Some existence results for a class of degenerate semilinear elliptic systems

Using a variational approach, we investigate a class of degenerate semilinear elliptic systems with measurable, unbounded nonnegative weights, where the domain is bounded or unbounded. Some existence results are obtained.     

Global stability of the attracting set of an enzyme-catalysed reaction system

The essential feature of enzymatic reactions is a nonlinear dependency of reaction rate on metabolite concentration taking the form of saturation kinetics. Recently, it has been shown that this feature is associated with the phenomenon of “loss of system coordination” [1]. In this paper, we study a system of ordinary differential equations representing a branched biochemical system of enzyme-mediated reactions. We show that this system can become very sensitive to changes in certain maximum enzyme activities. In particular, we show that the system exhibits three distinct responses: a unique, globally-stable steady-state, large amplitude oscillations, and asymptotically unbounded solutions, with the transition between these states being almost instantaneous. It is shown that the appearance of large amplitude, stable limit cycles occurs due to a “false” bifurcation or canard explosion. The subsequent disappearance of limit cycles corresponds to the collapse of the domain of attraction of the attracting set for the system and occurs due to a global bifurcation in the flow, namely, a saddle connection. Subsequently, almost all nonnegative data become unbounded under the action of the dynamical system and correspond exactly to loss of system coordination. We discuss the relevance of these results to the possible consequences of modulating such systems.     

Global existence of solutions to a parabolic–elliptic chemotaxis system with critical degenerate diffusion

This paper is devoted to the analysis of nonnegative solutions for a degenerate parabolic–elliptic Patlak–Keller–Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, thereby completing previous results on finite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.     

Spherically Symmetric Solutions to Compressible Hydrodynamic Flow of Liquid Crystals in N Dimensions

The paper is concerned with the system modeling the compressible hydrodynamic flow of liquid crystals with radially symmetric initial data and non-negative initial density in dimension N (N ≥ 2).The au...     

Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation

Using theory of global attractors for multi-valued semiflows, we prove the existence of a global attractor for the m-semiflow generated by a parabolic equation involving the nonlinear degenerate operator in a bounded domain.     

拟线性退化抛物型方程组解的整体存在性和爆破

该文研究光滑有界区域Ω( R^N (N≥ 1) 上具有齐次Dirichlet边界条件的拟线性退化抛物型方程组 u_t-div(|▽u|^p-2 ▽u) =av^α, v_t-div(|▽v|^q-2 ▽v) =bu^β 的非负解的性质, 其中p, q>2, α, β ≥ 1, a, b> 0是常数. 该文指出上述方程组的解是否在有限时刻爆破依赖于初值、系数 a 与 b以及 αβ 和 (p-1)(q-1)之间的关系.     

A certain class of systems of “reaction-diffusion” type

The Cauchy problem with bounded, nonnegative initial function is investigated for a quasilinear, degenerate, parabolic system of two equations, containing, in general, lower terms with non-power nonlinearities. Their form is such that the first component of the solution can become unbounded after a finite time interval, provided the second component remains strictly positive. However, the latter may tend to zero and even become identically zero after a finite time. Two theorems are proved regarding the existence of a global, bounded generalized solution of the considered problem. Examples are given, attesting to the sharpness of these theorems.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 78–88, 1980.     

Structure of the global attractor for a second order strongly damped lattice system

In this paper, we consider the dynamical behavior of a second order strongly damped lattice system where the coupled operator is nonnegative definite symmetric. Firstly, we prove the existence of a global attractor, and give an upper bound of Hausdorff dimension of the global attractor, which keeps bounded for large strongly damping. Then we show that when the damping term is linear and the damping is suitable large, the system has an unbounded one-dimensional global attractor, which is a restricted horizontal curve.     

Existence and Nonexistence of Global Classical Solutions to Porous Medium and Plasma Equations with Singular Sources

Let Ω be a bounded or unbounded domain in R^n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching phenomenon and the existence of global classical solution to the above problem are established. Also, the life span of the quenching solution is estimated or evaluated for some domains.     

Global bifurcation for quasivariational inequalities of reaction-diffusion type

We consider a reaction-diffusion system with implicit unilateral boundary conditions introduced by U. Mosco. We show that global continua of stationary spatially nonhomogeneous solutions bifurcate in the domain of parameters where bifurcation in the case of classical boundary conditions is excluded. The problem is formulated as a quasivariational inequality and the proof is based on the Leray-Schauder degree.     

