A NONLOCAL PARABOLIC SYSTEM WITH APPLICATION TO A THERMOELASTIC PROBLEM

A system modeling the thermoelastic bards contacts is studied. The problem is first transformed into an equivalent nonlocal parabolic systems using a transformation, and then the existence and uniqueness of the solutions are demonstrated via the theoretical potential representation theory of the parabolic equations. Finally some realistic situations in the applications are discussed using the results obtained in this paper.


Introduction.
In this paper we extend the results obtained in [3,8] for a nonlocal parabolic system for two dependent variables to a general system for n such variables.The result has application to the problem of thermoelastic contact of n rods.We consider first the existence, uniqueness and continuous dependence of the solutions of the nonlocal parabolic system of equations governing the temperature distribution in the rods.
Definition 1.1.A vector θ is said to be a solution to the problems (1.2)-(1.10)if θ i ∈ C 2 (Ω i × J) ∩ L ∞ (Ω i × J) and satisfies equation (1.2) together with the initial and boundary conditions almost everywhere.
2. An equivalent problem.The problem described by equations (1.1)-(1.10)can be reduced to an equivalent problem by setting Multiplying each of equations (2.1) by a i in turn and integrating over Ω i and summing we have where w = (w 1 ,w 2 ,...,w n ) T .If we add g to either side we have

1) has the unique inverse
(2.4) Proof.Equation (2.3) implies that and the result follows.

Preliminaries.
In this section we list several classical results from [4] and develop solutions to the problems (2.6)-(2.12).We refer to [4] for proofs of the following lemmas: solves the problems Here we have defined Lemma 3.2.Let w(x, t) be the solution of where where ψ 1 ,ψ 2 are the unique solutions in C[0,T ] of the Volterra system Note. (i) If F,G are piecewise continuous and bounded, then Lemma 3.2 holds with ψ 1 ,ψ 2 piecewise continuous and bounded.
(iii) If F and G ∈ L 1 (0,T ), then ψ 1 ,ψ 2 ∈ L 1 (0,T ) and the solution w ∈ L 1 [(0, 1) × (0,T )].Lemma 3.3.Let w(x, t) be the solution of where Extensions.We require the following corollaries of Lemmas 3.2 and 3.3 to adapt the solutions to the intervals of interest for the problems (2.6)-(2.12).Corollary 3.1.With the assumptions of Lemma 3.2 the solution of is given by ψ 1 2 (s) ds, (3.12)where ψ 1 1 ,ψ 2 1 are the unique solutions of the Volterra system for j = 2, 3,...,n− 1 is given by where is given by where ψ n 1 ,ψ n 2 are the unique solutions of the Volterra system , and proceed as in Corollary 3.1.

We now set
Clearly, once is determined uniquely, the solution to our problem is known.We will show that ψ satisfies a matrix Volterra system of the form where G, H are suitable vectors and A, M, B suitable matrices.We require the following lemma: and let the N × N matrices Then the system (3.26) has a unique solution ψ(t) ∈ [L(0,T )] N .In particular if ψ 1 ,ψ 2 are two solutions corresponding to data G 1 , H 1 , and G 2 ,H 2 respectively then there exists Proof.See [8].

Application to thermoelastic bars.
Consider n thermoelastic bars lying along the positive x axis with the ith bar, 1 ≤ i ≤ n, occupying the interval Ω i .We use the notation of Section 1.The equations describing the displacements and temperature distributions are given by ) for i = 1, 2,...,n, t ∈ J. u i (x, t) denotes the displacement and θ i (x, t) the temperature of the ith bar at position x and time t.σ i (x, t) represents the corresponding stress and c i , k i , α i are constants, i = 1, 2,...,n, denoting the heat capacity, conductivity and coefficient of thermal expansion, respectively, of the ith bar.θ 0 is a reference temperature, measured in degrees Kelvin, normally taken as the ambient temperature.
It is convenient to nondimensionalize the quantities of interest and we set where for i = 1, 2,...,n.The quantities λ i , µ i are the Lamé elastic constants.The intervals x ∈ Ω i are replaced with the corresponding intervals x ∈ Ω i , i = 1, 2,...,n.If the above quantities are substituted in equations (5.1), (5.2), and (5.3) and subsequently the hats dropped we have the equations in the following nondimensional form ) ) for i = 1, 2,...,n, where (5.9) Clearly equation (5.8) implies that (5.10) Since (5.10) holds this implies that if one end of any of the bars is free then σ i (t) = 0, i = 1, 2,...,n whereas if all of the bars are in contact then σ i (t) ≤ 0, i = 1, 2,...,n.
In addition to the governing equations we require the initial and boundary conditions.The conditions on the θ i (x, t) are given in equations (1.3) through (1.7).For the moment we require only the initial conditions, namely, together with the conditions (5.12) There are essentially two cases to consider depending on whether all bars are in contact or not.There are then subcases depending on how the bars are grouped in contact.
The difficulties are the same whether we consider n bars or three bars.The latter case simplifies and clarifies the procedure and we now confine our attention to that case.The generalization required for n bars then follows.
We consider then three bars lying along the positive x axis lying in the intervals and set ( We begin by considering the initial conditions.Set (5.17) From equations (5.11), θ i (x, 0), i = 1, 2, 3 are known, so that Θ i (x, 0) are known.There are two cases.Case I.If Using equations (5.7) and (5.12) we have If equation (5.18) does not hold but (5.19) does then whereas if (5.18) holds and (5.19) does not Thus if the middle bar is in contact with either of the end bars initially then the initial stresses, displacements and temperatures are known.If on the other hand the middle bar has no contact with the other two initially u 2 ( 2 , 0) is indeterminate and an additional initial condition must be added.If we define where then in all three of the above subcases and conversely if (5.27) holds, then one of these subcases does.
The general situation for t > 0 may be handled in the same manner except that Θ i (x, t), i = 1, 2, 3 are not known a priori.
Case I.Here equations (5.18) through (5.24) are replaced by the same equations with t = 0 replaced by the general time t.If both conditions replacing (5.18), ( 5 Since in this case, for i = 1, 2, 3, σ i (t) = 0 and u i (x, t) are given by the updated forms of equations (5.20) through (5.22), we may substitute in equations (5.6) to give (5.40) Case II.In this case we follow the procedure of equations (5.30) through (5.32) again replacing t = 0 with general t > 0. On substituting the updated values of the stresses σ i (t) into the expressions for the updated values of u i (x, t) we can substitute into equations (5.6) to obtain where Ω(t) is given in equation (5.25).Since, in this case, σ i (t) ≤ 0, i = 1, 2, 3 then Ω(t) ≤ 0. If Ω(t) > 0 we have Case I.This allows us to combine equations (5.40) and (5.41) in the form (5.43) then it is clear that equations (1.2) are a direct generalization of equations (5.42).

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: 21 in equations(3.11),and consider Lemma 3.2 in terms of the new variables x, t.