On Relaxation Hyperbolic Systems Violating the Shizuta–Kawashima Condition

In this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta–Kawashima ([SK]) coupling condition. This investigation is motivated by the fact that this condition is not satisfied by various physical systems, and almost all the time in several space dimensions. First, we explore the role of entropy functionals around equilibrium solutions, which may not be constant, proposing a stability condition for such solutions. Then we find strictly dissipative entropy functions for one dimensional 2 × 2 systems which violate the [SK] condition. Finally, we prove the existence of global smooth solutions for a class of systems such that condition [SK] does not hold, but which are linearly degenerated in the non-dissipative directions.     

Coagulation-Fragmentation Processes

We study the well-posedness of coagulation-fragmentation models with diffusion for large systems of particles. The continuous and the discrete case are considered simultaneously. In the discrete situation we are concerned with a countable system of coupled reaction-diffusion equations, whereas the continuous case amounts to an uncountable system of such equations. These problems can be handled by interpreting them as abstract vector-valued parabolic evolution equations, where the dependent variables take values in infinite-dimensional Banach spaces. Given suitable assumptions, we prove existence and uniqueness in the class of volume preserving solutions. We also derive sufficient conditions for global existence. Accepted: (August 18, 1999)     

On global solutions of functional differential equations

We construct a system of linear differential equations all solutions of which are global solutions of a system of functional differential equations. We substantiate the existence of this system and examine some properties of its solutions.     

Global Solutions of Systems of Nonlinear Impulsive Volterra Integral Equations in Banach Spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｔｈｅｏｒｙｏｆｉｍｐｕｌｓｉｖｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｈａｓｂｅｅｎｅｍｅｒｇｉｎｇａｓａｎｉｍｐｏｒｔａｎｔａｒｅａｏｆｉｎｖｅｓｔｉｇａｔｉｏｎｓｉｎｒｅｃｅｎｔｙｅａｒｓ (ｓｅｅ [1 ] ) .Ｉｎｐａｐｅｒ [2 ] ,ＧｕｏＤａ＿ｊｕｎｅｓｔａｂｌｉｓｈｅｄｓｏｍｅｅｘｉｓｔｅｎｃｅｔｈｅｏｒｅｍｓｏｆｅｘｔｒｅｍａｌｓｏｌｕｔｉｏｎｓｆｏｒｎｏｎｌｉｎｅａｒｉｍｐｕｌｓｉｖｅＶｏｌｔｅｒｒａｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎｓｏｎａｆｉｎｉｔｅｉｎ…     

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.     

Equilibria, Connecting Orbits and a Priori Bounds for Semilinear Parabolic Equations with Nonlinear Boundary Conditions

We consider a semilinear parabolic equation with a nonlinear non-dissipative boundary condition. In the one-dimensional case we describe bifurcation diagrams for positive and sign-changing equilibria and connecting orbits between these equilibria. We also show that the number of sign-changing stationary solutions strongly depends on the spatial dimension. The results are based on new a priori estimates of global solutions.     

Asymptotic behaviors of solutions for dissipative quantum Zakharov equations

The dissipative quantum Zakharov equations are mainly studied. The existence and uniqueness of the solutions for the dissipative quantum Zakharov equations are proved by the standard Galerkin approximation method on the basis of a priori estimate. Meanwhile, the asymptotic behavior of solutions and the global attractor which is constructed in the energy space equipped with the weak topology are also investigated.     

To the boundary and back—a numerical study

This study identifies the key parameters upon which energy absorption at artificial boundaries depends. A thorough numerical study is presented, of typical reflections from open computational boundaries, for problems governed by hyperbolic systems of equations. The emphasis is on systems, where it is often the combination of all boundary procedures that determine the quality of boundary treatment. We study dissipative numerical models which have so far not been analysed to the same extent as non-dissipative models and employ a Law-Wendroff-type scheme as a prototype. While it is widely accepted that dissipative models tend to give fewer problems than non-dissipative ones, we show a variety of cases where substantial reflections do occur even in ID and quasi-ID set-ups, where theory predicts best results. This can partly be explained by the vanishing of dissipation in the far field. Group velocity analysis, justifiable on the grounds of weak dissipation, predicts a pathological behaviour which is confirmed by numerical experiments. We demonstrate strong focusing of asymptotic errors generated at the artificial boundary. Internal reflections due to slowly expanding grids are shown for non-linear systems. The need for high-frequency boundary conditions naturally arises and combined low-high-frequency boundary recipes following Higdon, Vichnevetsky and Pariser are adapted to systems and tested. Partial cures are also discussed, mainly in terms of pointing out their theoretically limited potential.     

Global existence of solutions to the periodic initial value problems for two-dimensional Newton-Boussinesq equations

A class of periodic initial value problems for two-dimensional Newton- Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved.     

Some existence theorems of common and coincidence solutions for a class of systems of functional equations arising in dynamic programming

Some existence theorems of common and coincidence solutions for a class of moregeneral systems of functional equations arising in dynamic programming are shown.Theresults presented in this paper not only contain the corresponding results of[6,7]as specialcases,but also give an existence theorem of solutions for a class of functional equationssuggested by Wang recently.     

