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In this paper, we first investigate the regularity of finite energy solutions to the following Wolff type integral system: { u (x) = R 1 (x) W β , γ (v p u r) (x) , u (x) > 0 , x ∈ R N , v (x) = R 2 (x) W β , γ (u q v s) (x) , v (x) > 0 , x ∈ R N , where γ > 1 , β > 0 , β γ < N , W β , γ is the Wolff potential, R 1 , R 2 are double bounded functions in R N and p , q > max ⁡ { 1 , γ − 1 } , r , s ≥ 0 with p − s ≥ q − r > − γ + 1. We exploit the regularity lifting lemma to obtain the optimal integrability, boundedness and decaying property of finite energy solutions to the system. Secondly, we establish sharp pointwise estimates of positive finite energy solutions to the p -Laplacian and k -Hessian systems related to the above integral system, by using the previous regularity results, the interior Hölder estimates of solutions for the corresponding differential systems and a doubling lemma of Poláčik, Quittner and Souplet (Duke Math. J.,2007). These extend Vétois's decay results (Indiana Univ. Math. J.,2019) on the positive solutions of Laplacian systems to the p -Laplacian and k -Hessian cases. We remark that our methods do not need Harnack type inequalities and it can be applied to deal with the solutions without radial structures. As far as we know, this is the first attempt to derive sharp decay estimates for possibly non-radial solutions involving k -Hessian operators. We also obtain some decay estimates of the gradients at infinity. [ABSTRACT FROM AUTHOR]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the global wellposedness and large time behavior results of strong solutions for the compressible Oldroyd-B model derived by Barrett, Lu, Süli (Commun. Math. Sci., 15, 1265–1323, 2017). Exploiting the harmonic analysis tools (especially Littlewood-Paley theory), we first study the global well-posedness of the model with small initial data in spaces with low regularity. Then, under some suitable conditions involving only the low frequency of the initial data, we also obtain the optimal decay rates of the solutions. Compared with the result by Wang and Wen (Math. Models Methods Appl. Sci., 30, 139–179, 2020), the polymer number density is allowed to vanish and the stress tensor is near zero equilibrium. [ABSTRACT FROM AUTHOR]

The vanishing viscosity limit for the multi-dimensional compressible Navier-Stokes equations to the corresponding Euler equations is a difficult and challenging problem in the mathematics. Recently, L.Li, D.Wang and Y.Wang [17] verified that the solutions for the 2D compressible isentropic Navier-Stokes equations converge to the planar rarefaction wave solution for the corresponding 2D Euler equations as viscosity vanishes with a convergence rate ϵ 1 / 6 | ln ⁡ ϵ |. In this paper, the vanishing viscosity limit of 2D non-isentropic compressible Navier-Stokes equations is studied. In contrast to the work [17] , the convergence rate for the 2D full compressible Navier-Stokes equations is improved to ϵ 2 / 7 | ln ⁡ ϵ | 2 by choosing a different scaling argument and performing more detailed energy estimates. [ABSTRACT FROM AUTHOR]

Abstract: The dynamics of dilute electrons and plasma can be modeled by Vlasov–Poisson–Boltzmann equation, for which the equilibrium state can be a global Maxwellian. In this paper, we show that the rate of convergence to equilibrium is , by using a method developed for the Boltzmann equation without external force in [L. Desvillettes, C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math. 159 (2005) 245–316]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this paper is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro–micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers, namely the dehomogenized Loud''s systems. The local bifurcation diagram of the period function at the center is fully understood using the results of Chicone and Jacobs [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486]. Most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center. The techniques that we use here are different from the ones in [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486] because, while the period function extends analytically at the center, it has no smooth extension to the polycycle. At best one can hope that it has some asymptotic expansion. Another major difficulty is that the asymptotic development has to be uniform with respect to the parameters, in order to prove that a parameter is not a bifurcation value. We study also the bifurcations in the interior of the period annulus and we show that there exist three germs of curves in the parameter space that correspond to this type of bifurcation. Moreover we determine some regions in the parameter space for which the corresponding period function has at least one or two critical periods. Finally we propose a complete conjectural bifurcation diagram of the period function of the dehomogenized Loud''s systems. Our results can also be viewed as a contribution to the proof of Chicone''s conjecture [C. Chicone, review in MathSciNet, ref. 94h:58072]. [Copyright &y& Elsevier]

