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Abstract: In this paper, we study the existence of infinitely many periodic solutions for second order Hamiltonian systems , where is either asymptotically quadratic or superquadratic as . [Copyright &y& Elsevier]

Abstract: We consider an elliptic–hyperbolic model of phase transitions and we show that any Lax shock can be approximated by a traveling wave with a suitable choice of viscosity and capillarity. By varying viscosity and capillarity coefficients, we can cover any Lax shock which either remains in the same phase, or admits a phase transition. The argument used in this paper extends the one in our earlier works. The method relies on LaSalleʼs invariance principle and on estimating attraction region of the asymptotically stable of the associated autonomous system of differential equations. We will show that the saddle point of this system of differential equations lies on the boundary of the attraction region and that there is a trajectory leaving the saddle point and entering the attraction region. This gives us a traveling wave connecting the two states of the Lax shock. We also present numerical illustrations of traveling waves. [Copyright &y& Elsevier]

Abstract: We study the singularly perturbed state-dependent delay-differential equation which is a special case of the equation One knows that for every sufficiently small , Eq. (⁎) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as . In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size for small ε. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small ε. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the stationary problem of a prey–predator cross-diffusion system with a protection zone for the prey. We discuss the existence and non-existence of coexistence states of the two species by using the bifurcation theory. As a result, it is shown that the cross-diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive. We also study the asymptotic behavior of positive stationary solutions as the cross-diffusion coefficient of the prey tends to infinity. [Copyright &y& Elsevier]

Abstract: In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle''s invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle''s invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times. [Copyright &y& Elsevier]

Abstract: The paper deals with solutions of a differential inclusion constrained to a compact convex set Ω. Here F is a compact, possibly non-convex valued, Lipschitz continuous multifunction, whose convex closure co F satisfies a strict inward pointing condition at every boundary point . Given a reference trajectory taking values in an ε-neighborhood of Ω, we prove the existence of a second trajectory which satisfies . As shown by an earlier counterexample, this bound is sharp. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the initial boundary value problems of the system modeling the compressible viscoelastic fluid of Oldroyd type. The global in time solution is proved to exist uniquely near the equilibrium state in . [Copyright &y& Elsevier]

Abstract: We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman–Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) , and by Frederickson and Lazer (1969) . Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry''s results in Fabry (1995) , for scalar equations with double resonance with respect to the Dancer–Fučik spectrum. [Copyright &y& Elsevier]

Abstract: Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the sign of the derivative of the associated bifurcation function at a zero. In this paper we show that, for analytic systems, this result is of topological nature. This means that it is enough to impose a change of sign at the zero, without any assumption on the succesive derivatives. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter . We prove that under suitable assumptions, the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter satisfies the equation where [ABSTRACT FROM AUTHOR]

Abstract: The paper studies the longtime behavior of the Kirchhoff type equation with a strong dissipation . It proves that the related continuous semigroup possesses in the phase space with low regularity a global attractor which is connected. And an example is shown. [ABSTRACT FROM AUTHOR]

Abstract: This work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions. The word “hybrid” highlights the coexistence of continuous dynamics and discrete events. The underlying process has two components. One component describes the continuous dynamics, whereas the other is a switching process representing discrete events. One of the main features is the switching component depending on the continuous dynamics. In this paper, weak continuity is proved first. Then continuous and smooth dependence on initial data are demonstrated. In addition, it is shown that certain functions of the solutions verify a system of Kolmogorov''s backward differential equations. Moreover, rates of convergence of numerical approximation algorithms are dealt with. [Copyright &y& Elsevier]

Abstract: The subject matter of this paper concerns the asymptotic regimes for transport equations with advection fields having components of very disparate orders of magnitude. The main purpose is to derive the limit models: we justify rigorously the convergence towards these limit models and we investigate the well-posedness of them. Such asymptotic analysis arises in the magnetic confinement context, where charged particles move under the action of strong magnetic fields. In these situations we distinguish between a slow motion driven by the electric field and a fast motion around the magnetic lines. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the growth/decay rate of solutions of an abstract integral equation which frequently arises in quasilinear differential equations applying a variation-of-constants formula. These results are applicable to some abstract equations which appear in the theory of age-dependent population models and also to some quasilinear delay differential equations with bounded and unbounded delays. Examples are given to illustrate the sharpness of the results. [ABSTRACT FROM AUTHOR]

Abstract: The current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established. [Copyright &y& Elsevier]

