# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

2010 , Volume 13 , Issue 3

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2010, 13(3): 517-535 doi: 10.3934/dcdsb.2010.13.517 +[Abstract](346) +[PDF](1127.5KB)
Abstract:
In this article we establish the global well-posedness of a recent model proposed by Noguera, Fritz, Clément and Baronnet for simultaneously describing the process of nucleation, growth and ageing of particles in thermodynamically closed and initially supersaturated systems. This model, which applies to precipitation in solution, vapor condensation and crystallization from a simple melt, can be seen as a highly nonlinear age-dependent population problem involving a delayed birth process and a hysteresis damage operator.
2010, 13(3): 537-557 doi: 10.3934/dcdsb.2010.13.537 +[Abstract](497) +[PDF](474.5KB)
Abstract:
Some models in population dynamics with intra-specific competition lead to integro-differential equations where the integral term corresponds to nonlocal consumption of resources [8][9]. The principal difference of such equations in comparison with traditional reaction-diffusion equation is that homogeneous in space solutions can lose their stability resulting in emergence of spatial or spatio-temporal structures [4]. We study the existence and global bifurcations of such structures. In the case of unbounded domains, transition between stationary solutions can be observed resulting in propagation of generalized travelling waves (GTW). GTWs are introduced in [18] for reaction-diffusion systems as global in time propagating solutions. In this work their existence and properties are studied for the integro-differential equation. Similar to the reaction-diffusion equation in the monostable case, we prove the existence of generalized travelling waves for all values of the speed greater or equal to the minimal one. We illustrate these results by numerical simulations in one and two space dimensions and observe a variety of structures of GTWs.
2010, 13(3): 559-575 doi: 10.3934/dcdsb.2010.13.559 +[Abstract](427) +[PDF](233.9KB)
Abstract:
Recently, we derived a lattice model for a single species with stage structure in a two-dimensional patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay (IMA J. Appl. Math., 73 (2008), 592-618.). The important feature of the model is the reflection of the joint effect of the diffusion dynamics, the nonlocal delayed effect and the direction of propagation. In this paper we study the asymptotic stability of traveling wavefronts of this model when the immature population is not mobile. Under the assumption that the birth function satisfies monostable condition, we prove that the traveling wavefront is exponentially stable by means of weighted energy method, when the initial perturbation around the wave is suitably small in a weighted norm. The exponential convergent rate is also obtained.
2010, 13(3): 577-591 doi: 10.3934/dcdsb.2010.13.577 +[Abstract](304) +[PDF](175.8KB)
Abstract:
The main objective of this article is to study dynamic of the three-dimensional Boussinesq equations with the periodic boundary condition.We prove that when the Rayleigh number $R$ crosses the first critical Rayleigh number $R_c$, the Rayleigh-Bénard problem bifurcates from the basic state to an global attractor $\Sigma$, which is homeomorphic to $S^3$.
2010, 13(3): 593-608 doi: 10.3934/dcdsb.2010.13.593 +[Abstract](339) +[PDF](216.7KB)
Abstract:
We are concerned with the Cauchy problem for a viscous shallow water system with a third-order surface-tension term. The global existence and uniqueness of the solution in the space of Besov type is shown for the initial data close to a constant equilibrium state away from the vacuum by using the Friedrich's regularization and compactness arguments.
2010, 13(3): 609-622 doi: 10.3934/dcdsb.2010.13.609 +[Abstract](448) +[PDF](174.7KB)
Abstract:
In this paper, we consider the second-order nonlinear dynamic equation

$(p(t)y^{\Delta}(t))^{\Delta}+f(t, y^{\sigma})g(p(t)y^{\Delta})=0,$

on a time scale $\mathbb{T}$. Our goal is to establish necessary and sufficient conditions for the existence of certain types of solutions of this dynamic equation. We apply results from the theory of lower and upper solutions for related dynamic equations and use several results from calculus.