On a variational degenerate elliptic problem

In this paper we study a class of variational degenerate elliptic problems of the form in on , where is a bounded or unbounded domain in Received January 1999     

On 2x2 conservation laws with large data

We deal with a strictly hyperbolic system of two conservation laws in one spatial dimension. One of the eigenvalues of the system is of Temple type (rarefaction and shock curves coincide), the other eigenvalue is only required to be genuinely nonlinear.We consider the initial value problem for data of the following kind: the total variation of the Temple component is bounded, possibly large, while the total variation of the other component is small. For such data we prove global existence, uniqueness and L⊃-Lipschitz continuous dependence of solutions.AMS Subject Classification: Primary 35L65; Secondary 35D05, 35L45.     

On the Minimal Solutions of Variational Inequalities in Orlicz-Sobolev Spaces

In this paper, the author studies the existence of the minimal nonnegative solutions of some elliptic variational inequalities in Orlicz-Sobolev spaces on bounded or unbounded domains. She gets some comparison results between different solutions as tools to pass to the limit in the problems and to show the existence of the minimal solutions of the variational inequalities on bounded domains or unbounded domains. In both cases,coercive and noncoercive operators are handled. The sufficient and necessary conditions for the existence of the minimal nonnegative solution of the noncoercive variational inequality on bounded domains are established.     

一类竞争扩散系统的定态分歧与稳定性

本文研究一类竞争扩散系统,在方程所描述的模型中,两个相互竞争的物种栖息在同一有界区域内,相互制约的项是Holling-Tanner型的,在齐次Dirichlet边界条件下,应用谱分析和分歧理论的方法,研究了非负定态解的分歧及其稳定性。     

Unbounded branches of periodic solutions

We consider quasilinear ordinary differential equations of higher order that depend on a scalar parameter λ and consist of a stationary leading linear part at infinity and a periodic saturated nonlinearity with period independent of λ. We assume that the linear part becomes resonance at some λ = λ₀. If the leading nonlinear terms (positively homogeneous of zero order) are nondegenerate in a certain sense, then they determine the existence and the number of unbounded branches of periodic solutions for λ close to λ₀. We completely investigate the case of “double degeneration,” in which both the linear and the leading nonlinear terms are degenerate and the terms decaying at infinity play the key role. To take them into account, we develop a new method for the computation of exact asymptotics of the projections of bounded functional nonlinearities onto the two-dimensional resonance subspace.     

Sequential and continuum bifurcations in degenerate elliptic equations

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. In the problems we study, which do not represent Fredholm operators, we show that there is a critical parameter value at which an infinity of bifurcations occur from the trivial solution. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

     

On the two-dimensional tidal dynamics system: stationary solution and stability

ABSTRACT

In this work, we consider the two-dimensional stationary and non-stationary tidal dynamic equations and examine the asymptotic behavior of the stationary solution. We prove the existence and uniqueness of weak and strong solutions of the stationary tidal dynamic equations in bounded domains using compactness arguments. Using maximal monotonicity property of the linear and nonlinear operators, we also establish that the solvability results are even valid in unbounded domains. Later, we obtain a uniform Lyapunov stability of the steady state solution. Finally, we remark that the stationary solution is exponentially stable if we add a suitable dissipative term in the equation corresponding to the deviations of free surface with respect to the ocean bottom. This exponential stability helps us to ensure the mass conservation of the modified system, if we choose the initial data of the modified system as stationary solution.     

Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology

In this paper we will focus on a parabolic degenerate system with respect to unknown functions u and w on a bounded domain of the two dimensional Euclidean space. This system appears as a mathematical model for some biological processes. Global existence and uniqueness of a nonnegative classical Hölder continuous solution are proved. The last part of the paper is devoted to the study of the asymptotic behavior of the solutions.     

A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature

We prove a pointwise gradient bound for bounded solutions of Δu+F^′(u)=0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature.We also obtain some rigidity results when equality in the bound holds at some point.     

具有Radon测度初值的热敏电阻问题

本文证明具有有界非负Ｒａｌｏｎ测度初值的热敏电阻问题弱解整体存在。