The main purpose of this paper is to investigate the formal deficiency indices N± of a symmetric first-order systemJf′+Bf=λHfon an interval I, where I=R or I=R±. Here J,B,H are n×n matrix-valued functions and the Hamiltonian H&ges;0 may be singular even everywhere. We obtain two results for such a system to have minimal numbers (N±=0 if I=R resp. N±=n if I=R+) and a criterion for their maximality N±=2n for I=R+ (as well as the quasi-regularity). This covers the Kac–Krein and de Branges (Trans. Amer. Math. Soc. 99 (1961) 118) theorems on 2×2 canonical systems and some results from Kogan and Rofe–Beketov (Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75) 5). Some conditions for a canonical system to have intermediate formal deficiency indices are presented, too.We also obtain a generalization of the well known Titchmarsh–Sears theorem for second-order Sturm–Liouville-type equations. This contains results due to Lidskii and Krein as special cases.We present two approaches to the above problems: one dealing with formal deficiency indices and one dealing with (ordinary) deficiency indices. Our main (non-formal) approach is based on the investigation of a symmetric linear relation Smin which is naturally associated to a first-order system. This approach works in the framework of extension theory and therefore we investigate in detail the domain D(Smin&ast;) of Smin&ast;. In particular, we prove the so called regularity theorem for D(Smin&ast;).As a byproduct of the regularity result we obtain very short proofs of (generalizations of) the main results of the paper by Kogan and Rofe–Beketov (1974/75). [Copyright &y& Elsevier]

A class of chemotaxis-Stokes systems generalizing the prototype { n t + u ⋅ ∇ n = ∇ ⋅ ( n m − 1 ∇ n ) − ∇ ⋅ ( n ∇ c ) , c t + u ⋅ ∇ c = Δ c − n c , u t + ∇ P = Δ u + n ∇ ϕ , ∇ ⋅ u = 0 , is considered in bounded convex three-dimensional domains, where ϕ ∈ W 2 , ∞ ( Ω ) is given. The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that (0.1) m > 9 8 . Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state ( 1 | Ω | ∫ Ω n 0 , 0 , 0 ) in the large time limit. This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1) . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the inverse nodal problems for discontinuous Sturm–Liouville operators with Robin boundary conditions. We show that the potential q up to its mean value on the interval [ 0 , 1 ] and coefficients h , H , b a can be uniquely determined by the twin-dense nodal subset on the subinterval [ a 0 , b 0 ] , a 0 < 1 2 < b 0 . In addition, we still establish several uniqueness theorems by the left twin-dense nodal subset on the interval [ a 0 , 1 2 ] and additional information, respectively. [ABSTRACT FROM AUTHOR]

The Winfree model is the first mathematical model for synchronization of weakly coupled oscillators. Compared to the well-known Kuramoto model, the Winfree model does not conserve the total phase. This leads to rich dynamic features compared to those produced by other phase models. In this paper, we study the emergent dynamics of the Winfree model on a locally coupled static network. A randomly chosen phase configuration undergoes several dynamic phase transitions such as incoherence, partial locking, complete locking, partial oscillator death, and complete oscillator death, as the coupling strength increases. We provide several rigorous analytical results on the emergence of these dynamic features. We also provide several numerical simulations and compare their results to the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider an indefinite fractional elliptic problem. A corresponding Liouville-type theorem is established. Furthermore, we obtain a priori estimate for solutions in a bounded domain by blowing-up and rescaling. We also classify the solutions of some degenerate elliptic equation originated from the fractional Laplacian. [ABSTRACT FROM AUTHOR]

