A population explosion in an evolutionary game in spatial economics: Blow up radial solutions to the initial value problem for the replicator equation whose growth rate is determined by the continuous Dixit–Stiglitz–Krugman model in an urban setting

2015 ◽  
Vol 23 ◽  
pp. 26-46 ◽  
Author(s):  
Minoru Tabata ◽  
Nobuoki Eshima
Keyword(s):  
Growth Rate ◽  
Initial Value Problem ◽  
Blow Up ◽  
Evolutionary Game ◽  
Urban Setting ◽  
Spatial Economics ◽  
Replicator Equation ◽  
Radial Solutions ◽  
Initial Value ◽  
Population Explosion

scholarly journals Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Minoru Tabata ◽  
Nobuoki Eshima
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Equilibrium Solution ◽  
Global Solutions ◽  
Initial Time ◽  
Spatial Economics ◽  
Replicator Equation ◽  
Unique Global Solution ◽  
Initial Value ◽  
The Cauchy Problem

We study the initial-value problem for the replicator equation of theN-region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.

Stability of time-dependent rotational Couette flow Part 2. Stability analysis

1971 ◽  
Vol 48 (2) ◽  
pp. 365-384 ◽  
Author(s):  
C. F. Chen ◽  
R. P. Kirchner
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Kinetic Energy ◽  
Initial Value Problem ◽  
Basic Flow ◽  
The Other ◽  
Time Dependent ◽  
Stability Criteria ◽  
Initial Value ◽  
The Stability

The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results.

scholarly journals Effect of Radial Wavevector on Collisional Drift Waves in a Toroidal Heliac

1999 ◽  
Vol 52 (1) ◽  
pp. 71 ◽  
Author(s):  
J. L. V. Lewandowski ◽  
R. M. Ellem
Keyword(s):  
Growth Rate ◽  
Electrostatic Potential ◽  
Growth Rates ◽  
Initial Value Problem ◽  
Linear Growth ◽  
Field Model ◽  
Linear Growth Rate ◽  
Drift Waves ◽  
Initial Value ◽  
Magnetic Shear

A 3-field model for collisional drift waves, in the ballooning representation, for a low-pressure stellarator plasma is presented. In particular, the effect of a finite radial mode number (≡ θk) is studied, and the linear growth rates for the fluctuating plasma density, electrostatic potential and electron temperature are computed numerically by solving the 3-field model as an initial-value problem. Numerical results for a 3-field period stellarator with low global magnetic shear are then presented. It is found that, in a system with small global magnetic shear, the case θk = 0 yields the fastest linear growth rate.

Instability growth rate of two-phase mixing layers from a linear eigenvalue problem and an initial-value problem

2010 ◽  
Vol 22 (9) ◽  
pp. 092104 ◽  
Author(s):  
Anne Bagué ◽  
Daniel Fuster ◽  
Stéphane Popinet ◽  
Ruben Scardovelli ◽  
Stéphane Zaleski
Keyword(s):  
Growth Rate ◽  
Eigenvalue Problem ◽  
Initial Value Problem ◽  
Instability Growth Rate ◽  
Mixing Layers ◽  
Initial Value ◽  
Two Phase ◽  
Phase Mixing ◽  
Linear Eigenvalue Problem ◽  
Instability Growth

scholarly journals 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

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Minoru Tabata ◽  
Nobuoki Eshima
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Global Solutions ◽  
Wage Equation ◽  
Rural Setting ◽  
Nominal Wage ◽  
Replicator Equation ◽  
Real Wages ◽  
Initial Value ◽  
Urban Rural

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.

scholarly journals ON THE UNIQUENESS OF POSITIVE SOLUTIONS OF A QUASILINEAR EQUATION CONTAINING A WEIGHTED p-LAPLACIAN, THE SUPERLINEAR CASE

2008 ◽  
Vol 10 (03) ◽  
pp. 405-432 ◽  
Author(s):  
MARTA GARCÍA-HUIDOBRO ◽  
DUVAN A. HENAO
Keyword(s):  
Dirichlet Problem ◽  
Initial Value Problem ◽  
Quasilinear Equation ◽  
Radial Solutions ◽  
Initial Value ◽  
Superlinear Growth ◽  
Positive Radial Solutions ◽  
Locally Lipschitz ◽  
Comparison Arguments ◽  
Separation Results

We consider the quasilinear equation of the form [Formula: see text] where Δpu ≔ div (|∇u|p-2∇u) is the degenerate p-Laplace operator and the weight K is a positive C1function defined in ℝ+. We deal with the case in which f ∈ C[0,∞) has one zero at u0> 0, is non positive and not identically 0 in (0,u0), and is locally Lipschitz, positive and satisfies some superlinear growth assumption in (u0,∞). We carefully study the behavior of the solution of the corresponding initial value problem for the radial version of the quasilinear equation, as well as the behavior of its derivative with respect to the initial value. Combining, as Cortázar, Felmer and Elgueta, comparison arguments due to Coffman and Kwong, with some separation results, we show that any zero of the solutions of the initial value problem is monotone decreasing with respect to the initial value, which leads immediately the uniqueness of positive radial ground states, and the uniqueness of positive radial solutions of the Dirichlet problem on a ball.

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