The Existence and Uniqueness of Global Solutions to the Initial Value Problem for the System of Nonlinear Integropartial Differential Equations in Spatial Economics : The Dynamic Continuous Dixit-Stiglitz-Krugman Model in an Urban-Rural Setting

and Applied Analysis 3 where λ = λ(t, y) represents the density of workers at time t ≥ 0 and at point y ∈ D. By σ we denote the elasticity of substitution among varieties of manufactured goods. We assume that


Introduction
The new economic geography (NEG) is a new branch of spatial economics that was initiated by Krugman in the early 1990s.This new branch has attracted many social scientists and becomes one of the most important major branches of spatial economics at present.In 2008 Krugman received the Nobel Memorial Prize in Economic Sciences (officially Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel) for his great contribution to the NEG (see [1][2][3][4][5][6]).A large number of mathematical models have been built in the NEG.In particular there are many models described by nonlinear integropartial differential equations that are new and important in the theory of functional equations (see [1][2][3]).Hence the NEG is regarded as a new frontier of the theory of nonlinear integropartial differential equations.
The Krugman core-periphery model (the CP model) is the origin of the NEG (see [1,Chapter 5]) since various models are constructed as its extension.The CP model is a discrete model.In this model economic activities are conducted at two points.These two points represent a core region and a periphery region, respectively.An extension to the case of a finite set of points has been studied in [1].This model is called the Dixit-Stiglitz-Krugman model (DSK model).Its mathematical foundation is studied in [7][8][9][10][11][12][13].Moreover, in [14], we consider an extension of the CP model to the case of a bounded continuous domain where economic activities are conducted continuously in space.This model is called the continuous DSK model (cDSK model).
These models are static models with no population dynamics.It is very important in spatial economics to build population dynamics into them.Hence Krugman constructs the dynamic DSK model (dDSK model) by combining the DSK model with the replicator dynamics that is one of the most fundamental dynamics in evolutionary game theory 2 Abstract and Applied Analysis (see [15] and [16,Chapter 3]).His dynamic model is very important in spatial economics since it describes economies of agglomeration in the case where workers move from one point to another to seek higher real wages within a finite set of points at which economic activities are conducted (see [1, p. 62, p. 77]).Hence, by following this line, we consider the dynamic cDSK model (dcDSK model) in this paper; that is, we combine the cDSK model with the replicator dynamics.This dynamic model is regarded as a continuous version of the dDSK model and explains agglomeration of capital and concentration of workers when workers move in a bounded continuous domain where economic activities are conducted continuously in space.
Let us discuss the dcDSK model from the viewpoint of the theory of functional equations.The dcDSK model is described by the system of the nominal wage equation and the replicator equation.We refer to this system of equations as the dcDSK system.The nominal wage equation is a nonlinear integral equation that contains the density of nominal wages as an unknown function and the density of workers as a known function (see [14, (2.4)]).Hence, if we solve this equation under the condition that the density of workers is a given function, then we can obtain the density of nominal wages.However, the integral kernel of the nominal wage equation contains not only these densities but also the price index.We must note that the price index itself is a nonlinear integral operator acting on the density of nominal wages and the density of workers (see [14, (2.7)]).Therefore, we can say that the nominal wage equation is a double nonlinear integral equation (see [14,Remark 2.3] for mathematical difficulties caused by the double nonlinearity).
The replicator equation is a nonlinear integropartial differential equation whose unknown function denotes the density of workers.Its coefficient denotes the growth rate of worker population (see [1, (5.1), (5.2)] and [16, p. 73]).The coefficient is equal to the difference between the density of real wages and the average real wage, where the density of real wages is defined as the density of nominal wages deflated by a fractional power of the price index, and the average real wage is defined as the integral of the product of the density of workers and the density of real wages (see [1, (5,1), (5.6)]).Hence, the coefficient is regarded as a double nonlinear integral operator acting on the density of workers and the density of nominal wages.
Moreover, the coefficient of the replicator equation of the dcDSK system contains an unknown function implicitly in the sense that the coefficient is determined by solving the nominal wage equation under the condition that the unknown function is given, in contrast to the replicator equation whose coefficient explicitly contains an unknown function in evolutionary game theory (see [16, (3.3)]).If we can define an operator that maps the density of workers to the density of real wages by solving the nominal wage equation under the condition that the density of workers is a given function, then the replicator equation is regarded as a nonlinear integropartial differential equation whose coefficient contains the operator that acts on an unknown function.
For these reasons we deduce that the dcDSK system is an essentially new kind of system of nonlinear integropartial differential equations.Therefore, it is important to study this system not only in spatial economics but also in the theory of functional equations.In this paper we prove a sufficient condition for the initial value problem for the dcDSK system to have a unique global solution and obtain estimates of the solution.The main result is Theorem 4.

