma +i\tau )\Vert ^p\,d\tau \}^{1\over p}<+\infty$;

(c):

$\lim _{\gamma \rightarrow \alpha } \vartheta (\gamma +i\ell )=\vartheta (\alpha +i\ell )$ exists for almost all values of $\ell$ and

$$\begin{aligned} \int _{-\infty }^{+\infty }\Vert \vartheta (\alpha +i\tau )\Vert ^p\,d\tau <+\infty . \end{aligned}$$

The following result shows that every function in $H_p(\alpha , {\mathcal {Y}})$ may be represented by a generalized Laplace integral.

Proposition 3.2

(See [7, p. 230].) Let $\vartheta (\cdot )\in H_p(\alpha , {\mathcal {Y}})$ where $\alpha \ge 0$. Let $\gamma >\alpha$ and $\beta q>1$ where ${1\over p}+{1\over q}=1$. Then

$$\begin{aligned} \theta _\beta (t)=\displaystyle {1\over 2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty } e^{\lambda t}\lambda ^{-\beta }\vartheta (\lambda )\,d\lambda \end{aligned}$$

defines a continuous function on $(0,+\infty )$ to ${\mathcal {Y}}$ and

$$\begin{aligned} \vartheta (\lambda )=\lambda ^{\beta } \displaystyle \int _0^{+\infty }e^{-\lambda t}\theta _\beta (t)\,dt \end{aligned}$$

the integral being absolutely convergent for Re$\lambda >\alpha$.

For $p=1$ (see [7, p. 230]), we may take $\beta =0$, obtaining

$$\begin{aligned} \vartheta (\lambda )= \displaystyle \int _0^{+\infty }e^{-\lambda t}\theta _0(t)\,dt. \end{aligned}$$

Let $\vartheta (\cdot )$ be an ${\mathcal {Y}}$-valued function defined on a half-plane $P_\alpha :=\{\lambda \in {\mathbb {C}}{:}$ Re$\lambda >\alpha \}$ where ${\mathcal {Y}}$ is a complex Banach space and $\alpha \in {\mathbb {R}}$. Assume that the function $\vartheta (\cdot )$ satisfies Conditions (a) and (b) (except may be Condition (c)) of Definition 3.1, say

$$\begin{aligned} \vartheta (\cdot )\in \Xi _p(\alpha ,{\mathcal {Y}}). \end{aligned}$$

(3.12)

Then, naturally, for each $\alpha '>\alpha$, this function satisfies

$(a')$ :

$\vartheta (\cdot )$ is a function on complex numbers to ${\mathcal {Y}}$, which is holomorphic for Re$\lambda >\alpha '$;

$(b')$ :

$\sup _{\gamma>\alpha '}\{\int _{-\infty }^{+\infty }\Vert \vartheta (\gamma +i\tau )\Vert ^p\,d\tau \}^{1\over p}<\sup _{\gamma >\alpha }\{\int _{-\infty }^{+\infty }\Vert \vartheta (\gamma +i\tau )\Vert ^p\,d\tau \}^{1\over p}<+\infty$;

$(c')$ :

The holomorphy of $\vartheta (\cdot )$ on $P_\alpha$ implies its continuity on $P_\alpha$; thus,

$$\begin{aligned} \lim _{\gamma \rightarrow \alpha '} \vartheta (\gamma +i\ell )=\vartheta (\alpha '+i\ell )\ \hbox {exists for all values of}\ \ell \in {\mathbb {R}}. \end{aligned}$$

Furthermore,

$$\begin{aligned} \displaystyle \int _{-\infty }^{+\infty }\Vert \vartheta (\alpha '+i\tau )\Vert ^p\,d\tau\le & {} \displaystyle \sup _{\gamma>\alpha }\left\{ \int _{-\infty }^{+\infty }\Vert \vartheta (\gamma +i\tau )\Vert ^p\,d\tau \right. \\= & {} \left. \left( \sup _{\gamma >\alpha }\left\{ \displaystyle \int _{-\infty }^{+\infty }\Vert \vartheta (\gamma +i\tau )\Vert ^p\,d\tau \right\} ^{1/p}\right) ^p <+\infty \right. . \end{aligned}$$

This means that $\vartheta (\cdot )$ belongs to $H_p(\alpha ',{\mathcal {Y}})$ for every $\alpha '>\alpha$, and consequently, we deduce the following

Corollary 3.2

Let $\vartheta (\cdot )\in \Xi _p(\alpha , {\mathcal {Y}})$ where $\alpha \ge 0$. Let $\gamma >\alpha$ and $\beta q>1$ where ${1\over p}+{1\over q}=1$. Then, the conclusion of Proposition 3.2 holds true.

