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The present paper is devoted to the investigation of various properties of transition fronts in one-dimensional nonlocal equations in heterogeneous media of ignition type, whose existence has been established by the authors of the present paper in a previous work. It is first shown that transition fronts are continuously differentiable in space with uniformly bounded and uniformly Lipschitz continuous space partial derivative. This is the first time that space regularity of transition fronts in nonlocal equations is ever studied. It is then shown that transition fronts are uniformly steep. Finally, asymptotic stability, in the sense of exponentially attracting front-like initial data, of transition fronts is studied. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a previous paper by the authors. We are interested in the asymptotic behavior of eigenpairs on one dimensional linearized eigenvalue problem for Allen–Cahn equations as the diffusion coefficient tends to zero. We obtain the asymptotic profiles of all eigenfunctions by using the asymptotic formulas of corresponding eigenvalues, which have been obtained in the previous paper. Our results lead us to the concept of the classification of limiting eigenfunctions. In the case of Allen–Cahn equation it is provided by three special eigenfunctions, which correspond to the solutions of rescaled spectral problems on the whole line. [ABSTRACT FROM AUTHOR]

In this paper we continue to deal with the initial–boundary value problem for the coupled Keller–Segel–Stokes system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , ( x , t ) ∈ Ω × ( 0 , T ) , c t + u ⋅ ∇ c = Δ c − c + n , ( x , t ) ∈ Ω × ( 0 , T ) , u t + ∇ P = Δ u + n ∇ ϕ , ( x , t ) ∈ Ω × ( 0 , T ) , ∇ ⋅ u = 0 , ( x , t ) ∈ Ω × ( 0 , T ) , where Ω ⊂ R d is a bounded domain with smooth boundary and the chemotactic sensitivity S is not a scalar function but rather attains values in R d × d , and satisfies | S ( x , n , c ) | ≤ C S ( 1 + n ) − α with some C S > 0 and α > 0 . When d = 2 , our previous work (J. Differential Equations, 2015) has established the existence of global bounded classical solutions under the subcritical assumption α > 0 , which is consistent with the corresponding results of the fluid-free system, but the method seems to be invalid in the three-dimensional setting. In this paper, for the case d = 3 , we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption α > 1 2 , which is slightly stronger than the corresponding subcritical assumption α > 1 3 on the fluid-free system, where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions. The key idea here is to establish the general L p regularity of u from a rather low L p regularity of n , which will be obtained by a new combinational functional. [ABSTRACT FROM AUTHOR]

In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L 1 norm of potentials is known. We establish the minimization characterization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow. [ABSTRACT FROM AUTHOR]

Highlights • Asymptotics of relaxation solutions of DDE with large parameter is built. • Studying of dynamics of DDE is reduced to studying dynamics of special mapping. • Coexisting of stable relaxation periodic solutions is shown. Abstract In this paper, nonlocal dynamics of a system of two differential equations with a compactly supported nonlinearity and delay is studied. For some set of initial conditions asymptotics of solutions of considered system is constructed. By this asymptotics we build a special mapping. Dynamics of this mapping describes dynamics of initial system in general: it is proved that stable cycles of this mapping correspond to exponentially orbitally stable relaxation periodic solutions of initial system of delay differential equations. It is shown that amplitude, period of solutions of initial system, and number of coexisting stable solutions depend crucially on coupling parameter. Algorithm for constructing many coexisting stable solutions is described. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lamé system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with a time periodic competition–diffusion system { u t = u x x + u (r 1 (t) − a 1 (t) u − b 1 (t) v) , t > 0 , x ∈ R , v t = d v x x + v (r 2 (t) − a 2 (t) u − b 2 (t) v) , t > 0 , x ∈ R , where u (t , x) and v (t , x) denote the densities of two competing species, d > 0 is some constant, r i (t) , a i (t) and b i (t) are T -periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang (2013) [2] that this system admits periodic traveling fronts connecting two stable semi-trivial T -periodic solutions (p (t) , 0) and (0 , q (t)) associated to the corresponding kinetic system. In the present work, we first investigate the asymptotic behavior of periodic bistable traveling fronts with non-zero speeds at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then obtain some key estimates. As a result, by applying the super- and subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of the so-called entire solutions defined for all time and the whole space, which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment. [ABSTRACT FROM AUTHOR]

