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Communications on Pure & Applied Analysis

2011 , Volume 10 , Issue 5

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Preface
Zhaosheng Feng and  Wei Feng
2011, 10(5): i-ii doi: 10.3934/cpaa.2011.10.5i +[Abstract](18) +[PDF](104.4KB)
Abstract:
This issue of Communications on Pure and Applied Analysis, comprises a collection in the general area of nonlinear systems and analysis, and related applications in mathematical biology and engineering. During the past few decades people have seen an enormous growth of the applicability of dynamical systems and the new developments of related dynamical concepts. This has been driven by modern computer power as well as by the discovery of advanced mathematical techniques. Scientists in all disciplines have come to realize the power and beauty of the geometric and qualitative techniques developed during this period. More importantly, they have been able to apply these techniques to a various nonlinear problems ranging from physics and engineering to biology and ecology, from the smallest scales of theoretical particle physics up to the largest scales of cosmic structure. The results have been truly exciting: systems which once seemed completely intractable from an analytical point of view can now be studied geometrically and qualitatively. Chaotic and random behavior of solutions of various systems is now understood to be an inherent feature of many nonlinear systems, and the geometric and numerical methods developed over the past few decades contributed significantly in those areas.
$H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains
Antonio Vitolo
2011, 10(5): 1315-1329 doi: 10.3934/cpaa.2011.10.1315 +[Abstract](27) +[PDF](379.0KB)
Abstract:
In this paper we consider estimates of the Raleigh quotient and in general of the $H^{1,p}$-eigenvalue in quasicylindrical domains. Then we apply the results to obtain, by variational methods, existence and uniqueness of weak solutions of the Dirichlet problem for second-order elliptic equations in divergent form. For such solutions global boundedness estimates have been also established.
Existence of chaos in weakly quasilinear systems
Y. Charles Li
2011, 10(5): 1331-1344 doi: 10.3934/cpaa.2011.10.1331 +[Abstract](34) +[PDF](357.8KB)
Abstract:
The aim of this article is twofold: (1). develop a strategy to prove the existence of chaos in weakly quasilinear systems, (2). strengthen the existing results on chaos in partial differential equations. First, we study a sine-Gordon equation containing weakly quasilinear terms, and existence of chaos is proved. Then, we study a Ginzburg-Landau equation containing weakly quasilinear terms, and existence of chaos is proved under generic conditions. Finally, in the Appendix, we prove the existence of chaos in a reaction-diffusion equation.
Exterior differential systems and prolongations for three important nonlinear partial differential equations
Paul Bracken
2011, 10(5): 1345-1360 doi: 10.3934/cpaa.2011.10.1345 +[Abstract](40) +[PDF](329.9KB)
Abstract:
Partial differential systems which have applications to water waves will be formulated as exterior differential systems. A prolongation structure is determined for each of the equations. The formalism for studying prolongations is reviewed and the prolongation equations are solved for each equation. One of these differential systems includes the Camassa-Holm and Degasperis-Procesi equations as special cases. The formulation of conservation laws for each of the systems introduced is discussed and a single example for each is given. It is shown how a Bäcklund transformation for the last case can be obtained using the prolongation results.
Stability of linear differential equations with a distributed delay
Leonid Berezansky and  Elena Braverman
2011, 10(5): 1361-1375 doi: 10.3934/cpaa.2011.10.1361 +[Abstract](42) +[PDF](397.9KB)
Abstract:
We present some new stability results for the scalar linear equation with a distributed delay

$\dot{x}(t) + \sum_{k=1}^m \int_{h_k(t)}^t x(s) d_s R_k(t,s) =0, h_k(t)\leq t,$ su$p_{t\geq 0}(t-h_k(t))<\infty,$

where the functions involved in the equation are not required to be continuous.
The results are applied to integro-differential equations, equations with several concentrated delays and equations of a mixed type.