Proof of Proposition 3.1

By using the hypothesis $({\mathcal {A}}1(j,\varepsilon ))$ and Corollary 3.1 (resp. Lemma 3.5), we infer from (3.12) that $\vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\cdot )x$ (resp $\vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\cdot )$) belongs to $\Xi _1(\nu , {\mathcal {X}})$ for each $x\in H$ (resp. $\Xi _1(\nu , {\mathcal {L}}({\mathcal {X}}))$). This implies, according to Corollary 3.2, that the continuous function

$$\begin{aligned} \theta _j^\varepsilon (t):=\displaystyle {1\over 2\pi i}\int _{\gamma -i\infty }^{\gamma +i\infty } e^{\lambda t}\,\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda )\,d\lambda ,\ \gamma>\nu ,\ t>0, \end{aligned}$$

satisfies,

$$\begin{aligned} \displaystyle \int _0^{+\infty }e^{-\lambda t}\theta _j^\varepsilon (t)\,dt=\vartheta _j^\varepsilon (\mathcal T,\mathcal B)(\lambda ), \end{aligned}$$

where the integrals are considered in the strong sense (resp. ${\mathcal {L}}({\mathcal {X}})$). Keeping in mind that (Lemma 3.3) in the strong sense,

$$\begin{aligned} \int _0^{+\infty }e^{-\lambda t}\,{\mathcal {U}}_j^\varepsilon (t)\,dt=\vartheta ^\varepsilon _j({\mathcal {T}},{\mathcal {B}})(\lambda )\ \hbox {for Re}\lambda >\omega ({\mathcal {U}}), \end{aligned}$$

the uniqueness of the Laplace transform yields by continuity in the strong sense (see Lemma 3.1),

$$\begin{aligned} {\mathcal {U}}_j^\varepsilon (t)=\theta _j^\varepsilon (t)\ \hbox {for every}\ t>\varepsilon . \end{aligned}$$

Therefore, (3.11) follows in the strong sense and in ${\mathcal {L}}({\mathcal {X}})$; moreover, ${\mathcal {U}}_ {j}^\varepsilon (\cdot )$ is norm continuous on $(\varepsilon ,+\infty )$. This ends the proof. $\square$

Proof of Theorem 3.1

The result follows from Proposition 3.1 on the basis of Lemma 3.2. $\square$

As a consequence of Theorem 3.1 we obtain the following

Corollary 3.3

Assume that there exists $j\ge 0$ such that the hypothesis $({\mathcal {A}} 1(j))$ holds. Then, the semigroups $({\mathcal {U}}(t))_{t\ge 0}$ and $({\mathcal {V}}(t))_{t\ge 0}$ have the same critical type $\omega _{crit}$.

Remark 3.3

For each $j\ge 0$ and $\varepsilon >0$, let consider the following conditions

$$\begin{aligned} ({\mathcal {A}} 1'(j,\varepsilon )) \left\{ \begin{array}{l}\hbox {There exists}\ \nu \ge \max (\omega ({\mathcal {U}}),0)\ \hbox {satisfying}\\ \displaystyle \lim _{a\rightarrow +\infty } \ \sup _{\xi> \nu } \int _{|\tau |>a} \Vert \vartheta ^\varepsilon _j({\mathcal {T}},\mathcal B)(\xi +i\tau )\Vert \,d\tau =0 \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} ({\mathcal {A}} 1'(j))\ \hbox {The hypothesis}\ ({\mathcal {A}} 1'(j,\varepsilon ))\ \hbox {hols for every (small)}\ \varepsilon >0. \end{aligned}$$

Note that for every $j\ge 0$ and $\varepsilon >0$, Condition $({\mathcal {A}}1'(j,\varepsilon ))$ implies Condition $({\mathcal {A}}1(j,\varepsilon ))$. Therefore, for every $j\ge 0$, Condition $({\mathcal {A}}1'(j))$ implies condition $({\mathcal {A}} 1(j))$.

