# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems, 1999, 5 (4) : 697-728. doi: 10.3934/dcds.1999.5.697
 [1] Chang Zhang, Fang Li, Jinqiao Duan. Long-time behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 749-763. doi: 10.3934/dcdsb.2018041 [2] A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373 [3] H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119 [4] A. Kh. Khanmamedov. Long-time behaviour of wave equations with nonlinear interior damping. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1185-1198. doi: 10.3934/dcds.2008.21.1185 [5] Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025 [6] Dongxue Yan, Xianlong Fu. Long-time behavior of a size-structured population model with diffusion and delayed birth process. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021030 [7] Francesca Bucci, Igor Chueshov. Long-time dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 557-586. doi: 10.3934/dcds.2008.22.557 [8] Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15 [9] Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110 [10] Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541 [11] Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465 [12] Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018 [13] Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098 [14] Hongtao Li, Shan Ma, Chengkui Zhong. Long-time behavior for a class of degenerate parabolic equations. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2873-2892. doi: 10.3934/dcds.2014.34.2873 [15] Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675 [16] Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251 [17] Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767 [18] Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703 [19] C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139 [20] Renzhi Qiu, Shanjian Tang. The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 3-. doi: 10.1186/s41546-019-0037-3

2020 Impact Factor: 1.392