Abstract A well-known result by L. Markus [6] , later extended by D.A. Neumann [7] , states that two continuous flows on a surface are equivalent if and only if there is a surface homeomorphism preserving orbits and time directions of their separatrix configurations. In this paper we present several examples showing that, as originally formulated, the Markus–Neumann theorem needs not work. Besides, we point out the gap in its proof and show how to restate it in a correct (and slightly more general) way. [ABSTRACT FROM AUTHOR]

Abstract The motion of moderately thick spray can be modeled by a coupled system of equations consisting of the incompressible Navier–Stokes equations and the Vlasov–Boltzmann equation. In this paper, we study the initial value problem for the Navier–Stokes–Vlasov–Boltzmann equations. The existence of global weak solutions is established by the weak convergence method. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study a spatially heterogeneous predator–prey system where the interaction is governed by a Holling type II functional response, which has been studied in Du and Shi (2007) [14] . We further study the asymptotic profile of positive solutions and give a complete understanding of coexistence region. Moreover, a good understanding of the number, stability and asymptotic behavior of positive solutions is gained for large m . Finally, we further compare the difference of steady-state solutions between m > 0 and m = 0 . It turns out that the spatial heterogeneity of the environment and the Holling type II functional response play a very important role in this model. [ABSTRACT FROM AUTHOR]

Consider a function W ( x 1 , … , x d ) = ∑ k = 1 d W k ( x k ) , where each W k : R → R is a strictly increasing right continuous function with left limits. Given a matrix function A = diag { a 1 , … , a d } , let ∇ A ∇ W = ∑ k = 1 d ∂ x k ( a k ∂ W k ) be a generalized second-order differential operator. Our chief goal is to study the homogenization of generalized second-order difference operators, that is, we are interested in the convergence of the sequence of solutions of λ u N − ∇ N A N ∇ W N u N = f N to the solution of λ u − ∇ A ∇ W u = f , where the superscript N stands for some sort of discretization. In the continuous case we study the problem in the context of W -Sobolev spaces, whereas in the discrete case we develop the theoretical context in the present paper. The main result is a homogenization result. Under minor assumptions regarding weak convergence and ellipticity of these matrices A N , we show that every such sequence admits a homogenization. We provide two examples of matrix functions verifying these assumptions: the first one consists of fixing a matrix function A under minor regularity assumptions, and taking a convenient discretization A N ; the second one consists on the case where A N represents a random environment associated to an ergodic group, a case in which we then show that the homogenized matrix A does not depend on the realization ω of the environment. Finally, we provide an application geared towards the hydrodynamical limit of certain gradient processes. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a class of resonant Hamiltonian systems x ′ = J H x ( t , x ) in R 2 N . We will use saddle point reduction, Morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions, i.e., x ( t + T ) = Q x ( t ) for any t ∈ R with T > 0 and Q an symplectic orthogonal matrix. In the case: Q k ≠ I 2 N for any positive integer k , such a rotating periodic solution is just a quasi-periodic solution. Moreover, if H is even in x , we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a multidimensional scalar conservation law and construct an outer estimate for the domain of dependence of its Kružkov solution. The estimate can be represented as the controllability set of a specific differential inclusion. In addition, reachable sets of this inclusion provide outer estimates for the support of the wave profiles. Both results follow from a modified version of the classical Kružkov uniqueness theorem, which we also present in the paper. Finally, the results are applied to a control problem consisting in steering a distributed quantity to a given set. [ABSTRACT FROM AUTHOR]

On a manifold ( R n , e 2 u | d x | 2 ) , we say u is normal if the Q -curvature equation that u satisfies ( − Δ ) n 2 u = Q g e n u can be written as the integral form u ( x ) = 1 c n ∫ R n log ⁡ | y | | x − y | Q g ( y ) e n u ( y ) d y + C . In this paper, we show that the integrability assumption on the negative part of the scalar curvature implies the metric is normal. As an application, we prove a bi-Lipschitz equivalence theorem for conformally flat metrics. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if the solution of the planar N + 1 -body problem has N bodies rotating at the vertices of a regular N -gon then the N bodies have equal mass and the remaining body must be at the center. [ABSTRACT FROM AUTHOR]

In this paper, we consider the half-space problem for a parabolic–hyperbolic system arising from tumor angiogenesis. Under a mixed type boundary condition, we construct the Green's function for half-space problem by relating it with fundamental solution for an initial value problem. After differential equation method for the boundary operator and proper estimates for additional exponential factors generated by the asymmetry of characteristics, one reduces the Green's function (for initial-boundary value problem) into fundamental solution (for Cauchy problem) which could be estimated by singularity removal, long wave-short wave decomposition and weighted energy method outside cone. We finally obtain the pointwise estimate for Green's function which results in the pointwise convergence rate for nonlinear problem. The Green's function constructed is precise enough to avoid a priori estimate for derivatives by energy method. The half space problem studied here is a main ingredient for future study of shock profile with presence of boundary. [ABSTRACT FROM AUTHOR]

