
Previous Article
Smooth solution of the generalized system of ferromagnetic chain
 DCDS Home
 This Issue
 Next Article
A differencedifferential 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 
$dF_n$/$dt=\varphi(F_n)(F_{n1}F_n)\quad\qquad\qquad (\star)$
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 multidimensional 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 wavetrains. We show that diffusionwise 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.
[1] 
Chang Zhang, Fang Li, Jinqiao Duan. Longtime behavior of a class of nonlocal partial differential equations. Discrete & Continuous Dynamical Systems  B, 2018, 23 (2) : 749763. doi: 10.3934/dcdsb.2018041 
[2] 
A. Kh. Khanmamedov. Longtime behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 13731400. doi: 10.3934/cpaa.2009.8.1373 
[3] 
A. Kh. Khanmamedov. Longtime behaviour of wave equations with nonlinear interior damping. Discrete & Continuous Dynamical Systems  A, 2008, 21 (4) : 11851198. doi: 10.3934/dcds.2008.21.1185 
[4] 
H. A. Erbay, S. Erbay, A. Erkip. Longtime existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems  A, 2019, 39 (5) : 28772891. doi: 10.3934/dcds.2019119 
[5] 
Linghai Zhang. Longtime asymptotic behaviors of solutions of $N$dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems  A, 2002, 8 (4) : 10251042. doi: 10.3934/dcds.2002.8.1025 
[6] 
Francesca Bucci, Igor Chueshov. Longtime dynamics of a coupled system of nonlinear wave and thermoelastic plate equations. Discrete & Continuous Dynamical Systems  A, 2008, 22 (3) : 557586. doi: 10.3934/dcds.2008.22.557 
[7] 
Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Wellposedness and longtime behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems  A, 2007, 18 (1) : 1538. doi: 10.3934/dcds.2007.18.15 
[8] 
Andrey B. Muravnik. On the Cauchy problem for differentialdifference parabolic equations with highorder nonlocal terms of general kind. Discrete & Continuous Dynamical Systems  A, 2006, 16 (3) : 541561. doi: 10.3934/dcds.2006.16.541 
[9] 
Manuel Núñez. The longtime evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems  B, 2004, 4 (2) : 465478. doi: 10.3934/dcdsb.2004.4.465 
[10] 
Hongtao Li, Shan Ma, Chengkui Zhong. Longtime behavior for a class of degenerate parabolic equations. Discrete & Continuous Dynamical Systems  A, 2014, 34 (7) : 28732892. doi: 10.3934/dcds.2014.34.2873 
[11] 
Shan Ma, Chunyou Sun. Longtime behavior for a class of weighted equations with degeneracy. Discrete & Continuous Dynamical Systems  A, 2020, 40 (3) : 18891902. doi: 10.3934/dcds.2020098 
[12] 
Yang Liu. Longtime behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311326. doi: 10.3934/era.2020018 
[13] 
Vladimir Varlamov. Eigenfunction expansion method and the longtime asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems  A, 2001, 7 (4) : 675702. doi: 10.3934/dcds.2001.7.675 
[14] 
Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growthfragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251297. doi: 10.3934/krm.2016.9.251 
[15] 
Peter V. Gordon, Cyrill B. Muratov. Selfsimilarity and longtime behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767780. doi: 10.3934/nhm.2012.7.767 
[16] 
Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems  A, 2012, 32 (3) : 703715. doi: 10.3934/dcds.2012.32.703 
[17] 
C. I. Christov, M. D. Todorov. Investigation of the longtime evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139148. doi: 10.3934/proc.2013.2013.139 
[18] 
David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems  A, 2018, 38 (6) : 30993138. doi: 10.3934/dcds.2018135 
[19] 
Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems  A, 2012, 32 (1) : 331352. doi: 10.3934/dcds.2012.32.331 
[20] 
Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701716. doi: 10.3934/krm.2011.4.701 
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]