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A Difference-differential Analogue of the Burgers Equation and Some Models of Economic Development

Polterovich, Victor and Henkin, Gennadi (1998): A Difference-differential Analogue of the Burgers Equation and Some Models of Economic Development.

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Abstract

The paper is devoted to investigation of a number of difference-differential equations, among them the following one plays the central role: dFn/dt<=φ(Fn)(Fn-1 - Fn) (*) where, for every t, {Fn(t), n = 0, 1, 2, ...} is a probability distribution function, and φ is a positive function on [0, 1]. The equation (*) arose as a description of industrial economic development taking into account processes of creation and propagation of new technologies. The paper contains a survey of the earlier received results including a multy-dimensional generalization and an application to the economic growth theory.

If φ is decreasing then solutions of the Cauchy problem for (*) approach to a family of wave-trains. We show that diffusion-wise asymptotic behavior takes place if φ is increasing. For the nonmonotonic case a general hypothesis about asymptotic behavior is formulated and an analogue of a Weinberger's (1990) theorem is proved. It is argued that the equation can be considered as an analogue of Burgers equation.

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