Search

Abstract: Let X be a Banach space whose dual space is uniformly convex. We demonstrate that, for any demicontinuous, weakly Nagumo, k-pseudo-contractive mapping with closed domain, weakly generates a semigroup on . In this paper, we project the consequences of this result on fixed point theory. In particular, we show that if (id est, if T is strongly pseudo-contractive), then T has a unique fixed point. This implies that, if T is pseudo-contractive () and is closed, bounded, and convex, then T has at least one fixed point. Consequently, any demicontinuous pseudo-contractive mapping (for an appropriate C) has a fixed point, which has been an important open question in fixed point theory for quite some time. In a subsequent paper, we explore the consequences of the semigroup result on the existence of solutions to certain partial differential equations. The semigroup result directly implies the existence of unique global solutions to time evolution equations of the form where A is a combination of derivatives. The fixed point results from this paper imply the existence of solutions to partial differential equations of the form . [Copyright &y& Elsevier]

Abstract: In this paper, we study the stability of a single transonic shock wave solution to the hyperbolic conservation laws with a resonant moving source. Compared with the previous results [W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098; T.P. Liu, Nonlinear stability and instability of transonic flows through a nozzle, Comm. Math. Phys. 83 (2) (1982) 243–260] on this stability problem, in this paper, the transonic ith shock is assumed to be relatively strong and stable in the sense of Majda. Then the framework of [M. Lewicka, stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (4) (2000) 1515–1537; M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (5) (2001) 1094–1116 (electronic)] can be applied. A new criterion is obtained to test whether such a shock is time asymptotically stable or not. And by constructing the Liu–Yang functional, one can prove the stability of the shock under the stability condition. This is an extension of the result [S.-Y. Ha, T. Yang, stability for systems of hyperbolic conservation laws with a resonant moving source, SIAM J. Math. Anal. 34 (5) (2003) 1226–1251 (electronic); W.-C. Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (9) (1999) 1075–1098] to a more general case. [Copyright &y& Elsevier]

Abstract: To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties. [Copyright &y& Elsevier]

Abstract: In a previous paper we introduced various definitions of stability and instability for non-autonomous differential equations, and applied these to investigate the bifurcations in some simple models. In this paper we present a more systematic theory of local bifurcations in scalar non-autonomous equations. [Copyright &y& Elsevier]

In this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point. We prove the existence of many complicated dynamical objects for a large class of non-autonomous second order equations, ranging from attractive quasi-periodic torus to Newhouse sinks and Hénon-like attractors, and to rank one attractors with SRB measures and full stochastic behavior. This theory enables us to apply rigorously many profound dynamics theories on non-uniformly hyperbolic maps developed in the last forty years, including the Newhouse theory, the theory of SRB measures, the theory of Hénon-like attractors and the theory of rank one attractors, to the analysis of the strange attractors in a periodically perturbed Duffing equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we shall estimate the growing speed for higher Sobolev norms of the solutions to Schrödinger equations on Riemannian manifolds (), under some bilinear Strichartz estimate assumptions. [Copyright &y& Elsevier]

Abstract: We are concerned with the existence of bound states and ground states of the following nonlinear Schrödinger equation where the potential may vanish at infinity, is asymptotically linear at infinity, that is, as . For this kind of potential, it seems difficult to find solutions in , i.e. bound states of (0.1). If and with , Ambrosetti, Felli and Malchiodi [A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. 7 (2005) 117–144] showed that (0.1) has a solution in and (0.1) has no ground states if p is out of the above range. In this paper, we are interested in what happens if is asymptotically linear. Under appropriate assumptions on K, we prove that (0.1) has a bound state and a ground state. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew-product flow associated with a nonautonomous (random) competitive Kolmogorov system. [Copyright &y& Elsevier]

Abstract: Consider in this paper a linear skew-product system where or , and is a topological dynamical system on a compact metrizable space W, and where satisfies the cocycle condition based on θ and is continuously differentiable in t if . We show that ‘semi λ-exponential dichotomy’ of implies ‘λ-exponential dichotomy.’ Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ-stable direction and if is constant a.e., then Θ is almost λ-exponentially dichotomous. To prove this, we first use Liao''s spectrum theorem, which gives integral expression of the Lyapunov exponents, and then use the semi-uniform ergodic theorem by Sturman and Stark, which allows one to derive uniform estimates from nonuniform ones. As a consequence, we obtain the open-and-dense hyperbolicity of eventual -cocycles based on a uniquely ergodic endomorphism, and of -cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively. On the other hand, in the sense of -topology we obtain the density of -cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition. [Copyright &y& Elsevier]