Abstract: Consider a Navier–Stokes liquid filling the three-dimensional space exterior to a moving rigid body and subject to an external force. Using a coordinates system attached to the body, the equations of the fluid can be written in a time-independent domain, which results in a perturbed Navier–Stokes system where the extra terms depend on the velocity of the rigid body. In this paper, we consider the related whole space problem and construct a strong solution with finite kinetic energy for the corresponding steady-state equations. For this, appropriate conditions on the external force have to be imposed (for instance, that it is a function with compact support and null average) together with a smallness condition involving the viscosity of the fluid. First, a linearized version of the problem is analysed by means of the Fourier transform, and then a strong solution to the full nonlinear problem is obtained by a fixed point procedure. We also show that such a solution satisfies the energy equation and is unique within a certain class. [Copyright &y& Elsevier]

Abstract: We consider a mechanical system which is controlled by means of moving constraints. Namely, we assume that some of the coordinates can be directly assigned as functions of time by implementing frictionless constraints. This leads to a system of ODE''s whose right hand side depends quadratically on the time derivative of the control. In this paper we introduce a simplified dynamics, described by a differential inclusion. We prove that every trajectory of the differential inclusion can be uniformly approximated by a trajectory of the original system, on a sufficiently large time interval, starting at rest. Under a somewhat stronger assumption, we show this second trajectory reaches exactly the same terminal point. [Copyright &y& Elsevier]

Abstract: The compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the diffusive Leslie predator–prey model with large intrinsic predator growth rate, and investigate the change of behavior of the model when a simple protection zone for the prey is introduced. As in earlier work [Y. Du, J. Shi, A diffusive predator–prey model with a protection zone, J. Differential Equations 229 (2006) 63–91; Y. Du, X. Liang, A diffusive competition model with a protection zone, J. Differential Equations 244 (2008) 61–86] we show the existence of a critical patch size of the protection zone, determined by the first Dirichlet eigenvalue of the Laplacian over and the intrinsic growth rate of the prey, so that there is fundamental change of the dynamical behavior of the model only when is above the critical patch size. However, our research here reveals significant difference of the model''s behavior from the predator–prey model studied in [Y. Du, J. Shi, A diffusive predator–prey model with a protection zone, J. Differential Equations 229 (2006) 63–91] with the same kind of protection zone. We show that the asymptotic profile of the population distribution of the Leslie model is governed by a standard boundary blow-up problem, and classical or degenerate logistic equations. [Copyright &y& Elsevier]

Abstract: In this paper, we show the existence of the strong solutions for the coupled suspension bridge equations. Furthermore, existence of the strong global attractors is investigated using a new semigroup scheme. Since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal. [Copyright &y& Elsevier]

Abstract: This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where is a prime p or a power of a prime , and show that these systems can have a focus order ; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too. [Copyright &y& Elsevier]

Abstract: Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor, and orbits are nowhere dense. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that all directed feedback loops are positive. Several new results for cooperative systems are obtained in the process. Connections with biological models are discussed. [Copyright &y& Elsevier]

Abstract: This paper deals with existence and exponential decay of homoclinic orbits in the first-order Hamiltonian system where the Hamiltonian function is nonperiodic in and superquadratic in . With certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems. [Copyright &y& Elsevier]

Abstract: This paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three-dimensional whole space. For a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity. For this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces. Moreover we characterize the pyramidal traveling front in another way, that is, we write it as a combination of two-dimensional V-form waves on the edges. This characterization also uniquely determines a pyramidal traveling front. [Copyright &y& Elsevier]

Abstract: This paper is contributed to the elliptic equation where , , , and is a constant. We study the structure of positive radial solutions of (0.1) and obtain the uniqueness of solution decaying faster than at ∞ if μ is small enough under some assumptions on K and f, where m is the slow decay rate. [Copyright &y& Elsevier]

Abstract: In this paper we study a class of Caffarelli–Kohn–Nirenberg inequalities without restricting the pertinent parameters. In particular, we determine the values of the corresponding optimal constants and the functions that achieve them, i.e., minimizers of a suitable functional. By studying a corresponding Euler–Lagrange equation, we also determine infinitely many sign-changing solutions at higher energy levels in addition to the found ground-state solutions. [Copyright &y& Elsevier]