The System of Equations
Let us introduce the notations.Let  be a domain of a Euclidean space.By  1 () we denote the Banach space of all Lebesgue summable functions of  ∈ .By  ∞ () we denote the Banach space of all essentially bounded functions of  ∈ .By () we denote the Banach space of all uniformly bounded continuous functions of  ∈ .
The dcDSK model consists of a monopolistically competitive sector (manufacturing) and a perfectly competitive sector (agriculture) (see [1, p. 61]).Hence the income consists of agricultural income and manufacturing income; that is, it has the following form (see [1, (5.3)]): where  and (1−) denote the share of manufacturing expenditure and the share of agricultural expenditure, respectively, and we denote the density of farmers at point  ∈  by  = ().We assume that  = () is a given function such that (see [14, (2.12) Note that this function is independent of the time variable  ≥ 0.
The function  = (, ) represents the iceberg form of transport costs (see [17]).We refer to this function as the transport cost function.We reasonably accept the following assumption (see [14,Assumption 2.1]).
Considering (6), and noting that the right-hand side of (4) is a nonlinear integral operator acting on the density of workers  = (, ) and the density of nominal wages  = (, ), we see that the right-hand side of (3) is a double nonlinear integral operator acting on these densities.

Result and Discussion
We consider the initial value problem by imposing the following initial condition on the dcDSK system (3), (10), and : where  0 =  0 () is a given function of  ∈ .This function denotes the initial density of workers.The following assumption is imposed on this function in [1, pp.61-63] and [14, (2.11), (2.13)].
Hence, we accept this assumption in this paper also.Let  > 0 be a constant.If a function (, , ) = ( (, ) ,  (, ) ,  (, )) (16) belongs to and satisfies the dcDSK system for a.e.(, ) ∈ [0, ] ×  and the initial condition ( 15), then we say that the function ( 16) is a solution to the initial value problem in [0, ].If a function ( 16) is defined for a.e.(, ) ∈ [0, +∞) ×  and is a solution to the initial value problem in [0, ] for each  > 0, then we say that the solution is global.No boundary condition needs to be imposed on the density of workers, since the evolution of the density of workers can be determined uniquely in (17) by the initial condition (15) as done in Section 5.
We define a function  = (,), (, ) ∈ (0, 1) × (1, +∞) in order to state a sufficient condition for the initial value problem to have a unique global solution.Consider the quadratic equation where  denotes an unknown quantity.It follows from ( 5) that this equation has a positive solution and a negative solution.
We denote the positive solution by  = (, ).We see easily that By making use of this positive solution, we define the following quadratic equation: where V denotes an unknown quantity.This quadratic equation has a positive solution and a negative solution, since (, ) 1/ > 0. We denote the positive solution by  = (, ).We see easily that The following lemma is proved in [12, Lemma 3.2] (see [12, (3.3)-(3.8)]).
Remark 5. We impose ( 7) on the dcDSK model; that is, we consider the dcDSK model in an urban-rural setting.In this paper we cannot treat the case that is, we cannot consider the dcDSK model in an urban setting (see [1, p. 331]) because it follows from Lemma 3, (i), that (41) cannot be substituted in ( 22).The DSK model with (41) is studied in [13].The cDSK model with (41) is studied in [14, Theorem 3.2].

Solutions of the Nominal Wage Equation
Let us solve the nominal wage equation (3) under the condition that the density of workers is a given function.
In this section we do not deal with the replicator equation.Hence, we have no need to consider the time evolution of the density of workers, the density of nominal wages, and the density of real wages.Therefore, for simplicity of symbols, we omit the time variable  from these densities, and we denote them by  = (),  = (), and  = () in ( 3), ( 4), (6), and (10) in this section.We refer to these equations with the same numbers.No confusion should arise.We assume that  = () is a given function that satisfies the same condition as Assumption 2 as follows: Proposition 6.If , , and C satisfy ( 5), (7), and (36), then the following statements (i) and (ii) hold.
Remark 10. (i) If  is not a continuous domain but a finite set of points, then the nominal wage equation is not a nonlinear integral equation but a nonlinear equation in an Euclidean space whose dimension is equal to the number of points of  (see [12, (2.5)]).This subject is treated in [12].In [12] we prove a result similar to Propositions 6 and 9 by analyzing this nonlinear equation in the Euclidean space.However, in this paper, Propositions 6 and 9 are proved in the Banach spaces in contrast to the finite dimensional proof done in [12].Propositions 6 and 9 are similar to, but essentially different from, the results obtained in [12].
(ii) The inequalities ( 5), (7), and (36) are a sufficient condition for the nominal wage equation (3) to have a solution in  ∞ + () (see Proposition 6).The inequalities ( 5), (7), and (22) are a sufficient condition for this solution to be unique.We make use of ( 22) in order to obtain (95) in the proof of Lemma 8.
(iii) It is proved in [14, Theorem 3.1, (iii)] that if  > 1 and C > 0 are sufficiently small, then (3) has a unique solution.However, the condition ( 22) is not accepted in [14].Recalling (35), we see that the result of [14] can be regarded as a corollary of Proposition 9.
Note that Propositions 6 and 9 are proved on the basis of (42) and (43).Hence, we need Lemma 13, (i), (ii), in order to make use of Propositions 6 and 9 in the proof of Lemma 13, (iii)-(viii).
Abstract and Applied Analysis 11 The following lemma gives necessary conditions for the initial value problem to have a solution that belongs to (17).
Proof.Recalling ( 12), (13), and the definition of ( 17 Hence, integrating both sides of (11) with respect to  ∈  and recalling ( 12) and ( 13), we see easily that where be a solution that belongs to (17).Substitute this solution and (135) in (11).Subtracting both sides of the equalities thus obtained from each other, we obtain the following equation in the same way as (123): Note that both (135) and ( 144) satisfy ( 25) and ( 28