Let us consider the following hypothesis.

$$\begin{aligned} ({\mathcal {A}}2) \left\{ \begin{array}{l}\hbox {For every}\ \varepsilon>0,\ \hbox {there exist}\ \nu>\max (\omega ({\mathcal {U}}),0) \ \hbox {and }\ C_\varepsilon >0 \ \hbox {satisfying }\\ \\ |Im \lambda |\,\Vert {\mathcal {B}}\vartheta ^\varepsilon _0(\mathcal T,{\mathcal {B}})(\lambda ){\mathcal {B}}\Vert \le C_\varepsilon \ \hbox {for every}\ \lambda \in {\mathcal {R}}_\nu .\end{array} \right. \end{aligned}$$

Now we state another consequence of Theorem 3.1. This result is very useful in applications.

Corollary 3.4

Assume that $({\mathcal {A}} 2)$ holds true. Then for each $j\ge 4$, the operator ${\mathcal {U}}_j(t)$ is norm continuous for each $t>0$ and therefore, the semigroups $({\mathcal {U}}(t))_{t\ge 0}$ and $(\mathcal V(t))_{t\ge 0}$ have the same critical type $\omega _{crit}$. Furthermore (3.11) holds in the strong sense and in ${\mathcal {L}}({\mathcal {X}})$, for each $\varepsilon >0$, $\xi >\nu$ and $j\ge 4$.

Proof

From $({\mathcal {A}}2)$, we have for $\varepsilon >0$, $j\ge 4$, Re$\lambda \ge \nu$ and Im$\lambda \ne 0$,

$$\begin{aligned} \Vert \vartheta _{j}^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda )\Vert\le & {} \Vert \vartheta ^\varepsilon _0({\mathcal {T}},{\mathcal {B}})(\lambda )[{\mathcal {B}}\vartheta ^\varepsilon _0({\mathcal {T}},{\mathcal {B}})(\lambda )]^{4}\Vert \, \Vert [{\mathcal {B}}\vartheta ^\varepsilon _0({\mathcal {T}},\mathcal B)(\lambda )]^{j-4}\Vert \\\le & {} \,\Vert {\mathcal {B}} \vartheta ^\varepsilon _0({\mathcal {T}},{\mathcal {B}})(\lambda )\mathcal B\Vert ^2\,\Vert \vartheta ^\varepsilon _0({\mathcal {T}},\mathcal B)(\lambda )\Vert ^3\, \Vert [{\mathcal {B}}\vartheta ^\varepsilon _0(\mathcal T,{\mathcal {B}})(\lambda )]^{j-4}\Vert \\\le & {} \displaystyle {C_\varepsilon ^2\over {Im^2\lambda }}\,\Vert \mathcal B\Vert ^{j-4}\,\Vert \vartheta ^\varepsilon _0({\mathcal {T}},\mathcal B)(\lambda )\Vert ^{j-1}. \end{aligned}$$

Since the operator ${\mathcal {B}}$ is assumed to be bounded, then by the fact that $\Vert \vartheta ^\varepsilon _0({\mathcal {T}},{\mathcal {B}})(\lambda )\Vert$ is uniformly bounded on ${\mathcal {R}}_\nu$ (apply (3.9) with $\omega '=\nu$ and ${\omega ={\omega ({\mathcal {U}})+\nu \over 2}}$), there exists ${\tilde{C}}_\varepsilon >0$ verifying

$$\begin{aligned} \begin{array}{ll} \displaystyle \lim _{a\rightarrow +\infty }\sup _{\xi> \nu }\int _{|\tau |>a} \Vert \vartheta _{j}^\varepsilon ({\mathcal {T}},\mathcal B)(\xi +i\tau )\Vert \,d\tau&\le {\tilde{C}}_\varepsilon \displaystyle \lim _{a\rightarrow +\infty }\displaystyle \int _{|\tau |>a} {d\tau \over \tau ^2}=0. \end{array} \end{aligned}$$