Abstract: We construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder, which is close to a uniform transonic shock wave. In other words, suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily, if we perturbed the supersonic incoming flow and the infinite nozzle a little bit, we can obtain a transonic wave near the uniform one. As a consequence, we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall. Based on the theory of [G.Q. Chen, M. Feldman, Existence and stability of multi-dimensional transonic flows through an infinite nozzle of arbitrary cross-sections, Arch. Ration. Mech. Anal. 184 (2007) 185–242], the crucial parts of this paper are to derive the uniform Schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder. [Copyright &y& Elsevier]

Abstract: In this paper we consider the stationary Navier–Stokes equations in a bounded domain with a variable viscosity. We prove that one can uniquely determine the viscosity function from the knowledge of boundary data. [Copyright &y& Elsevier]

Abstract: In this paper we study the limit cycles of the Liénard differential system of the form , or its equivalent system , . We provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles. [Copyright &y& Elsevier]

Abstract: In this paper a half space problem for the one-dimensional Boltzmann equation with specular reflective boundary condition is investigated. It is shown that the solution of the Boltzmann equation time-asymptotically converges to a global Maxwellian under some initial conditions. Furthermore, a time-decay rate is also obtained. [Copyright &y& Elsevier]

Abstract: For finite-dimensional bifurcation problems, it is well known that it is possible to compute normal forms which possess nice symmetry properties. Oftentimes, these symmetries may allow for a partial decoupling of the normal form into a so-called “radial” part and an “angular” part. Analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its unfoldings. In this paper, we are interested in the case where such bifurcations occur in retarded functional differential equations, and we revisit the realizability and restrictions problem for the class of radial equations by nonlinear delay-differential equations. Our analysis allows us to recover and considerably generalize recent results by Faria and Magalhães [T. Faria, L.T. Magalhães, Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities, J. Dynam. Differential Equations 8 (1996) 35–70] and by Buono and Bélair [P.-L. Buono, J. Bélair, Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differential Equations 189 (2003) 234–266]. [Copyright &y& Elsevier]

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 study the asymptotic behaviour as of the radial solution of the problem where is an annulus of , . [Copyright &y& Elsevier]

Abstract: In this paper, we study mathematical properties of an integro-differential equation that arises as a particular limit case in the study of individual cell-based model. We obtain global well-posedness for some classes of interaction potentials and finite time blow-up for others. The existence of space homogeneous steady states as well as long-time asymptotics for the solutions of the problem is also discussed. [Copyright &y& Elsevier]

Abstract: In this paper we consider the differential inclusion problemwhere is radially symmetric, and stands for the generalized gradient of a locally Lipschitz function . Under suitable oscillatory assumptions on the potential at zero or at infinity, we show the existence of infinitely many, radially symmetric solutions of (DI). No symmetry requirement on is needed. Our approach is based on a non-smooth Ricceri-type variational principle, developed by Marano and Motreanu (J. Differential Equations 182 (2002) 108–120). [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows. [Copyright &y& Elsevier]

Abstract: This work is a continuation of our previous work (Kong, J. Differential Equations 188 (2003) 242–271) “Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities”. In the present paper we prove the global structure instability of the Lax''s Riemann solution , containing rarefaction waves, of general quasilinear hyperbolic system of conservation laws. Combining the results in (Kong, 2003), we prove that the Lax''s Riemann solution of general quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a class of pseudo monotone semiflows, which only enjoy some weak monotonicity properties and are defined on product-ordered topological spaces. Under certain conditions, several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium, which improve and extend some corresponding results already known. Some applications to delay differential equations are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we study the local existence and uniqueness of classical solutions to a wide class of systems of chemotaxis equations. These systems are essentially quasi-linear strongly coupled partial differential equations. We also study the maximal interval of existence in time of solutions. The results are illustrated in application to a number of partial differential equation models arising in biology. [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]

In this paper, the detailed asymptotic behaviour of the solutions of a kinetic equation for quantum particles is studied. It is shown that this behaviour is sensitive not only to the total mass of the initial data but also to its precise behaviour near the origin. In some cases, solutions develop a Dirac mass at the origin for long times in a self-similar manner that is analysed in detail. [Copyright &y& Elsevier]