In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov–Kuznetsov (GZK) equations of form u t + ( F ( u ) ) x x x + ( G ( u ) ) x y y + ( H ( u ) ) x = 0 . As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov–Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with the existence of entire solutions for a quasilinear equation in , depending on a real parameter λ, which involves a general variable exponent elliptic operator A in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value with the property that admits nontrivial nonnegative entire solutions if and only if . Furthermore, under the further assumption that the potential of A is uniform convex, we give the existence of a second independent nontrivial nonnegative entire solution of , when . Our results extend the previous work of Autuori and Pucci (2013) [6] from the case of constant exponents p, q and r to the case of variable exponents. More interesting, we weaken the condition to the simple request that . Furthermore, we extend the previous work of Alama and Tarantello (1996) [2] from Dirichlet Laplacian problems in bounded domains of to the case of a general variable exponent differential equation in the entire , and also remove the assumption . Hence the results of this paper are new even in the canonical case . [Copyright &y& Elsevier]

This paper deals with the spatial propagation for reaction–diffusion cooperative systems. It is well-known that the solution of a reaction–diffusion equation with monostable nonlinearity spreads at a finite speed when the initial condition decays to zero exponentially or faster, and propagates fast when the initial condition decays to zero more slowly than any exponentially decaying function. However, in reaction–diffusion cooperative systems, a new possibility happens in which one species propagates fast although its initial condition decays exponentially or faster. The fundamental reason is that the growth sources of one species come from the other species. Simply speaking, we find a new interesting phenomenon that the spatial propagation of one species is accelerated by the other species. This is a unique phenomenon in reaction–diffusion systems. We present a framework of fast propagation for reaction–diffusion cooperative systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the degenerate initial–boundary value problem to the one-dimensional nonlinear wave equation u t t = ( ( 1 + u ) a u x ) x which arises in a number of various physical contexts. The global existence of smooth solutions to the degenerate problem was established under relaxed conditions on the initial–boundary data by the characteristic decomposition method. Moreover, we show that the solution is uniformly C 1 , α continuous up to the degenerate boundary and the degenerate curve is C 1 , α continuous for α ∈ ( 0 , min ⁡ { a 1 + a , 1 1 + a } ) . [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ground state solution for an indefinite variational problem of the type ( P ) { − Δ u + ( V ( x ) − W ( x ) ) u = f ( x , u ) in R N , u ∈ H 1 ( R N ) , where N ≥ 2 , V , W : R N → R and f : R N × R → R are continuous functions verifying some technical conditions and f possesses a critical growth. Here, we will consider the case where the problem is asymptotically periodic, that is, V is Z N -periodic, W goes to 0 at infinity and f is asymptotically periodic. [ABSTRACT FROM AUTHOR]

In a Hilbert space setting, we study the asymptotic behavior, as time t goes to infinity, of the trajectories of a second-order differential equation governed by the Yosida regularization of a maximally monotone operator with time-varying positive index λ ( t ) . The dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient γ ( t ) . A suitable tuning of the parameters γ ( t ) and λ ( t ) makes it possible to prove the weak convergence of the trajectories towards zeros of the operator. When the operator is the subdifferential of a closed convex proper function, we estimate the rate of convergence of the values. These results are in line with the recent articles by Attouch–Cabot [3] , and Attouch–Peypouquet [8] . In this last paper, the authors considered the case γ ( t ) = α t , which is naturally linked to Nesterov's accelerated method. We unify, and often improve the results already present in the literature. [ABSTRACT FROM AUTHOR]

Optimal partial mass transport, which is a variant of the optimal transport problem, consists in transporting effectively a prescribed amount of mass from a source to a target. The problem was first studied by Caffarelli and McCann (2010) [6] and Figalli (2010) [12] with a particular attention to the quadratic cost. Our aim here is to study the optimal partial mass transport problem with Finsler distance costs including the Monge cost given by the Euclidian distance. Our approach is different and our results do not follow from previous works. Among our results, we introduce a PDE of Monge–Kantorovich type with a double obstacle to characterize active submeasures, Kantorovich potential and optimal flow for the optimal partial transport problem. This new PDE enables us to study the uniqueness and monotonicity results for the active submeasures. Another interesting issue of our approach is its convenience for numerical analysis and computations that we develop in a separate paper [14] (Igbida and Nguyen, 2018). [ABSTRACT FROM AUTHOR]