Now, the result follows from the remark above and Theorem 3.1. $\square$

3.2 Compactness of ${\mathcal {U}}_j(\cdot )$

For $j\ge 0$, let us consider the following hypothesis

$$\begin{aligned} ({\mathcal {A}}3(j)) \left\{ \begin{array}{l}\hbox {There exists}\ \nu>\max (\omega ({\mathcal {U}}),0)\ \hbox {such that}\\ \hbox {for every (small)}\ \varepsilon >0, \vartheta ^\varepsilon _j({\mathcal {T}},{\mathcal {B}})(\lambda )\ \hbox {is compact for all}\ \lambda \in {\mathcal {R}}_\nu . \end{array} \right. \end{aligned}$$

The second main result of this section is the following

Theorem 3.2

Let $j\ge 0$ and assume that $({\mathcal {A}} 1(j))$ and $({\mathcal {A}} 3(j))$ hold true. Then, ${\mathcal {U}}_{j}(t)$ is compact for every $t> 0$.

Proof

From the hypothesis $({\mathcal {A}} 1(j))$ and according to Proposition 3.1 we get, in the strong sense and in ${\mathcal {L}}({\mathcal {X}})$, for each $\varepsilon >0$ and $\xi >\nu$,

$$\begin{aligned} {\mathcal {U}}^\varepsilon _j(t)={ 1\over 2 \pi i}\displaystyle \int _{\xi -i\infty }^{\xi +i\infty } \exp ({\lambda t})\,\vartheta _j^\varepsilon ({\mathcal {T}},{\mathcal {B}})(\lambda )\ d\lambda ,\quad t> 0. \end{aligned}$$

Now the use of the convex compactness property [24] gives that $\mathcal U^\varepsilon _j(t)$ is compact for every $t> 0$ and $\varepsilon >0$. Now, the result follows from Lemma 3.2. $\square$

Consequently, we get

Corollary 3.5

Assume that there exists $j\ge 0$ such that the hypotheses $({\mathcal {A}} 1(j))$ and $({\mathcal {A}} 3(j))$ hold. Then, the semigroups $({\mathcal {U}}(t))_{t\ge 0}$ and $({\mathcal {V}}(t))_{t\ge 0}$ have the same essential type $\omega _{ess}$.

Now we state another consequence of Theorem 3.2. This result is very useful in applications.

Corollary 3.6

Assume that the hypothesis $({\mathcal {A}} 2)$ holds and there exists $j\ge 4$ such that the hypothesis $({\mathcal {A}} 3(j))$ holds. Then the operator ${\mathcal {U}}_j(t)$ is compact for each $t>0$ and therefore, the semigroups $({\mathcal {U}}(t))_{t\ge 0}$ and $(\mathcal V(t))_{t\ge 0}$ have the same essential type $\omega _{ess}$.

4 Application to $L^1$-neutron transport theory

In this section, we consider the following concrete initial value problem governing the time evolution of the distribution of neutrons in nuclear reactor, namely the neutron transport equation (see [13, 14] and the references therein)

(4.1)

where $(x,v)\in \Omega \times V$ with $\Omega$ and V are smooth open subsets of ${\mathbb {R}}^N$ $(N\ge 1)$, $\Omega$ and V denote respectively the space of positions and the space of velocities, dx and $d\mu$ denote respectively the Lebesgue measure on ${\mathbb {R}}^N$ and a positive Radon measure on ${\mathbb {R}}^N$ with support V. The function $\psi (x,v)$ represents the number (or probability) density of particles having the position x and the velocity v. The functions $\sigma (\cdot )$ and $\kappa (\cdot ,\cdot ,\cdot )$ are called, respectively, the collision frequency and the scattering kernel, and,

$$\begin{aligned} \Gamma ^-=\{(x,v)\in \partial \Omega \times V:\ v \ \hbox {is incomming at}\ x\in \partial \Omega \}. \end{aligned}$$

The operators A, T and K the integral part of A, are called respectively the transport operator, the streaming operator and the collision operator.