Abstract: In this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane. We study a strip of hyperbolas occurring in some quadratic systems. We deal with the cyclicity of the degenerate graphics from the programme, set up in [F. Dumortier, R. Roussarie, C. Rousseau, Hilbert''s 16th problem for quadratic vector fields, J. Differential Equations 110 (1994) 86–133], to solve the finiteness part of Hilbert''s 16th problem for quadratic systems. Techniques from geometric singular perturbation theory are combined with the use of the Bautin ideal. We also rely on the theory of Darboux integrability. [Copyright &y& Elsevier]

Abstract: Under some nondegeneracy conditions we give asymptotic formulae for the stability parameter of a family of singular-limit Hill''s equation which depends on three parameters. We use the blow-up techniques introduced in [R. Martínez, A. Samà, C. Simó, Analysis of the stability of a family of singular-limit linear periodic systems in . Applications, J. Differential Equations 226 (2006) 652–686]. The main contribution of this paper concerns the study of the nondegeneracy conditions. We give a geometrical interpretation of them, in terms of heteroclinic orbits for some related systems. In this way one can determine values of the parameters such that the nondegeneracy conditions are satisfied. As a motivation and application we consider the vertical stability of homographic solutions in the three-body problem. [Copyright &y& Elsevier]

Abstract: We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, and a stress–strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. In this paper, we first prove a local (in time) well-posedness result for (a suitable initial–boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial–boundary value problem, we indeed obtain a global well-posedness theorem. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the initial–boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation in the half-space Here is an unknown function of and , are two given constant states and the nonlinear function is assumed to be a strictly convex function of u. We first show that the corresponding boundary layer solution of the above initial–boundary value problem is global nonlinear stable and then, by employing the space–time weighted energy method which was initiated by Kawashima and Matsumura [S. Kawashima, A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985) 97–127], the convergence rates (both algebraic and exponential) of the global solution to the above initial–boundary value problem toward the boundary layer solution are also obtained for both the non-degenerate case and the degenerate case . [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine–Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47–84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257–289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101–132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson''s equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities. [Copyright &y& Elsevier]

Abstract: In this paper, we use Borel''s procedure to construct Gevrey approximate solutions of an initial value problem for involutive systems of Gevrey complex vector fields. As an application, we describe the Gevrey wave-front set of the boundary values of approximate solutions in wedges of Gevrey involutive structures . We prove that the Gevrey wave-front set of the boundary value is contained in the polar of a certain cone contained in where X is a maximally real edge of . We also prove a partial converse. [Copyright &y& Elsevier]

Abstract: In this paper we focus on the initial–boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial–boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role. [Copyright &y& Elsevier]

Abstract: We study the linear differential equation , on , where the coefficient is strictly positive and continuous on I, and satisfies the Hartman–Wintner condition at . The four main results of the paper are: (i) a criterion for rectifiable oscillations of , characterized by the integrability of on I; (ii) a stability result for rectifiable and unrectifiable oscillations of , in terms of a perturbation on ; (iii) the s-dimensional fractal oscillations (for which we assume also when , , and ); and (iv) the co-existence of rectifiable and unrectifiable oscillations in the absence of the Hartman–Wintner condition on . Explicit examples related to the above results are given. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to establish the well-posedness and the regularity of solutions of the initial–boundary value problems for general higher order parabolic equations in infinite cylinders with the bases containing conical points. [Copyright &y& Elsevier]

Abstract: For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré''s classical results [H. Poincaré, Sur l''intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161–191; 11 (1897) 193–239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235–255] on periodic systems. Then in the space of analytic autonomous systems in with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler–Lagrange equation we provide a new approach to its proof. [Copyright &y& Elsevier]

Abstract: This paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle-node and two hyperbolic saddles and for the non-generic cases of , and , , where and are the hyperbolicity ratio of the saddles and , respectively. For the case of , , we suppose that the connection from to and the connection from to keep unbroken. We prove that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity is linearly dependent on the order of the neutral saddle if is contractive and . We also show that the nearer is close to 1, the more the limit cycles are bifurcated. For the case of , , we obtain that these graphics and ensembles are of finite cyclicity respectively if is of finite order and the hp-connection from to keeps unbroken. [Copyright &y& Elsevier]

