#### Abstract

This paper deals with the mathematical analysis of a retarded partial integrodifferential equation that belongs to the class of thermostatted kinetic equations with time delay. Specifically, the paper is devoted to the proof of the existence and uniqueness of mild solutions of the related Cauchy problem. The main result is obtained by employing integration along the characteristic curves and successive approximations sequence arguments. Applications and perspective are also discussed within the paper.

#### 1. Introduction

The derivation and analysis of mathematical frameworks have recently gained much attention in the applied sciences and specifically for the modeling of complex phenomena occurring in biology, chemistry, vehicular traffic, crowd/swarm dynamics, economics, and social systems. At the origin of complex dynamics, there are interactions that occur in nonlinear fashion and randomly among the elements (particles, cells, and pedestrians) composing the complex system [1]. Complex systems are also characterized by emergent properties, which are properties of the system as a whole which do not exist at the individual element level.

Different ordinary differential equations and partial differential equations have been derived with the aim of obtaining an accurate description of complex phenomena. Moreover, thermostatted kinetic frameworks have been developed for the mathematical modeling of complex systems in physics and life sciences [2–5]. The goal of the thermostatted kinetic models is the possibility of modeling the interactions among the particles at the microscopic scale. In particular, these frameworks allow modeling the ability of the particles to express strategies.

The present paper deals with further developments of the thermostatted kinetic theory proposed in [6]. Specifically, this paper is devoted to the mathematical analysis of a partial integrodifferential equation with thermostat and time delay that belongs to the class of the thermostatted kinetic theory frameworks. The paper focuses on the proof of the existence and uniqueness of mild solutions of the related Cauchy problem. The main result is obtained by employing integration along the characteristic curves and successive approximations sequence arguments. Applications and perspective are also discussed within the paper.

The time delay is introduced in order to take into account the fact that most of the emerging behaviours occurring in complex systems at a certain time are strictly related to the interactions among the particles of the system at a previous time. In the pertinent literature, mathematical models with time delays have been proposed only in ODE-based models; see, among others, [7–16]. The introduction of the time delay has provoked the onset of fluctuations and Hopf bifurcation; see [17–20].

It is worth stressing that, to the best of our knowledge, this is the first time that time delay is introduced into a thermostatted partial integrodifferential equation (kinetic).

The contents of the present paper are developed through four more sections, which follow this introduction. In detail, Section 2 deals with the retarded partial integrodifferential equation and the related Cauchy problem; Section 3 is concerned with some preliminary results that are needed for the proof of the main result that is outlined in Section 4. Finally, Section 5 is devoted to a critical analysis of the proposed mathematical equation including research perspective and applications.

#### 2. The Retarded Integrodifferential Equation

This paper is devoted to the result about the existence and uniqueness of a solution of the following Cauchy problem: where is the unknown function, , , and is the following integral operator: with , the time delay, and the initial datum.

In the partial integrodifferential equation with time delay
is the conservative interaction operator, which is splitted into the gain (of particles into state ) operator and the loss (of particles into state ) operator :
Bearing all of the above in mind, and under suitable integrability assumptions on , the th-order moment of is defined as follows:
In particular, the zero-order, first-order, and second-order moments represent the* density* (mass),* mean activation* (linear momentum), and* activation energy* (kinetic energy), respectively. The term
is a damping operator that allows the control of the activation energy. This term is based on the Gaussian isokinetic thermostat (the interested reader is referred, among others, to [21–23]).

In what follows, we assume that for (initial function).

#### 3. Preliminary Results

This section is concerned with some preliminary results that are at the basis of the main result of the present paper.

Lemma 1. *The gain operator (4) satisfies, for all functions and , the following identity:
*

*Proof. *It is obtained by straightforward calculations.

The main result is based on the following assumptions on the probability density function .(A_{1})The probability density function satisfies, for all , the following identity:
which models the conservation of particles.(A_{2}) The probability density function is an even function with respect to ; then, in particular,
(A_{3}) The probability density function satisfies, for all , the following identity:

Lemma 2. *If the function satisfies assumptions (9), (10), and (11), then
*

*Proof. *Condition (9) implies
Bearing in mind condition (10), we have
Finally, condition (11) implies
Therefore, the proof of the lemma is concluded.

Bearing the previous lemma in mind, the evolution equation for the 1st-order moment of is stated in the following results.

Theorem 3. *Assume that assumptions (9), (10), and (11) hold. If there exists a nonnegative solution of the partial integrodifferential equation (3) such that as , then the 1st-order moment is solution of the following delayed first-order nonlinear ordinary differential equation:
*

*Proof. *The integral operator can be written as follows:
Multiplying both sides of by , integrating over , and considering (10), we have
Since
then we have the proof.

Corollary 4. *Assume that assumptions (9), (10), and (11) hold. If there exists a nonnegative solution of the Cauchy problem (1) such that *(i)*,*(ii)* as ,**
then the 1st-order moment reads as follows:
**
where
*

*Proof. *The proof is obtained by coupling the delayed differential equation (18) with the initial condition . If or , then we have for all . Otherwise, the unique solution is function (22).