Let us consider the Banach space

$$\begin{aligned} X=L^1(\Omega \times V , dx\otimes d\mu (v)). \end{aligned}$$

We assume that

$$\begin{aligned} \sigma (\cdot )\in L^\infty (V),\ K\in {\mathcal {L}}(X)\ \hbox {and}\ \Omega \ \hbox {is bounded and convex}. \end{aligned}$$

From [25], the unbounded operator

$$\begin{aligned} \left\{ \begin{array}{l}T\psi (x,v)=\displaystyle -v{\partial \psi \over \partial x}(x,v)-\sigma (v)\,\psi (x,v),\ \ \psi \in D(T)\\ D(T)=\{\psi \in X:\ \displaystyle {\partial \psi \over \partial x}\in X\ \hbox {and}\ \psi _{|_{\Gamma ^-}}=0 \} \end{array}\right. \end{aligned}$$

(4.2)

generates the following explicit positive $C_0$-semigroup

$$\begin{aligned} U(t)\psi (x,v)=\left\{ \begin{array}{ll}e^{-\sigma (v)t}\psi (x-vt,v)\ &{}\hbox { if}\ t<\tau (x,v)\\ \\ 0&{}\ \hbox { otherwise},\end{array}\right. \end{aligned}$$

where $\tau (x,v)=\inf \{t>0:\ x-tv\notin \Omega \}$. The type of $(U(t))_{t\ge 0}$ is given [16, Formulae (2.4), p. 10] by

$$\begin{aligned}\omega (T)=\left\{ \begin{array}{ll}-\infty &{}\,\, \hbox {if}\ 0\notin \overline{V}\\ \\ -\!\!\!\!\!\!\!\!\!\!\displaystyle \lim _{\;\;\;\;\;\;\;\;V\ni v\rightarrow 0}\!\!\!\!\!\!\!\hbox {inf}\,\; \sigma (v)&{}\,\,\hbox {if}\ 0\in \overline{V}.\end{array}\right. \end{aligned}$$

Let us write the evolution problem (4.1) as an abstract Cauchy problem:

$$\begin{aligned} \begin{array}{ll} \displaystyle {\frac{\partial {\varphi }}{\partial {t}}} =(T+K)\varphi , \;\;\; \varphi (0)=\varphi _0\in X, \end{array} \end{aligned}$$

where the streaming operator T is the closed unbounded operator with dense domain defined by (4.2), and the collision operator K is the bounded integral operator having the form

$$\begin{aligned} \begin{array}{ll} X\ni \psi \mapsto K\varphi :=\displaystyle \int _{\mathbb V}\kappa (x,v,v')\;\psi (x,v')\,d\mu (v')\in X. \end{array} \end{aligned}$$

(4.3)

Note that from the classical perturbation theory, the perturbed (transport) operator $A:=T+K$ generates also a strongly continuous semigroup $(V(t))_{t\ge 0}$ in X given by the Dyson–Phillips expansion

$$\begin{aligned} V(t)=\sum _{j=0}^{m-1} U_j(t)+ R_m(t), \end{aligned}$$

(4.4)

where

$$\begin{aligned} U_0(t)= U(t),\ \ U_j(t)=\int _0^t U(t-s) K U_{j-1}(s)\,ds ,\ \ (j\ge 1) \end{aligned}$$

and

$$\begin{aligned} R_m(t)=\sum _{j= m} ^{+\infty }U_j(t),\ (m\ge 1). \end{aligned}$$

Following [13, 14], we adopt the following definition

Definition 4.1

A collision operator $K\in {\mathcal {L}}(X)$ is said to be regular if it belongs to the closure in the operator norm topology of the class of collision operators with kernels in the form

$$\begin{aligned} \kappa (x,v,v')={\displaystyle \sum _{i\in I}}\alpha _i(x)f_i(v)g_i(v'), \end{aligned}$$

with $\alpha _i(.)\in L^\infty (\Omega )$, $f_i(.)\in L^1(V,d\mu )$, $g_i(.)\in L^\infty (V,d\mu )$, $i\in I$, where I is finite.

For every $(x,s)\in \Omega \times (0,+\infty )$, we define the map

$$\begin{aligned} \begin{array}{ll} f_{x,s}:&{}V\longrightarrow \Omega \\ &{}v\longmapsto (x-sv)\,\chi _{[0,\tau (x,v)]}(s). \end{array} \end{aligned}$$

Let us assume as in [10] that the Radon measure $d\mu$ on ${\mathbb {R}}^N$ satisfies the following geometrical properties,

$$\begin{aligned} \displaystyle \int _{\alpha _1\le |v|\le \alpha _2}d\mu (v)\int _0^{\alpha _3}\chi _A(tv)\, dt\rightarrow 0\ \hbox {as}\ |A|\rightarrow 0 \end{aligned}$$

(4.5)

for every $\alpha _1<\alpha _2<+\infty$ and $\alpha _3<+\infty$, where |A| is the Lebesgue measure of A and $\chi _A$ denotes the characteristic function of A.

