Search

Abstract: In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if is a domain, for some and verifies where , and with some positive constants B and M, then there exists a constant independent of ε such that . [Copyright &y& Elsevier]

Abstract: In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45] to prove that any large solution of , , , must satisfywhereand is any smooth extension of the boundary normal section of a at , i.e.,Subsequently, stands for the outward unit normal at . Therefore, the theory can be extended to cover the general case when . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the structure of the set of radially symmetric solutions for the equationwith . Here the nonlinear term is assumed to be a smooth function of that is positive for and is equal to 0 for . Then any radial solution , of the equation is shown to be classified into one of several types according to its behavior as and . Under the assumption that is supercritical for small and is subcritical for large , we clarify the entire structure of the set of solutions of various types. The Pohozaev identity plays a crucial role in the investigation of the structure. [Copyright &y& Elsevier]

Abstract: The paper deals with polynomial Liénard equations of type , i.e. planar vector fields associated to a scalar second order differential equation , with f and g polynomials of respective degree m and n. It is shown that, besides compactifying the phase plane, or the Liénard plane, one can also compactify and desingularize the space of Liénard equations of type for each separately, by adding both singular perturbation problems and Hamiltonian perturbation problems. [Copyright &y& Elsevier]

Abstract: Let be the solution of the heat equation on subject to on . The main goal of this paper is to characterize such a nonnegative measure on that induces a bounded embedding from the Sobolev space , into the Lebesgue space , . [Copyright &y& Elsevier]

Abstract: In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541–1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342–368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction–diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Cauchy problem of the long-wave–short-wave resonance equations. By making use of a Strichartz-type inequality for the solutions, decomposing suitably the solution semigroup into a decay parts and a more regular parts, and ruling out the “vanishing” and “dichotomy” of the solutions, we prove the existence of the global attractor and the asymptotic smoothing effect of the solutions. [Copyright &y& Elsevier]

Abstract: The Euler equations with frictional force have been extensively studied. Since the Boltzmann equation is closely related to the equations of gas dynamics, we study, in this paper, the Boltzmann equation with frictional force when the external force is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro–micro decomposition for the Boltzmann equation introduced in Liu et al. [Energy method for the Boltzmann equation, Physica D 188 (3–4) (2004) 178–192] and Liu and Yu [Boltzmann equation: micro–macro-decompositions and positivity of shock profiles, Comm. Math. Phys. 246(1) (2004) 133–179] through energy estimates. [Copyright &y& Elsevier]

Abstract: This paper concerns the global existence and the large time behavior of strong and classical solutions to the two-dimensional (2D) Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the 2D Stokes approximation equations for the compressible flows together with the space-periodicity boundary condition or the no-stick boundary condition or Cauchy problem for arbitrarily large initial data. First, we prove that the density is bounded from above independent of time in all these cases. Secondly, we show that for the space-periodicity boundary condition or the no-stick boundary condition, if the initial density contains vacuum at least at one point, then the global strong (or classical) solution must blow up as time goes to infinity. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction–diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption. [Copyright &y& Elsevier]

Abstract: The asymptotic behavior of solutions of the damped compressible Euler equations is conjectured to obey to the famous porous media equations (PMES). The previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic. In present paper, we prove that the entropy weak solution with vacuum, obtained by the compensated compactness theory, converges strongly in space to the unique similarity solution of the related PME, as time goes to infinity. [Copyright &y& Elsevier]

Abstract: This paper is concerned with analyzing the limiting behavior of the least energy solutions for the Hénon equation. We study the problems in bounded domains with smooth or non–smooth boundaries, and in particular we examine the effect of the boundary on the limiting profile of the solutions. [Copyright &y& Elsevier]

Abstract: This paper is concerned with existence and stability of traveling curved fronts for the Allen–Cahn equation in the two-dimensional space. By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front. [Copyright &y& Elsevier]

Abstract: The problem of a classification of robust homoclinic cycles in low-dimensional spaces has been frequently asked in recent years. In this paper, we resume the results in and and we solve the problem in in the case of orientation-preserving group actions. [Copyright &y& Elsevier]

Elaydi and Yakubu showed that a globally asymptotically stable(GAS) periodic orbit in an autonomous difference equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous difference equations via the concept of skew-product dynamical systems. We show that for a k-periodic difference equation, if a periodic orbit of period r is GAS, then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our methods are then applied to prove a conjecture by J. Cushing and S. Henson concerning a non-autonomous Beverton–Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to study the existence of single-peaked positive solutions of the singularly perturbed elliptic equation where is a symmetric uniformly elliptic matrix and is a positive potential, possibly unbounded from above. If , then solutions concentrate at non-degenerate critical points of . [Copyright &y& Elsevier]

Abstract: In this paper, we consider the positive solution of the Cauchy problem for the equationand give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the Cauchy problem. Furthermore, the life span of solutions are also studied. [Copyright &y& Elsevier]

Abstract: This paper uses Lie symmetry group methods to study PDEs of the form We show that when the drift function f is a solution of a family of Ricatti equations, then symmetry techniques can be used to find a fundamental solution. [Copyright &y& Elsevier]

The purpose of this paper is to prove well-posedness for a problem that describes the dynamics of a set of points by means of a system of parabolic equations. It has been seen in Velázquez (Point dynamics in a singular limit of the Keller–Segel model. (1) motion of the concentration regions, SIAM J. Appl. Math., to appear) that the considered model is the limit of a singular perturbation problem for a system of the Keller–Segel type. [Copyright &y& Elsevier]

In this paper, we answer affirmatively an open problem (cf. Theorem 4′ in Ferrero and Gazzola (J. Differential Equations 177 (2001) 494): Let Ω∋0 be an open-bounded domain, Ω⊂RN(N⩾5) and assume that 0⩽μ<(N-2/2)2-(N+2/N)2, then, for all λ>0 there exists a nontrivial solution with critical level in the range (0,1/NSμN/2) for the problem -Δu-μ u/|x|2=λu+|u|2*-2u in Ω; u=0 on ∂Ω. [Copyright &y& Elsevier]

This paper is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. Our main result is that if -A is the generator of an analytic semigroup, then there exists a family of regularizing operators for such an ill-posed Cauchy problem by using the quasi-reversibility method, fractional powers and semigroups of linear operators. The applications to ill-posed partial differential equations are also given. [Copyright &y& Elsevier]

This paper deals with multifrequency slow–fast systems. It is shown that, under a suitable change of coordinates, the system can be reduced to a simple form such that slow motions are described by autonomous equations except for exponential error of perturbations. Hence, the fast and slow motions are decoupled. The Newton rapid iteration is used. In addition, for a perturbation, only the smallness condition is needed. [Copyright &y& Elsevier]

In this paper, we study the singular limit of the Porous Medium equation ut=Δum+g(x,u), as m→∞, in a bounded domain with Neumann boundary condition. [Copyright &y& Elsevier]

In this paper, we present a new technique of proving the existence of an infinite number of symmetric periodic solutions. This technique combines the covering relations method introduced by Zgliczyn´ski (J. Differential Equations 171 (2001) 173; Methods in Nonlinear Anal. 8 (1996) 169; Nonlinearity 10 (1997) 243) with fixed set iteration method described in Lamb (Reversing symmetries in dynamical systems, Ph.D. Thesis, Universiteit van Amsterdam, 1994). As an example we present a computer-assisted proof of symbolic dynamics in the Kuramoto–Sivashinsky equations. Moreover, we prove the existence of an infinite number of symmetric and an infinite number of nonsymmetric periodic solutions. [Copyright &y& Elsevier]