October  1999, 5(4): 697-728. doi: 10.3934/dcds.1999.5.697

A difference-differential analogue of the Burgers equations and some models of economic development

1. 

Université Pierre et marie Curie, Paris, France

2. 

CEMI, Academy of Science, Moscow, Russian Federation

Received  January 1999 Revised  June 1999 Published  July 1999

The paper is devoted to investigation of a number of difference-deiiferential equations, among them the following one plays the central role:

$dF_n$/$dt=\varphi(F_n)(F_{n-1}-F_n)\quad\qquad\qquad (\star)$

where, for every $t, \{F_n(t), n=0,1,2,\ldots\}$ is a probability distribution function, and $\varphi$ is a positive function on $[0, 1]$. The equation $(\star)$ arose as a description of industrial economic development taking into accout processes of creation and propagation of new technologies. The paper contains a survey of the earlier received results including a multi-dimensional generalization and an application to the economic growth theory.
If $\varphi$ is decreasing then solutions of Cauchy problem for $(\star)$ approach to a family of wave-trains. We show that diffusion-wise asymptotic behavior takes place if $\varphi$ is increasing. For the nonmonotonic case a general hypothesis about asymtotic behavior is formulated and an analogue of a Weinberger's (1990) theorem is proved. It is argued that the equation can be considereded as an analogue of Burgers equation.

Citation: Gennadi M. Henkin, Victor M. Polterovich. A difference-differential analogue of the Burgers equations and some models of economic development. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 697-728. doi: 10.3934/dcds.1999.5.697
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