Theorem 5. *Let be an odd number and . Assume that assumptions (9), (10), and (11) hold. Then, the th-order moment of satisfies the following delayed ordinary differential equation:
*

*Proof. *The proof follows by multiplying both sides of (3) by , taking into account assumptions (9), (10), and (11), and performing integration by parts on the thermostat term.

According to Corollary 4, (3) can be rewritten as follows: and, after integrating along the characteristics, (25) reads as follows: where being Bearing all of the above in mind, the integral form of (26) is where .

#### 4. Existence of Mild Solutions

*Definition 6. *A function is said to be a mild solution to the Cauchy problem (1) on the time interval if and is solution to the following integral equation:
where

Lemma 7. *Let be the following successive approximations sequence:
**
where is a nonnegative function such that . Then, admits, as goes to infinity, a nonnegative limit such that .*

*Proof. *Since is a nonnegative function, then , . Moreover,
and .

We now prove by induction that the sequence is monotone, and specifically
and .

Assume as the induction hypothesis that, for some , we have and . Then,
Taking into account property (8), equation (36) thus reads as follows:
and by using the induction hypothesis we conclude that the sequence is monotone. Moreover,
Taking into account property (12), we have
and by using the induction hypothesis we have
Bearing all of the above in mind, we conclude that the sequence has a nonnegative limit such that , as . Then, the Levi theorem implies that satisfies the following equation:
whose unique solution is . Therefore, the lemma is completely proved.

The main result of this paper is the following.

Theorem 8. *Let be a given nonnegative function such that . Then, there exists a unique nonnegative mild solution
**
to the Cauchy problem (1).*

*Proof. *Since Lemma 7 states that solves (31), in order to prove that is a mild solution of (1), it is enough to show that . In order to prove that , we consider the following successive approximations sequence:
or the following equivalent form of (43):
The assumption on implies that the zero-order moment . Assume now as induction hypothesis that , for some . Integrating both sides of (44) over with respect to , and using (28), we obtain
Taking into account property (12) and by using the induction hypothesis, the right-hand side of (45) thus reads as follows:
Therefore, for all , we have .

Multiplying both sides of (44) by , and integrating over with respect to , we have
Taking into account (12), (13) and repeating the computations developed in [6], it is easy to prove by induction on that
Multiplying both sides of (44) by and integrating over with respect to , we have
and according to [6] it is easy to prove by induction on that
Let . Then,
Since and by construction , we have
Therefore, in , and since is bounded, then .

We now prove the uniqueness of the solution. Let be any solution to (31). The positivity of the operators and implies that, for all , , and then . Since , we thus have .

Corollary 9. *Let be a given nonnegative function such that *(i)*;*(ii)*.**
Then, the mild solution of the Cauchy problem (1) belongs to where
*

*Proof. *It is sufficient to remember (24) and note that assumption (ii) implies that
which allows us to conclude that .

#### 5. Applications and Research Perspectives

The main goal of the present paper refers to the proof of the global existence of mild solutions of a thermostatted partial integrodifferential equation with time delay. As already mentioned in Section 2, this equation can be proposed for the modeling of complex systems in nature and society where only the interactions at the microscopic scales are affected by time delay. Specifically, the partial integrodifferential equation with time delay can be proposed as a general mathematical framework for the modeling of complex systems composed by a large number of interacting particles and subjected to the external force field . The overall state of the system is described by the distribution function (statistical description). Particles are able to express a specific function; this ability of the particles is modeled by the variable . Moreover, is the encounter rate between particles with states (or ) and . Finally, is the probability density that a particle with state ends up into the state after the interaction with the particle with state . The action of the external force field is controlled by the thermostat term that, as shown in Section 4, maintains constant the first-order and the second-order moments (number density and activation energy of the system).

From the applications viewpoint, we consider a simple model for the evolution of malignancy in tumor cells. Specifically, we assume that the variable models the magnitude of the malignancy of tumor cells and is a delta Dirac function (deterministic output of a pair interaction) depending on the microscopic state of the interacting particles: and finally we assume that the malignancy of tumor cells increases when cells interact with each other with rate . Accordingly, we have where is a positive parameter. Bearing all of the above in mind, the integral operator reads as follows: and then delayed equation (55) now reads as follows:

However, thermostatted equation (55) does not include the role of the space and velocity variables; then, applications refer to the modeling of complex phenomena that are homogeneous in space and velocity. The mathematical analysis performed in the present paper has to be thus generalized for taking also into account the dynamics described by these variables. Moreover, (55) refers to complex systems where the mutual interactions do not produce modification in the number density (conservative interactions).

Research perspectives include the possibility of performing an asymptotic analysis by parabolic (low-field) and hyperbolic (high-field) scalings (see [24–34]) with the aim of obtaining the dynamics of the system at the macroscopic scale. This is a work in progress and results will be presented in due course.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

Carlo Bianca and Annie Lemarchand were partially supported by L’Agence Nationale de la Recherche (ANR T-KiNeT Project).