Duffing--van der Pol--type oscillator system and its first integrals
Zhaosheng Feng , Guangyue Gao and  Jing Cui
2011, 10(5): 1377-1391 doi: 10.3934/cpaa.2011.10.1377 +[Abstract](42) +[PDF](366.3KB)
Abstract:
In this paper, under certain parametric conditions we are concerned with the first integrals of the Duffing--van der Pol--type oscillator system, which include the van der Pol oscillator and the damped Duffing oscillator etc as particular cases. We apply the Lie symmetry method to find two nontrivial infinitesimal generators and use them to construct canonical variables. Through the inverse transformations we obtain the first integrals of the original oscillator system under the given parametric conditions, and some particular cases such as the damped Duffing equation and the van der Pol oscillator system are discussed accordingly.
Periodic solutions for $p$-Laplacian systems of Liénard-type
Wenbin Liu and  Zhaosheng Feng
2011, 10(5): 1393-1400 doi: 10.3934/cpaa.2011.10.1393 +[Abstract](30) +[PDF](334.8KB)
Abstract:
In this paper, we study the existence of periodic solutions for $n-$dimensional $p$-Laplacian systems by means of the topological degree theory. Sufficient conditions of the existence of periodic solutions for $n-$dimensional $p$-Laplacian systems of Liénard-type are presented.
Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential
D. Motreanu , V. V. Motreanu and  Nikolaos S. Papageorgiou
2011, 10(5): 1401-1414 doi: 10.3934/cpaa.2011.10.1401 +[Abstract](41) +[PDF](364.1KB)
Abstract:
A nonautonomous second order system with a nonsmooth potential is studied. It is assumed that the system is asymptotically linear at infinity and resonant (both at infinity and at the origin), with respect to the zero eigenvalue. Also, it is assumed that the linearization of the system is indefinite. Using a nonsmooth variant of the reduction method and the local linking theorem, we show that the system has at least two nontrivial solutions.
Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems
Yuncheng You
2011, 10(5): 1415-1445 doi: 10.3934/cpaa.2011.10.1415 +[Abstract](46) +[PDF](542.3KB)
Abstract:
In this paper we prove the existence of a global attractor, an $(H,E)$ global attractor, and an exponential attractor for the cubic autocatalytic reaction-diffusion systems represented by the reversible Gray-Scott equations. The two pairs of oppositely signed nonlinear terms feature the challenge in conducting various estimates. A new rescaling and grouping estimation method is introduced and combined with the other approaches to achieve the proof of dissipation, asymptotic compactness, and discrete squeezing property in all the stages.
Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam
Emine Kaya , Eugenio Aulisa , Akif Ibragimov and  Padmanabhan Seshaiyer
2011, 10(5): 1447-1462 doi: 10.3934/cpaa.2011.10.1447 +[Abstract](43) +[PDF](597.1KB)
Abstract:
In this work we consider the dynamical response of a non-linear beam with viscous damping, perturbed in both the transverse and axial directions. The system is modeled using coupled non-linear momentum equations for the axial and transverse displacements. In particular we show that for a class of boundary conditions (beam clamped at the extremes) and uniformly distributed load, there exists a non-uniform equilibrium state. Different models of damping are considered: first, third and fifth order dissipation terms. We show that in all cases in the presence of the damping forces, the excited beam is stable near the equilibrium for any perturbation. An energy estimate approach is used in order to identify the space in which the solution of the perturbed system is stable.
Global attractors of reaction-diffusion systems modeling food chain populations with delays
Wei Feng , C. V. Pao and  Xin Lu
2011, 10(5): 1463-1478 doi: 10.3934/cpaa.2011.10.1463 +[Abstract](43) +[PDF](749.4KB)
Abstract:
In this paper, we study a reaction-diffusion system modeling the population dynamics of a four-species food chain with time delays. Under Dirichlet and Neumann boundary conditions, we discuss the existence of a positive global attractor which demonstrates the presence of a positive steady state and the permanence effect in the ecological system. Sufficient conditions on the interaction rates are given to ensure the persistence of all species in the food chain. For the case of Neumann boundary condition, we further obtain the uniqueness of a positive steady state, and in such case the density functions converge uniformly to a constant solution. Numerical simulations of the food-chain models are also given to demonstrate and compare the asymptotic behavior of the time-dependent density functions.
On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis
Sze-Bi Hsu and  Feng-Bin Wang
2011, 10(5): 1479-1501 doi: 10.3934/cpaa.2011.10.1479 +[Abstract](29) +[PDF](495.1KB)
Abstract:
In this paper we construct a mathematical model of two microbial populations competing for a single-limited nutrient with internal storage in an unstirred chemostat. First we establish the existence and uniqueness of steady-state solutions for the single population. The conditions for the coexistence of steady states are determined. Techniques include the maximum principle, theory of bifurcation and degree theory in cones.
A particle method and numerical study of a quasilinear partial differential equation
Roberto Camassa , Pao-Hsiung Chiu , Long Lee and  W.-H. Sheu
2011, 10(5): 1503-1515 doi: 10.3934/cpaa.2011.10.1503 +[Abstract](32) +[PDF](640.8KB)
Abstract:
We present a particle method for studying a quasilinear partial differential equation (PDE) in a class proposed for the regularization of the Hopf (inviscid Burger) equation via nonlinear dispersion-like terms. These are obtained in an advection equation by coupling the advecting field to the advected one through a Helmholtz operator. Solutions of this PDE are "regularized" in the sense that the additional terms generated by the coupling prevent solution multivaluedness from occurring. We propose a particle algorithm to solve the quasilinear PDE. "Particles" in this algorithm travel along characteristic curves of the equation, and their positions and momenta determine the solution of the PDE. The algorithm follows the particle trajectories by integrating a pair of integro-differential equations that govern the evolution of particle positions and momenta. We introduce a fast summation algorithm that reduces the computational cost from $O(N^2)$ to $O(N)$, where $N$ is the number of particles, and illustrate the relation between dynamics of the momentum-like characteristic variable and the behavior of the solution of the PDE.
Structural parameter optimization of linear elastic systems
David Russell
2011, 10(5): 1517-1536 doi: 10.3934/cpaa.2011.10.1517 +[Abstract](28) +[PDF](548.9KB)
Abstract:
The design of elastic structures to optimize strength and economy of materials is a fundamental problem in structural engineering and related areas of applied mathematics. In this article we explore a finite dimensional framework for approximate solution of such design problems based on linear elasticity with a range of elastic coefficients assumed available as design parameters. Solution methods for related optimization problems based on the matrix trace norm are suggested and analyzed, providing existence and uniqueness theorems. Results of computations for sample problems are presented and compared with parallel results in the literature based on other approaches.
Fluctuation and extinction dynamics in host-microparasite systems
Kaifa Wang and  Yang Kuang
2011, 10(5): 1537-1548 doi: 10.3934/cpaa.2011.10.1537 +[Abstract](32) +[PDF](457.7KB)
Abstract:
Although experimental and observational studies have shown that microparasites can induce the deterministic reduction, fluctuation and extinction scenarios for its host population, most existing host-parasite interaction models fail to produce such rich dynamical behaviors simultaneously. We explore the effects of explicit dynamics of parasites under logistic host growth and different infection rate function. Our results show that the explicit dynamics of parasites and standard incidence function can induce the host density fluctuation and extinction scenario in the case of logistic host growth.

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