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Abstract and Applied Analysis
Volume 2015, Article ID 760136, 12 pages
Research Article

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

1Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan
2Department of Statistics, Oita University, Oita 879-5593, Japan

Received 18 May 2014; Accepted 29 July 2014

Academic Editor: Juan J. Nieto

Copyright © 2015 Minoru Tabata and Nobuoki Eshima. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Assume that economic activities are conducted in a bounded continuous domain where workers move toward regions that offer higher real wages and away from regions that offer below-average real wages. The density of real wages is calculated by solving the nominal wage equation of the continuous Dixit-Stiglitz-Krugman model in an urban-rural setting. The evolution of the density of workers is described by an unknown function of the replicator equation whose growth rate is equal to the difference between the density of real wages and the average real wage. Hence, the evolution of the densities of workers and real wages is described by the system of the nominal wage equation and the replicator equation. This system of equations is an essentially new kind of system of nonlinear integropartial differential equations in the theory of functional equations. The purpose of this paper is to obtain a sufficient condition for the initial value problem for this system to have a unique global solution.