In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non-necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide a growth lemma and a Harnack inequality which can be used to prove higher regularity estimates. [ABSTRACT FROM AUTHOR]

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger–Wang, Chau–Weinkove and the author solved this global problem in W 4 , p under some restrictions on the sign or integrability of the affine mean curvature. We remove these restrictions in this paper and obtain W 4 , p solution to the second boundary value problem when the affine mean curvature belongs to L p with p greater than the dimension. Our self-contained analysis also covers the case of Abreu's equation. [ABSTRACT FROM AUTHOR]

Finding cyclicity of homoclinic or heteroclinic loops in quadratic systems is an open problem, which is related to the second part of Hilbert's 16th problem. This paper mainly considers the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle. It is proved that the cyclicity of the triangle in quadratic systems is at least three. [ABSTRACT FROM AUTHOR]

The paper deals with homogenization of Navier–Stokes-type system describing electrorheological fluid with random characteristics. Under non-standard growth conditions we construct the homogenized model and prove the convergence result. The structure of the limit equations is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for a higher order shallow water type equation u t − u t x x + ∂ x 2 j + 1 u − ∂ x 2 j + 3 u + 3 u u x − 2 u x u x x − u u x x x = 0 , where x ∈ T = R / 2 π and j ∈ N + . Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in H s ( T ) with s ≥ − j − 2 2 for arbitrary initial data. By using the I -method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in H s ( T ) with 2 j + 1 − j 2 2 j + 1 < s ≤ 1 . Our results improve the results of A.A. Himonas, G. Misiolek (Himonas and Misiolek, 1998 [11] , 2000 [12] ). [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to study singularly perturbed control systems. Firstly, we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics. Secondly, we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system. [Copyright &y& Elsevier]

Abstract: Motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by Smith and Sontag, this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one. First, by fixing a solution of a base system converging to an equilibrium, we establish both the Nonordering of Limit Sets and the Limit Set Dichotomy for the solutions of the cascade system. Combining these tools with the idea of limiting equation, we then prove the Sequential Limit Set Trichotomy and hence the quasiconvergence in generic meaning. The generic convergent result is finally obtained by improving the Limit Set Dichotomy. [Copyright &y& Elsevier]

Abstract: In this paper, we study the smoothness effect of Cauchy problem for the spatially homogeneous Landau equation in the hard potential case and the Maxwellian molecules case. We obtain the analytic smoothing effect for the solutions under rather weak assumptions on the initial datum. [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, the authors establish the existence of partially regular weak solutions to the Landau–Lifshitz equations coupling with static Maxwell systems in 3 dimensions by Ginzburg–Landau approximation. It is proved that the Hausdorff measure of the singular set is locally finite. This extends the similar results of Ding and Guo [S. Ding, B. Guo, Hausdorff measure of the singular set of Landau–Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (1) (2004) 95–117] from the stationary solutions to weak solutions and the results of Wang [C. Wang, On Landau–Lifshitz equations in dimensions at most four, Indiana Univ. Math. J. 55 (5) (2006) 1615–1644] from Landau–Lifshitz equations to Landau–Lifshitz–Maxwell equations. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients. [Copyright &y& Elsevier]

Abstract: This paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field (one fast, multiple slow directions). Orbits connecting both sections are canard orbits, i.e. they first move rapidly towards the attracting part of a critical surface, then travel a distance near this critical surface, even beyond the point where the orbit enters the repelling part of the critical surface, and finally repel away from the surface. We prove that the transition map is smooth. In a transcritical situation however, where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold, the smoothness of the transition map may be limited, due to resonance phenomena that are revealed by blowing up the turning point! We present a polynomial example in . [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]