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: This paper shows the existence and the uniqueness of the positive solution of the singular boundary value problem where f is a continuous non-decreasing function such that , and h is a non-negative function satisfying the Keller–Osserman condition. Moreover, it also ascertains the exact blow-up rate of at in the special case when there exist and such that for sufficiently large u. Naturally, the blow-up rate of the problem in such a case equals its blow-up rate for the very special, but important, case when for all . So, our results are substantial improvements of some previous findings of [J. López-Gómez, Uniqueness of large solutions for a class of radially symmetric elliptic equations, in: S. Cano-Casanova, J. López-Gómez, C. Mora-Corral (Eds.), Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, World Scientific, 2005, pp. 75–110] and [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439]. [Copyright &y& Elsevier]

Abstract: In this paper we study well-posedness and stability of a free boundary problem modeling the growth of multi-layer tumors under the action of external inhibitors. An important feature of this problem is that the surface tension of the free boundary is taken into account. We first reduce this free boundary problem into an evolution equation in little Hölder space and use the well-posedness theory for differential equations in Banach spaces of parabolic type (i.e., equations which are treatable by using the analytic semi-group theory) to prove that this free boundary problem is locally well-posed for initial data belonging to a little Hölder space. Next we study flat solutions of this problem. We obtain all flat stationary solutions and give a precise description of asymptotic stability of these stationary solutions under flat perturbations. Finally we investigate asymptotic stability of flat stationary solutions under non-flat perturbations. By carefully analyzing the spectrum of the linearized stationary problem and employing the theory of linearized stability for differential equations in Banach spaces of parabolic type, we give a complete analysis of stability and instability of all flat stationary solutions under small non-flat perturbations. [Copyright &y& Elsevier]

Abstract: In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide. [Copyright &y& Elsevier]

Abstract: In this paper, we study the structure of the Fucík spectrum of −Δ, the set of points in for which the equation has a nontrivial solution. For this, we first prove a variant implicit function theorem. Then we use the variant implicit function theorem to obtain curves in the Fucík spectrum under a local nondegenerate assumption. Solvability of the equation is also investigated. Some nodal domain theorems are proved. [Copyright &y& Elsevier]

Abstract: Let Ω be a bounded domain in , , with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω where ν is an exterior normal to ∂Ω and a nonlinearity f of subcritical growth. Under rather strong conditions on f, it has been known that for small , there exists a solution of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on ∂Ω as . In this paper, we obtain the same result under some conditions on f (Berestycki–Lions conditions), which we believe to be almost optimal. [Copyright &y& Elsevier]

Abstract: In this paper, we prove the existence of eigenvalues for the problem where , , λ is a real parameter and the indefinite weight h is a nonnegative measurable function on which may be singular at 0 and/or 1, and on any compact subinterval in . We derive similar properties of eigenvalues as known in linear case () or continuous case if h satisfies when and when , respectively. For the result, we establish the -regularity of all solutions at the boundary for the above problem as well as the following problem: where , for , f is odd and is bounded above as . [Copyright &y& Elsevier]

Abstract: In this paper, we study the “triply” degenerate problem: on , on Ω and “ on some part of the boundary ,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition and the range condition . Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111–139]. [Copyright &y& Elsevier]

Abstract: For a supersonic Euler flow past a straight-sided wedge whose vertex angle is less than the extreme angle, there exists a shock-front emanating from the wedge vertex, and the shock-front is usually strong especially when the vertex angle of the wedge is large. In this paper, we establish the well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge whose boundary slope function has small total variation, when the total variation of the incoming flow is small. In this case, the Lipschitz wedge perturbs the flow, and the waves reflect after interacting with the strong shock-front and the wedge boundary. We first obtain the existence of solutions in BV when the incoming flow has small total variation by the wave front tracking method and then establish the stability of the solutions with respect to the incoming flows. In particular, we incorporate the nonlinear waves generated from the wedge boundary to develop a Lyapunov functional between two solutions containing strong shock-fronts, which is equivalent to the norm, and prove that the functional decreases in the flow direction. Then the stability is established, so is the uniqueness of the solutions by the wave front tracking method. Finally, the uniqueness of solutions in a broader class, the class of viscosity solutions, is also obtained. [Copyright &y& Elsevier]

Abstract: This paper focuses on developing new Riccati type conditions for an abnormal time scale symplectic system (). These conditions provide characterizations of the nonnegativity (with and without a certain “image condition”) and positivity of the quadratic functionals associated with such a system. The novelty of these conditions rely on the natural conjoined basis of () in which is not necessarily invertible, and thus the system () could be abnormal. These results are new even in the special case of continuous time, as are some of them in the discrete time setting. [Copyright &y& Elsevier]