Let us assume that there exists a positive measure $d\nu$ on $\mathbb R^N$ satisfying

$$\begin{aligned} \left\{ \begin{array}{l} f_{x,s}(d\mu )=(1/s^N)\,d\nu \ \hbox { for every}\ x\in \Omega \ \hbox {and}\ s\in (0,+\infty ),\,\hbox { and}\\ \hbox {the canonical injection}\ j:L^1(\Omega ,dx)\longrightarrow L^1(\Omega ,d\nu ) \ \hbox {is continuous}. \end{array}\right. \end{aligned}$$

(4.6)

Here, $f_{x,s}(d\mu )$ denotes the image of the measure $\mu$ on ${\mathbb {R}}^N$ by the map $f_{x,s}$.

Remark 4.1

The set of Radon measures $\mu$ on ${\mathbb {R}}^N$ satisfying condition (4.6) is non-empty, in fact, it contains the usual Lebesgue measure dv on ${\mathbb {R}}^N$.

For any Borelian subset $V_0\subset V$, we consider the following geometrical property.

Definition 4.2

Let $n_0\in {\mathbb {N}}^*$. A Borelian subset $V_0$ of V is said to be $(\Omega -\Omega )$-finite of order $n_0$, if it has a finite $\mu -$measure and for each $x\in (\Omega -\Omega ):=\{x-y : x, y\in \Omega \}$, the set

$$\begin{aligned} \Delta _x(V_0):=\{t\in (0,+\infty )\ \hbox {such that}\ (x/t)\in V_0\} \end{aligned}$$

is a union of at most $n_0$ intervals.

Let us assume the following geometrical assumption on the velocity space.

$$\begin{aligned} \left\{ \begin{array}{l} \hbox {The velocity space}\ V\ \hbox {can be written} \\ \hbox {as a partition of}\ (\Omega -\Omega )\hbox {-finite subsets of order}\ n_0. \end{array}\right. \end{aligned}$$

(4.7)

Remark 4.2

If $d\mu$ denotes the Lebesgue measure on ${\mathbb {R}}^N$, then both of the subsets $B(a,r):=\{v-a\in {\mathbb {R}}^N\ \hbox {such that}\ \Vert v\Vert \le r\}$, $r>0$, $a\in {\mathbb {R}}^N$, and $C_{r_1,r_2}:=\{v\in {\mathbb {R}}^N\ \hbox {such that}\ r_1\le \Vert v\Vert \le r_2\}$, $0\le r_1\le \Vert v\Vert \le r_2$, are $(\Omega -\Omega )$-finite subsets of order one [and hence each of them satisfies the geometrical assumption (4.7)]. Therefore, since $\displaystyle \cup _{n\in {\mathbb {N}}}C_{n,n+1}={\mathbb {R}}^N$, this velocity space satisfies also the geometrical assumption (4.7).

The following theorem is the main result of this section.

Theorem 4.1

Assume that $\Omega$ is bounded, $0\le \sigma (\cdot )$ and the Radon measure $d\mu$ satisfies conditions (4.5) and (4.6). If the collision operator is regular and the velocity space satisfies the assumption (4.7), then for every $m\ge 4$, the operator $U_m(t)$ is compact for each $t\ge 0$ and therefore, the semigroups $(U(t))_{t\ge 0}$ and $(V(t))_{t\ge 0}$ have the same critical type $\omega _{crit}$ and the same essential type $\omega _{ess}$.

Remark 4.3

The result of Theorem 4.1 completes the results of [14, Theorem 10 and Corollary 11] and that of [12, Theorem 2.1]. Furthermore, the result of Theorem 4.1 can be generalized directly to $L^p$-spaces, $1\le p<+\infty$, and thus our result improves Sbihi and Mokhtar-Kharroubi’s results.