We classify all ( − 1 ) -homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier–Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles, parameterizing them as a four dimensional surface with boundary in appropriate function spaces. Then we establish smoothness properties of the solution surface in the four parameters. The smoothness properties will be used in a subsequent paper where we study the existence of ( − 1 ) -homogeneous axisymmetric solutions with non-zero swirl on S 2 ∖ { S , N } , emanating from the four dimensional solution surface. [ABSTRACT FROM AUTHOR]

In this paper, we find that the unbounded spaces Y σ ( R N ) ( 0 < σ < 2 m − 1 ) can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted L 1 – L ∞ estimates for the solutions. [ABSTRACT FROM AUTHOR]

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): x ˙ = y − 12 x 2 + 16 y 2 , y ˙ = − x − 16 x y , and (r20): x ˙ = y + 4 x 2 , y ˙ = − x + 16 x y , and the periodic orbits of the quadratic isochronous centers ( S 1 ) : x ˙ = − y + x 2 − y 2 , y ˙ = x + 2 x y , and ( S 2 ) : x ˙ = − y + x 2 , y ˙ = x + x y . The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system ( S 1 ) and ( S 2 ) are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line y = 0 . It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively 4 n − 3 ( n ≥ 4 ) and 4 n + 3 ( n ≥ 3 ) counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers ( S 1 ) and ( S 2 ) are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., 229 (2005), no. 1, 62–94] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions a , b , A , B . The functions a ( k ) and b ( k ) are defined via a nonlinear Fourier transform of the initial data, whereas A ( k ) and B ( k ) are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study an optimal control problem in the Mayer's form in the space of probability measures on R n endowed with the Wasserstein distance. Our aim is to study optimality conditions when the knowledge of the initial state and velocity is subject to some uncertainty, which are modeled by a probability measure on R d and by a vector-valued measure on R d , respectively. We provide a characterization of the value function of such a problem as unique solution of an Hamilton–Jacobi–Bellman equation in the space of measures in a suitable viscosity sense. Some applications to a pursuit-evasion game with uncertainty in the state space is also discussed, proving the existence of a value for the game. [ABSTRACT FROM AUTHOR]

In this paper, we study the quenching rate of the solution for a nonlocal parabolic problem which arises in the study of the micro-electro mechanical system. This question is equivalent to the stabilization of the solution to the transformed problem in self-similar variables. First, some a priori estimates are provided. In order to construct a Lyapunov function, due to the lack of time monotonicity property, we then derive some very useful and challenging estimates by a delicate analysis. Finally, with this Lyapunov function, we prove that the quenching rate is self-similar which is the same as the problem without the nonlocal term, except the constant limit depends on the solution itself. [ABSTRACT FROM AUTHOR]

In this paper we establish bifurcation theory of limit cycles for planar C k smooth autonomous differential systems, with k ∈ N . The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and C ∞ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the radial singular solution of Δ 2 u = − 1 u p , in R N with u > 0 , p > 0 . By using some elementary ordinary differential equation arguments, we prove the above equation admits no radial singular solution for N = 3 , p ≥ 3 . In addition, the exact asymptotic behaviors of the radial singular solution as r → 0 is established for p = 1 , N ≥ 4 , which have a significant difference with the explicit singular solution. As a product, the singularity of any radial singular solution is of type II if p = 1 , N ≥ 4 . [ABSTRACT FROM AUTHOR]

In this paper we study the dynamical behavior of the differential equation x ″ + a x + − b x − = f ( t ) , where x + = max ⁡ { x , 0 } , x − = max ⁡ { − x , 0 } , a and b are two different positive constants, f ( t ) is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see [11] ). Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation. [ABSTRACT FROM AUTHOR]

In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves. [ABSTRACT FROM AUTHOR]