Abstract: This paper is devoted to the periodic problem for quasilinear parabolic hemivariational inequalities at resonance as well as at nonresonance. By use of the theory of multi-valued pseudomonotone operators, the notion of generalized gradient of Clarke and the property of the first eigenfunction, we build a Landesman–Lazer theory in the nonsmooth framework of quasilinear parabolic hemivariational inequalities. [Copyright &y& Elsevier]

Abstract: Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree , with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly . The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l. [Copyright &y& Elsevier]

Abstract: The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with . The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n. We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions. [Copyright &y& Elsevier]

Abstract: This paper deals with a class of linear equations with boundary degeneracy. According to the degenerate ratio, the equations are divided into weakly degenerate ones and strongly degenerate ones, which should be supplemented by different Dirichlet boundary value conditions. After establishing some necessary existence, nonexistence and comparison principles, we investigate the optimal Hölder continuity of weak solutions in these two cases utilizing the Harnack inequality and the Morrey theorem, respectively. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique periodic solution is analyzed; the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution. [Copyright &y& Elsevier]

Abstract: In this paper, we study a free boundary problem for compressible spherically symmetric Navier–Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in -norm and weighted -norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of curved front solution of a partial differential equation coming from a mathematical model of stroke. The equation is of reaction–diffusion type in a cylinder of radius R and of diffusion and absorption type outside of the cylinder. We prove the nonexistence of a travelling front when R is small enough and the existence if R is large enough using a recent energy method. We construct the travelling front as the limit in time of a solution with a well-chosen initial condition, in a travelling referential. [Copyright &y& Elsevier]

Abstract: In this paper we study unique continuation properties of solutions to higher (fifth) order nonlinear dispersive models. The aim is to show that if the difference of two solutions of the equations, , decays sufficiently fast at infinity at two different times, then . [Copyright &y& Elsevier]

Abstract: Consider the periodic solutions of autonomous Hamiltonian systems on the given compact energy hypersurface . If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones. [Copyright &y& Elsevier]

Abstract: In this paper, we study the biharmonic equation in the half-space , with . We prove in theory, with , existence and uniqueness results. We consider data and give solutions which live in weighted Sobolev spaces. [Copyright &y& Elsevier]

Abstract: We discuss the generation and motion of interfaces for Lotka–Volterra competition–diffusion system with large interaction. An asymptotic analysis of solutions shows that the two competing species are segregated and an interface appears on the common boundary of their habitats. The motion of the interface is governed by a free boundary problem. In this paper we establish a mathematical theory for the formation of interfaces (at the initial stage) by using an upper and lower solutions method. In addition, combining our results and a known result for the motion of interfaces (after the initial stage), we obtain some information on the generation and motion of interfaces for given almost any smooth initial data. [Copyright &y& Elsevier]

Abstract: In this paper we prove that if the potential has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), then under very mild conditions on the perturbation term g, for every there exists such that has at least k distinct weak solutions in , for every with . Moreover, information about the location of such solutions is also given. In fact, there exists a positive real number , which does not depend on λ, such that the -norm of each of those k solutions is not greater than σ. [Copyright &y& Elsevier]

Abstract: The well-known Schiffer conjecture saying that for a smooth bounded domain , if there exists a positive Neumann eigenvalue such that the corresponding Neumann eigenfunction u is constant on the boundary of Ω, then Ω is a ball. In this paper, we shall prove that the Schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding Neumann eigenfunction is constant on the boundary. We also prove a similar result to the Berenstein conjecture for the overdetermined Dirichlet eigenvalue problem. [Copyright &y& Elsevier]

Abstract: Let X be a vector field without singularities. In this paper, we show that X is in the interior of the set of vector fields with the shadowing property if and only if X satisfies both Axiom A and the strong transversality condition; that is, X is structurally stable. [Copyright &y& Elsevier]

