Asymptotics of dissipative nonlinear evolution equations with ellipticity: different end states

In this paper, we consider the global existence and asymptotic behaviors of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:

(E)     

Evolution on a Smooth Landscape: The Role of Bias

The role of mutational bias in evolution on a smooth landscape is investigated. We consider both a finite-length genome where the bias increases linearly with the fitness, and an infinite genome with a fixed bias. We present simulations of finite populations in a waiting time model, showing both the nonequilibrium dynamics and the equilibrium fitness distributions that are reached. We compute the equilibrium analytically in several cases, using approximate direct solution of the master equations and truncated hierarchies.     

Leading logarithmic large-x resummation of off-diagonal splitting functions and coefficient functions

We analyze the iterative structure of unfactorized partonic structure functions in the large-x   limit, and derive all-order expressions for the leading-logarithmic off-diagonal splitting functions Pgq$P_{\mathrm{gq}}$ and Pqg$P_{\mathrm{qg}}$ and the corresponding coefficient functions C_?,q$C_{?,\mathrm{q}}$ and C_2,g$C_{2,\mathrm{g}}$ in Higgs- and gauge-boson exchange deep-inelastic scattering. The splitting functions are given in terms of a new function not encountered in perturbative QCD so far, and vanish maximally in the supersymmetric limit CA−CF→0$C_A-C_F\to 0$. The coefficient functions do not vanish in this limit, and are given by simple expressions in terms of the above new function and the well-known leading-logarithmic threshold exponential. Our results also apply to the evolution of parton fragmentation functions and semi-inclusive e⁺e⁻$e^{+}{e}^{-}$ annihilation.     

Influence of the variance of degree distributions on the evolution of cooperation in complex networks

We study how initial network structure affects the evolution of cooperation in a spatial prisoner’s dilemma game. The network structure is characterized by various statistical properties. Among those properties, we focus on the variance of the degree distribution, and inquire how it affects the evolution of cooperation by three methods of imitation. For every method, it was found that a scale-free network does not always promote the evolution of cooperation, and that there exists an appropriate value of the variance, at which cooperation is optimal.     

Evolution of Chinese airport network

With the rapid development of the economy and the accelerated globalization process, the aviation industry plays a more and more critical role in today’s world, in both developed and developing countries. As the infrastructure of aviation industry, the airport network is one of the most important indicators of economic growth. In this paper, we investigate the evolution of the Chinese airport network (CAN) via complex network theory. It is found that although the topology of CAN has remained steady during the past few years, there are many dynamic switchings inside the network, which have changed the relative importance of airports and airlines. Moreover, we investigate the evolution of traffic flow (passengers and cargoes) on CAN. It is found that the traffic continues to grow in an exponential form and has evident seasonal fluctuations. We also found that cargo traffic and passenger traffic are positively related but the correlations are quite different for different kinds of cities.     

A generalization of Chernoff's product formula for time-dependent operators

In this article we provide a set of sufficient conditions that allow a natural extension of Chernoff's product formula to the case of certain one-parameter family of functions taking values in the algebra L(B) of all bounded linear operators defined on a complex Banach space B. Those functions need not be contraction-valued and are intimately related to certain evolution operators U(t,s)_0?s?t?T on B. The most direct consequences of our main result are new formulae of the Trotter-Kato type which involve either semigroups with time-dependent generators, or the resolvent operators associated with these generators. In the general case we can apply such formulae to evolution problems of parabolic type, as well as to Schrödinger evolution equations albeit in some very special cases. The formulae we prove may also be relevant to the numerical analysis of non-autonomous ordinary and partial differential equations.     

Blow-up of solutions to parabolic equations with nonstandard growth conditions

We study the phenomenon of finite time blow-up in solutions of the homogeneous Dirichlet problem for the parabolic equation

     

Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and an integral equation . We characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.     

Asymptotic Behavior of Solutions to the Cahn-Hilliard Equation in Spherically Symmetric Domains

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. The asymptotic behavior of solutions to the Cahn-Hilliard equation with Dirichlet boundary conditions and the associated stationary problem have been studied. In particular, it is proved that the only possible stable equilibrium solutions in spherically symmetric domains are spherically symmetric and monotone in the radial direction.