In a Hilbert space H , we study the asymptotic behavior, as time variable t goes to +∞, of nonautonomous gradient-like inertial dynamics, with a time-dependent viscosity coefficient. Given Φ : H → R a convex differentiable function, γ ( ⋅ ) a time-dependent positive damping term, we consider the second-order differential equation x ¨ ( t ) + γ ( t ) x ˙ ( t ) + ∇ Φ ( x ( t ) ) = 0 . This system plays a central role in mechanics and physics in the asymptotic stabilization of nonlinear oscillators. Its importance in optimization was recently put to the fore by Su, Boyd, and Candès. They have shown that in the particular case γ ( t ) = 3 t , this is a continuous version of the fast gradient method initiated by Nesterov, with Φ ( x ( t ) ) − min H ⁡ Φ = O ( 1 t 2 ) as t → + ∞ in the worst case. Recently, in the case γ ( t ) = α t with α > 3 , Attouch and Peypouquet have improved this result by showing the convergence of the trajectories to optimal solutions, and Φ ( x ( t ) ) − min H ⁡ Φ = o ( 1 t 2 ) as t → + ∞ . For these questions, and the design of fast optimization methods, the tuning of the damping parameter γ ( t ) is a subtle question, which we deal with in this paper in general. We obtain convergence rates for the values, and convergence results of the trajectories under general conditions on γ ( ⋅ ) which unify, and often improve the results already present in the literature. We complement these results by showing that they are robust with respect to perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an integrable two-component peakon system, which was proposed by Xia, Qiao and Zhou. We present a precise blow-up scenario and a new blow-up result for strong solutions to the system. Moreover, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

This paper studies the Neumann problem of a nonlocal Allen–Cahn equation in an interval. A main result finds a symmetry breaking (secondary) bifurcation point on the bifurcation curve of solutions with odd-symmetry. Our proof is based on a level set analysis for the associated integral map. A method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point. We also show some numerical simulations concerning the global bifurcation structure. [ABSTRACT FROM AUTHOR]

In this paper we consider nonlocal problems in thin domains. First, we deal with a nonlocal Neumann problem, that is, we study the behavior of the solutions to f ( x ) = ∫ Ω 1 × Ω 2 J ϵ ( x − y ) ( u ϵ ( y ) − u ϵ ( x ) ) d y with J ϵ ( z ) = J ( z 1 , ϵ z 2 ) and Ω = Ω 1 × Ω 2 ⊂ R N = R N 1 × R N 2 a bounded domain. We find that there is a limit problem, that is, we show that u ϵ → u 0 as ϵ → 0 in Ω and this limit function verifies ∫ Ω 2 f ( x 1 , x 2 ) d x 2 = | Ω 2 | ∫ Ω 1 J ( x 1 − y 1 , 0 ) ( U 0 ( y 1 ) − U 0 ( x 1 ) ) d y 1 , with U 0 ( x 1 ) = ∫ Ω 2 u 0 ( x 1 , x 2 ) d x 2 . In addition, we deal with a double limit when we add to this model a rescale in the kernel with a parameter that controls the size of the support of J . We show that this double limit exhibits some interesting features. We also study a nonlocal Dirichlet problem f ( x ) = ∫ R N J ϵ ( x − y ) ( u ϵ ( y ) − u ϵ ( x ) ) d y , x ∈ Ω , with u ϵ ( x ) ≡ 0 , x ∈ R N ∖ Ω , and deal with similar issues. In this case the limit as ϵ → 0 is u 0 = 0 and the double limit problem commutes and also gives v ≡ 0 at the end. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the quasilinear fully parabolic Keller–Segel system under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary, . The diffusivity is assumed to satisfy some further technical conditions such as algebraic growth and , which says that the diffusion is allowed to be not only non-degenerate but also degenerate. The global-in-time existence and uniform-in-time boundedness of solutions are established under the subcritical condition that for with , and . When , this paper represents an improvement of Tao and Winkler [17], because the domain does not necessarily need to be convex in this paper. In the case and , uniform-in-time boundedness is an open problem left in a previous paper [7]. This paper also gives an answer to it in bounded domains. [Copyright &y& Elsevier]

Abstract: In this paper, we study the relativistic Boltzmann equation in the spatially flat Robertson–Walker spacetime. For a certain class of scattering kernels, global existence of classical solutions is proved. We use the standard method of Illner and Shinbrot for the global existence and apply the splitting technique of Guo and Strain for the regularity of solutions. The main interest of this paper is to study the evolution of matter distribution, rather than the evolution of spacetime. We obtain the asymptotic behaviour of solutions and will understand how the expansion of the universe affects the evolution of matter distribution. [Copyright &y& Elsevier]