Jibin Li and
2010, 13(3): 623-631 doi: 10.3934/dcdsb.2010.13.623 +[Abstract](508) +[PDF](184.9KB)
Abstract:
The paper is devoted to four kinds of fifth-order nonlinear wave equations including the Caudrey-Dodd-Gibbon equation, Kupershmidt equation, Kaup-Kupershmidt equation and Sawada-Kotera equation. The exact soliton solution and quasi-periodic solutions are found by using Cosgrove's work and the method of dynamical systems. The geometrical explanations of these solutions are also discussed. To guarantee the existence of the above solutions, the parameter conditions are determined.
2010, 13(3): 633-646 doi: 10.3934/dcdsb.2010.13.633 +[Abstract](390) +[PDF](965.9KB)
Abstract:
In this paper, we investigate the traveling wave solutions of $K(m, n)$ equation $u_t+a(u^m)_{x}+(u^n)_{x x x}=0$ by using the bifurcation method and numerical simulation approach of dynamical systems. We obtain some new results as follows: (i) For $K(2, 2)$ equation, we extend the expressions of the smooth periodic wave solutions and obtain a new solution, the periodic-cusp wave solution. Further, we demonstrate that the periodic-cusp wave solution may become the peakon wave solution. (ii) For $K(3, 2)$ equation, we extend the expression of the elliptic smooth periodic wave solution and obtain a new solution, the elliptic periodic-blow-up solution. From the limit forms of the two solutions, we get other three types of new solutions, the smooth solitary wave solutions, the hyperbolic 1-blow-up solutions and the trigonometric periodic-blow-up solutions. (iii) For $K(4, 2)$ equation, we construct two new solutions, the 1-blow-up and 2-blow-up solutions.
2010, 13(3): 647-664 doi: 10.3934/dcdsb.2010.13.647 +[Abstract](368) +[PDF](456.3KB)
Abstract:
This paper is concerned with time periodic solutions of Hamilton-Jacobi equations in which the hamiltonian is increasing wrt to the unknown variable. When the uniqueness of the periodic solution is not guaranteed, we define a notion of extremal solution and propose two different ways to attain it, together with the corresponding numerical simulations. In the course of the analysis, the ode case, where we show that things are rather explicit, is also visited.
2010, 13(3): 665-684 doi: 10.3934/dcdsb.2010.13.665 +[Abstract](594) +[PDF](645.2KB)
Abstract:
In this article, a fully discrete finite element method is considered for the viscoelastic fluid motion equations arising in the two-dimensional Oldroyd model. A finite element method is proposed for the spatial discretization and the time discretization is based on the backward Euler scheme. Moreover, the stability and optimal error estimates in the $L^2$- and $H^1$-norms for the velocity and $L^2$-norm for the pressure are derived for all time $t>0.$ Finally, some numerical experiments are shown to verify the theoretical predictions.
2010, 13(3): 685-708 doi: 10.3934/dcdsb.2010.13.685 +[Abstract](377) +[PDF](299.2KB)
Abstract:
In this paper, we introduce an efficient Legendre-Gauss collocation method for solving nonlinear delay differential equations with variable delay. We analyze the convergence of the single-step and multi-domain versions of the proposed method, and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner. We also make numerical comparison with other methods.
2010, 13(3): 709-728 doi: 10.3934/dcdsb.2010.13.709 +[Abstract](354) +[PDF](242.6KB)
Abstract:
This paper is concerned with monotone traveling wave solutions of reaction-diffusion systems with spatio-temporal delay. Our approach is to use a new monotone iteration scheme based on a lower solution in the set of the profiles. The smoothness of upper and lower solutions is not required in this paper. We will apply our results to Nicholson's blowflies systems with non-monotone birth functions and show that the systems admit traveling wave solutions connecting two spatially homogeneous equilibria and the wave shape is monotone. Due to the biological realism, the positivity of the monotone traveling wave solutions can be directly obtained by the construction of suitable upper-lower solutions.

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