For every $\varepsilon >0$, small, we consider the operator [see (3.6)]

$$\begin{aligned} \vartheta _0^\varepsilon (T, K)(\lambda ),\ \hbox {Re}\lambda >\omega (U). \end{aligned}$$

To prove Theorem 4.1, we need the following

Proposition 4.1

[15, Theorem 4.4] Assume that $\Omega$ is bounded and the collision operator K is regular. If the Radon measure $d\mu$ satisfies (4.5), then

$$\begin{aligned} K\vartheta _0^\varepsilon (T, K)(\lambda )K\ \hbox {is weakly compact on}\ X. \end{aligned}$$

Remark 4.4

Let K be a collision operator given by (4.3). It can be written in the form

$$\begin{aligned} K=[Re K^+-Re K^-]+i[Im K^+-Im K^-], \end{aligned}$$

where $Re K^\pm$ and $Im K^\pm$ denote, respectively, the collision operators with kernels

$$\begin{aligned}Re\,\kappa ^\pm (x,v,v')=\max \{\pm Re\,\kappa (x,v,v'),0\}\end{aligned}$$

and

$$\begin{aligned}Im\,\kappa ^\pm (x,v,v')=\max \{\pm Im\,\kappa (x,v,v'),0\}.\end{aligned}$$

Therefore, since the collision operator K is assumed to be regular and since $V(t)-U(t)$ depends linearly and continuously on K, we can assume that K is positive and regular. Now, since the semigroup $(U^H(t))_{t\ge 0}$ is positive, similar reasoning as in [14, p. 1240] using domination arguments, by linearity and according to Condition (4.7), we can assume that K is a one rank collision operator with kernel in the form

$$\begin{aligned}\kappa (x,v,v')=\chi _{_{V_0}}(v),\end{aligned}$$

where $V_0$ is a Borelian $(\Omega -\Omega )$-finite subset of V having the order $n_0$, i.e., $K=K_0$, where

$$\begin{aligned} K_0:X_p\ni \psi \longrightarrow K\psi (x,v)=\chi _{_{V_0}}(v)\int _V\psi (x,v')\,d\mu (v')\in X_p. \end{aligned}$$

(4.8)

For every $\omega >\omega (U)$, we consider the half-plane of $\mathbb C$:

$$\begin{aligned} {\mathcal {R}}_\omega :=\{\lambda \in {\mathbb {C}}:\ \hbox {Re}\lambda \ge \omega \}. \end{aligned}$$

We are going to prove the following auxiliary result.

Proposition 4.2

Assume that $\sigma (\cdot )={\underline{\sigma }}\in {\mathbb {R}}$ and let $\varepsilon >0$ and $V_0$ be a Borelian $(\Omega -\Omega )$-finite subset of V having the order $n_0$. If the Radon measure $d\mu$ satisfies Condition (4.6), then for every $\omega > \omega (U)$, there exists $C_\varepsilon >0$ such that

$$\begin{aligned} |\lambda +{\underline{\sigma }}|\,\Vert K_0\vartheta _0^\varepsilon (T, K)(\lambda ) K_0\Vert \le C_\varepsilon \ \hbox {for every}\ \lambda \in {\mathcal {R}}_\omega , \end{aligned}$$

and consequently,

$$\begin{aligned}|Im\,\lambda |\,\Vert K_0\vartheta _0^\varepsilon (T, K)(\lambda ) K_0\Vert \le C_\varepsilon \ \hbox {for every}\ \lambda \in \mathcal R_\omega . \end{aligned}$$

Before proving this result, we need the following

Lemma 4.1

Assume that $\sigma (\cdot )={\underline{\sigma }}\in {\mathbb {R}}$ and let $\varepsilon >0$ and $V_0$ be a Borelian $(\Omega -\Omega )$-finite subset of V having the order $n_0$. Then, for every $\omega > \omega (U)$, there exists $C_{\varepsilon ,n_0,1}>0$ such that

$$\begin{aligned}|\lambda +\underline{\sigma }| \left| \int _\varepsilon ^{+\infty } e^{-(\lambda +\underline{\sigma })s}\,\chi _{_{V_0}}\left( {x-x'\over s}\right) {ds\over s^N}\right| \le C_{\varepsilon ,n_0,1}\end{aligned}$$

uniformly on $x, x'\in \Omega$ and $\lambda \in \mathcal R_\omega$.