Abstract: Relying on the notion of set-valued Lie bracket introduced in an earlier paper, we extend some classical results valid for smooth vector fields to the case when the vector fields are just Lipschitz. In particular, we prove that the flows of two Lipschitz vector fields commute for small times if and only if their Lie bracket vanishes everywhere (i.e., equivalently, if their classical Lie bracket vanishes almost everywhere). We also extend the asymptotic formula that gives an estimate of the lack of commutativity of two vector fields in terms of their Lie bracket, and prove a simultaneous flow box theorem for commuting families of Lipschitz vector fields. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with a Cauchy problem for a semilinear heat equation with and . We show that if and , then for each positive integer there exist , , with and a global -solution u of (P) such that u is a regular solution of (P) for and blows up at for . [Copyright &y& Elsevier]

Abstract: We let Ω be a smooth bounded domain of and a sequence of functions such that in . We consider a sequence of functions such that in Ω for all . We address in this paper the question of the asymptotic behavior of the ''s when . The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the ''s are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. [Copyright &y& Elsevier]

Abstract: In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball. [Copyright &y& Elsevier]

Abstract: In this paper we study the global attractors for wave equations with nonlinear interior damping. We prove the existence, regularity and finite dimensionality of the global attractors without assuming a large value for the damping parameter, when the growth of the nonlinear terms is critical. [Copyright &y& Elsevier]

Abstract: In this paper we consider nonlinear boundary value problems whose simplest model is the following: where is a summable function in Ω (bounded open set in , ), , and . [Copyright &y& Elsevier]

Abstract: We investigate singular and degenerate behavior of solutions of the unstable free boundary problem First, we construct a solution that is not of class and whose free boundary consists of four arcs meeting in a cross-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is nontrivial. We also show existence of a degenerate solution. This answers two of the open questions in the paper [R. Monneau, G.S. Weiss, An unstable elliptic free boundary problem arising in solid combustion, http://arxiv.org/abs/math.AP/0507315, submitted for publication]. [Copyright &y& Elsevier]

Abstract: In this paper we prove rigorous results on persistence of invariant tori and their whiskers. The proofs are based on the parameterization method of [X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J. 52 (2) (2003) 283–328; X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2) (2003) 329–360]. The invariant manifolds results proved here include as particular cases of the usual (strong) stable and (strong) unstable manifolds, but also include other non-resonant manifolds. The method lends itself to numerical implementations whose analysis and implementation is studied in [A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, preprint, 2005; A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical implementation and examples, preprint, 2005]. The results are stated as a posteriori results. Namely, that if one has an approximate solution which is not degenerate, then, one has a true solution not too far from the approximate one. This can be used to validate the results of numerical computations. [Copyright &y& Elsevier]

Abstract: For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton–Jacobi equations. In the present paper, we consider a class of infinite horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value functions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability. [Copyright &y& Elsevier]

Abstract: Resonant and nonresonant Hopf bifurcations from relative equilibria posed in two spatial dimensions, in systems with Euclidean symmetry, have been extensively studied in the context of spiral waves in a plane and are now well understood. We investigate Hopf bifurcations from relative equilibria posed in systems with compact symmetry where is the group of rotations in three dimensions/on a sphere. Unlike the case the skew product equations cannot be solved directly and we use the normal form theory due to Fiedler and Turaev to simplify these systems. We show that the normal form theory resolves the nonresonant case, but not the resonant case. New methods developed in this paper combined with the normal form theory resolves the resonant case. [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: This paper analyzes the dynamics of a superlinear indefinite parabolic system. As a byproduct, a number of new results related to population dynamics and economy are obtained. Among them, it is shown that the presence of refuge areas in competitive environments is an optimal mechanism to avoid extinction, and that incorporating local symbiosis in competitive environments increases productivity and allows avoiding extinction of the ‘weaker’ species. Undoubtedly, a paradigm of global markets and possibly of Earth biodiversity. Our analysis combines a series of well-known results for systems with some very recent pioneering findings within the context of superlinear indefinite equations. [Copyright &y& Elsevier]