Reaction-diffusion systems in nonconvex domains: Invariant manifold and reduced form

A study is made of systems of weakly coupled, semilinear, parabolic equations, namely reaction-diffusion systems, subject to the homogeneous Neumann boundary conditions in parametrized nonconvex domains inR ². It is assumed that the domain approaches a union of two disjoint domains as the parameter varies. Under some conditions the long-time behavior of bounded solutions is discussed and the existence of a finite-dimensional invariant manifold is shown, together with its attractivity. This manifold is represented by a graph of some function defined in a possibly large bounded region of the phase space, and the original system is reduced to an ODE system on it. Since an explicit form of the reduced ODE system is given, its dynamics can be studied in detail, which in turn reveals the global dynamics of the original reaction-diffusion system. One can thereby prove, among other things, the existence of asymptotically stable equilibrium solutions of the original system having large spatial inhomogeneity. The existence and stability of a spatially inhomogeneous periodic solution of large amplitude are also discussed.     

Solutions for a system of nonlinear random integral and differential equations under weak topology

I.IntroductionTheexistenceandcomparisonresultsofsolutionsfornonlinearVolterraintegralequationsinstrongtopologyofBanachspaceshavebeenobtainedbyVaughnI"'"],LakshmikanthamIl3]andLakshmikantham-Leela114l.TheexistenceresultsofweaksolutionsfortheCauchyproblemof…     

Bifurcations of travelling wave solutions for a coupled nonlinear wave system

Introduction FangShaomeiandGuoBoling[1]consideredthefollowingtimeperiodicproblemof dampedcouplednonlinearwaveequations:ut f(u)x-αuxx βuxxx 2vvx=G1(u,v) h1(x),vt-γvxx 2(uv)x g(v)x=G2(u,v) h2(x),(1)whereα,β,γareconstants,andγ>0,β≠0.Undertheperiodicboundaryconditions,the authorsobtainedtheuniqueexistenceofstrongsolutionsfortheabovesystem.InthispaperweshallconsiderbifurcationbehaviorofthetravellingwavesolutionsofEq.(1)inthecaseGi(u,v)≡0,hi(u,v)≡0(i=1,2).Letξ=x-ct,u=u(x-ct),where cis…     

Behaviors of Solutions for a Singular Prey–Predator Model and its Shadow System

We study the asymptotic behaviors and quenching of the solutions for a two-component system of reaction–diffusion equations modeling prey–predator interactions in an insular environment. First, we give a global existence result for the solutions to the corresponding shadow system. Then, by constructing some suitable Lyapunov functionals, we characterize the asymptotic behaviors of global solutions to the shadow system. Also, we give a finite time quenching result for the shadow system. Finally, some global existence results for the original reaction–diffusion system are given.     

Long time behavior of solutions to coupled Burgers-complex Ginzbury-Landau (Burgers-CGL) equations for flames governed by sequential reaction

This paper studies the existence and long time behavior of the solutions to the coupled Burgers-complex Ginzburg-Landau （Burgers-CGL） equations, which are derived from the nonlinear evolution of the coupled long-scale oscillatory and monotonic instabilities of a uniformly propagating combustion wave governed by a sequential chem- ical reaction, having two flame fronts corresponding to two reaction zones with a finite separation distance between them. This paper firstly shows the existence of the global solutions to these coupled equations via subtle transforms, delicate a priori estimates and a so-called continuity method, then prove the existence of the global attractor and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the attractor.     

Unbounded Branches of Classical Injective Solutions to the Forced Displacement Problem in Nonlinear Elastostatics

We consider classical solutions of the equilibrium equations of nonlinear elastostatics under prescribed boundary displacements and body forces, both of which depend on a loading parameter. We employ recent results on global continuation [HS] that establish general existence of global continua of solutions in nonlinear elastostatics. Under reasonable hypotheses on the stored energy function we prove the existence of an unbounded branch of globally injective solutions to the boundary value problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

SOLUTIONS FOR A SYSTEM OF NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS

ＳＯＬＵＴＩＯＮＳＦＯＲＡＳＹＳＴＥＭＯＦＮＯＮＬＩＮＥＡＲＲＡＮＤＯＭＩＮＴＥＧＲＡＬＡＮＤＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＤｉｎｇＸｉｅｐｉｎｇ（丁协平）ＷａｎｇＦａｎ（王凡）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｓｉｃｈ...     

Bifurcations and chaos of an inclined cable

The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kova?i? and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.     

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H ¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.     

Traction problem in electro-elasticity and bifurcation theory

This paper is the first of a series of two. It will deal with the problem of static traction problem with minor deformations for a material which is governed by the electrostriction phenomenon. Two approaches to this problem will be described. We can consider either the equilibrium equations which are naturally non-linear, or the equations after linearization. The linearization of equations must be done near a natural state. Locally, under some conditions, we can establish the existence and the uniqueness of the solutions. We use the local theorem of implicit functions. The problem can be approached more globally. If we consider the non-linear equations, we can use a natural principle of these equations: the independence of the choice of the observer, also known as objectivity property. This property makes it possible for us to take into account an action of the rotations group of the Euclidean space, and consequently to take into account all the trivial solutions. It is then possible to prove within the space of all configurations the existence of the non-linear equations solutions and to find their number.This work presents a thorough and detailed approach to a non-linear theory, the geometric arguments of which make it possible for us to prove the existence of all the solutions and to study their stability in the aggregate; this last aspect will be developed in the second paper. Not only can this theory anticipate the eventual existence of a stable solution, it can also anticipate that an unstable solution in terms of the elasticity can, thanks to the effect of an electric field, become stable in terms of the electro-elasticity.