Abstract: In this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems. More details, we will prove that any complete analytic integrable diffeomorphism in with B having eigenvalues not modulus 1 and is locally analytically conjugate to its normal form. Meanwhile, we also prove that any complete analytic integrable differential system in with A having nonzero eigenvalues and is locally analytically conjugate to its normal form. Furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow. We note that parts of our results are the improvement of Moserʼs one in J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math. 9 (1956) 673–692 and of Poincaréʼs one in H. Poincaré, Sur lʼintégration des équations différentielles du premier order et du premier degré, II, Rend. Circ. Mat. Palermo 11 (1897) 193–239. These results also improve the ones in Xiang Zhang, Analytic normalization of analytic integrable systems and the embedding flows, J. Differential Equations 244 (2008) 1080–1092 in the sense that the linear part of the systems can be nonhyperbolic, and the one in N.T. Zung, Convergence versus integrability in Poincaré–Dulac normal form, Math. Res. Lett. 9 (2002) 217–228 in the way that our paper presents the concrete expression of the normal form in a restricted case. [Copyright &y& Elsevier]

The efficient conditions guaranteeing the existence of a T -periodic solution to the second order differential equation u ″ = h ( t ) g ( u ) are established in the paper. Here, g is a positive and decreasing function which has a strong singularity at the origin, and the weight h ∈ L ( R / T Z ) is a sign-changing function. The obtained results have the form of relation between the multiplicities of the zeroes of the weight function h and the order of the singularity of the nonlinear term. The approach is based on Leray–Schauder degree theory, proving that no T -periodic solution of a certain homotopy appears on the boundary of an unbounded open set during the deformation to an autonomous problem. [ABSTRACT FROM AUTHOR]

The geometric Lorenz attractor is an attractor set constructed in such a way that it satisfies the main qualitative properties evidenced on the Lorenz system equations, particularly the fact that this attractor is a robustly transitive set. In this paper we prove the C 1 -robust transitivity by using geometric properties for singular hyperbolic sets and without the assumption of the uniformly linearizing coordinates around the singularity. [ABSTRACT FROM AUTHOR]

This paper is devoted to pointwise second-order necessary optimality conditions for the Mayer problem arising in optimal control theory. We first show that with every optimal trajectory it is possible to associate a solution p ( ⋅ ) of the adjoint system (as in the Pontryagin maximum principle) and a matrix solution W ( ⋅ ) of an adjoint matrix differential equation that satisfy a second-order transversality condition and a second-order maximality condition. These conditions seem to be a natural second-order extension of the maximum principle. We then prove a Jacobson like necessary optimality condition for general control systems and measurable optimal controls that may be only “partially singular” and may take values on the boundary of control constraints. Finally we investigate the second-order sensitivity relations along optimal trajectories involving both p ( ⋅ ) and W ( ⋅ ) . [ABSTRACT FROM AUTHOR]

This paper studies the global (in time) regularity and large time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. When there is no kinematic viscosity in the momentum equation, the global regularity problem is not easy due to the lack of suitable bounds on the derivatives. The idea here is to fully exploit the structure of the system and control the vorticity via the evolution equation of a combined quantity of the vorticity and the micro-rotation angular velocity. To understand the large time behavior, we overcome two main difficulties, the lack of kinematic viscosity and the presence of linear terms. Classical tools such as the Fourier splitting method of Schonbek and Kato's approach for the decay of small solutions do not apply here. We introduce a diagonalization process to eliminate the linear terms and rely on the uniform bounds for the first derivatives of the solutions to generate suitable decay rates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and uniqueness of strong solutions to the initial boundary value problem for the 3D compressible Navier–Stokes equations without heat conductivity in a bounded domain with slip boundary. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in H 2 ( Ω ) . Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the behavior of certain solutions to the Swift–Hohenberg equation on a one-dimensional torus T . Combining results from Γ-convergence and ODE theory, it is shown that solutions corresponding to initial data that is L 1 -close to a jump function v , remain close to v for large time. This can be achieved by regarding the equation as the L 2 -gradient flow of a second order energy functional, and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v . [ABSTRACT FROM AUTHOR]

In this paper, the formation of singularities for the nonlocal Whitham-type equations is studied. It is shown that if the lowest slope of flows can be controlled by its highest value with the bounded Whitham-type integral kernel initially, then the finite-time blow-up will occur in the form of wave-breaking. This refined wave-breaking result is established by analyzing the monotonicity and continuity properties of a new system of the Riccati-type differential inequalities involving the extremal slopes of flows. Our theory is illustrated via the Whitham equation, Camassa–Holm equation, Degasperis–Procesi equation, and their μ -versions as well as hyperelastic rod equation. [ABSTRACT FROM AUTHOR]