Proof

Since $V_0$ is a Borelian $(\Omega -\Omega )$-finite subset of V having the order $n_0$, for every $x,x'\in \Omega$, there exists $m_0(x-x')<n_0$ satisfying

$$\begin{aligned}\int _\varepsilon ^{+\infty }\! e^{-(\lambda +\underline{\sigma })s}\,\chi _{_{V_0}}\left( {x-x'\over s}\right) {ds\over s^N}=\sum _{j=1}^{m_0(x-x')}\int _{a_j(x-x')}^{b_j(x-x')} e^{-(\lambda +\underline{\sigma })s}\,\chi _{_{V_0}}\left( {x-x'\over s}\right) {ds\over s^N},\end{aligned}$$

where the $(a_j,b_j)$ are bounded intervals included in $(\varepsilon ,+\infty )$. Simple integrations by part to the integrals

$$\begin{aligned} \int _{a_j(x-x')}^{b_j(x-x')}\,e^{-(\lambda +{\underline{\sigma }})s}\,\chi _{_{V_0}}\left( {x-x'\over s}\right) {ds\over s^N}, \ 1\le j\le m_0, \end{aligned}$$

yields

$$\begin{aligned}\begin{array}{ll}\displaystyle \int _\varepsilon ^{+\infty }\! e^{-(\lambda +\underline{\sigma })s}\,\chi _{_{V_0}} \left( {x-x'\over s}\right) {ds\over s^N}&{}=\displaystyle \sum _{j=1}^{m_0(x-x')}\left[ -{1\over \lambda +\underline{\sigma }}\,e^{-(\lambda +\underline{\sigma })s}\, {1\over s^{N}}\right] _{a_j(x-x')}^{b_j(x-x')}\\ &{}\ \ \ -\displaystyle \sum _{j=1}^{m_0(x-x')} \!\int _{a_j(x-x')}^{b_j(x-x')}\! {1\over \lambda +\underline{\sigma }}\,e^{-(\lambda +\underline{\sigma })s}\, {N ds\over s^{N+1}}. \end{array}\end{aligned}$$

Consequently, by passing to the module, we get

$$\begin{aligned}\begin{array}{ll}|\lambda +\underline{\sigma }|\displaystyle \left| \int _\varepsilon ^{+\infty } e^{-(\lambda +\underline{\sigma })s}\,\chi _{_{V_0}}\left( {x-x'\over s}\right) {ds\over s^N}\right| &{}\le \displaystyle {2m_0(x-x')\over \varepsilon } + \int _\varepsilon ^{+\infty } {N ds\over s^{N+1}}\\ &{}\le \displaystyle {2n_0+1\over \varepsilon ^N} :=C_{\varepsilon ,n_0,1}, \end{array}\end{aligned}$$

uniformly on $x, x'\in \Omega$ and $\lambda \in \mathcal R_\omega .$ This ends the proof. $\square$

Proof of Proposition 4.2

Note that according to [16, p. 12], the operator $K_0\vartheta _0^\varepsilon (T, K_0)(\lambda )K_0$ is an integral operator from $L^1(\Omega \times V,d\nu (x)\otimes d\mu (v))$ onto X, whose kernel is

$$\begin{aligned}\displaystyle \chi _{_{V_0}}(v)\,\chi _{\big \{|x-x'|\le \tilde{s}(x,{x-x'\over |x-x'|})\big \}}\int _\varepsilon ^{+\infty } e^{-(\lambda +\underline{\sigma })s}\,\chi _{_{V_0}}\left( {x-x'\over s}\right) {ds\over s^N},\end{aligned}$$

where $\tilde{s}(x,v)=\inf \{s>0 : x-sv\notin \Omega \}$. Therefore, the result follows from Lemma 4.1 on the basis of Assumption (4.6).$\square$

Finally, we are ready to give the

Proof of Theorem4.1

Since $0\le \sigma (\cdot )$, the semigroup $(U(t))_{t\ge 0}$ is positive and the collision operator K can be chosen to be the positive collision operator $K_0$ given by (4.8) (see, Remark 4.4). By using comparison arguments, we need only to prove the result for $\sigma (\cdot )=0$. Now, the result follows from Propositions 4.1 and 4.2 by applying Corollary 3.6 of Theorem 3.2. $\square$