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The paper [39] uses the Craig-Wayne-Bourgain method to construct solutions of an elliptic problem involving parameters. The results of [39] include regularity assumptions on the perturbation and involve excluding parameters. The paper [39] also constructs response solutions to a quasi-periodically perturbed (ill-posed evolution) problem. In this paper, we use several classical methods (freezing of coefficients, alternative methods for nonlinear elliptic equations) to extend the results of [39]. We weaken the regularity assumptions on the perturbation and we describe the phenomena that happens for all parameters. In the ill-posed problem, we use a recently developed time-dependent center manifold theorem which allows to reduce the problem to a finite-dimensional ODE with quasi-periodic dependence on time. The bounded and sufficiently small solutions of these ODE give solutions of the ill-posed PDE. [ABSTRACT FROM AUTHOR]

We discuss the recent paper by A. Darós and L.K. Arruda (2019) [6]. Our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space. [ABSTRACT FROM AUTHOR]

We provide generalizations of a class of stochastic Gronwall inequalities that has been studied by von Renesse and Scheutzow (2010), Scheutzow (2013), Xie and Zhang (2020) and Mehri and Scheutzow (2021). This class of stochastic Gronwall inequalities is a useful tool for SDEs. Our focus are convex generalizations of the Bihari-LaSalle type. The constants we obtain are sharp. In particular, we provide new sharp constants for the stochastic Gronwall inequalities. The proofs are connected to a domination inequality by Lenglart (1977), an inequality by Pratelli (1976) and a characterization of Lenglart's concept of domination via the Snell envelope. The inequalities we study appear for example in connection with exponential moments of solutions to path-dependent SDEs: For non-path-dependent SDEs, criteria for the finiteness of exponential moments are known. To be able to extend these proofs to the path-dependent case, a convex generalization of a stochastic Gronwall inequality seems necessary. Using the results of this paper, we obtain a criterion for the finiteness of exponential moments which is similar to that known for non-path-dependent SDEs. Stochastic Gronwall inequalities can also be applied to study other types of SDEs than path-dependent SDEs: An estimate of this paper is applied by Agresti and Veraar (2023) to prove global well-posedness for reaction-diffusion systems with transport noise. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following elliptic system of Hamiltonian type on a bounded domain: (0.1) { − Δ u = K 1 (| y |) | v | p − 1 v , in B 1 (0) , − Δ v = K 2 (| y |) | u | q − 1 u , in B 1 (0) , u = v = 0 on ∂ B 1 (0) , where K 1 (r) and K 2 (r) are positive bounded functions, B 1 (0) is the unit ball in R N , (p , q) is a pair of positive numbers lying on the critical hyperbola: 1 p + 1 + 1 q + 1 = N − 2 N. We investigate the local uniqueness and the nondegeneracy of the bubbling solution for problem (0.1). Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of Greens function. Our results are quite different from single equation, which is mainly caused by the non-cooperative nonlinear terms and (p , q) lying on the critical hyperbola. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems of Hamiltonian type with interact critical exponents. [ABSTRACT FROM AUTHOR]

In this paper, we study the following (p , q) -Laplacian equation with L p -constraint: { − Δ p u − Δ q u + λ | u | p − 2 u = f (u) , in R N , ∫ R N | u | p d x = c p , u ∈ W 1 , p (R N) ∩ W 1 , q (R N) , where 1 < p < q < N , Δ i = div (| ∇ u | i − 2 ∇ u) , with i ∈ { p , q } , is the i -Laplacian operator, λ is a Lagrange multiplier and c > 0 is a constant. The nonlinearity f is assumed to be continuous and satisfying weak mass supercritical conditions. The purpose of this paper is twofold: to establish the existence of ground states, and to reveal the basic behavior of the ground state energy E c as c > 0 varies. Moreover, we introduce a new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball of radius c p in L p (R N). The analysis developed in this paper allows to provide the general growth assumptions imposed to the reaction f. [ABSTRACT FROM AUTHOR]

This paper concerns static and evolution equations involving degenerate curls. The topic is motivated by the mathematical problems of Meissner states of anisotropic superconductors, and of the nonlinear magneto-static Maxwell equations with strong anisotropy. Two types of degenerate curls are studied, among them one is the directional curl along a unit length vector field, which has been introduced by the author before, and the other is the matrix-curl, which is introduced in this paper. Because of the degeneracy produced by the degenerate curl of the unknown vector field in the leading term of the equation, the boundary condition should be prescribed on part of the boundary according to the nature of the degenerate curl. We formulate the Dirichelt type and natural type boundary conditions, and give existence of solutions both for the static equations and for the evolution equations. [ABSTRACT FROM AUTHOR]

In a previous paper (Feng et al., J. Differential Equations (2023)), we studied a seasonally interactive model between closed seasons and open seasons with Michaelis-Menten type harvesting, in which we assumed that the harvesting quantity was relatively large (0 < κ = l G c E < 1) and obtained a length threshold of the closed season T ¯ ⁎ depending on the harvesting parameter and the seasonal fluctuation period. It was shown that the origin is globally asymptotically stable if and only if T ¯ ≤ T ¯ ⁎ , and there exists a unique globally asymptotically stable T -periodic solution if and only if T ¯ > T ¯ ⁎. In this paper, we continue to investigate the periodic dynamics of this model when the harvesting quantity is relatively small (κ = l G c E ≥ 1). By finding another smaller length threshold T ¯ ⁎ ∈ (0 , T ¯ ⁎) , we determine the number of periodic solutions and study their stability, which imply the occurrence of bifurcation of periodic solutions. Our results demonstrate that designing the closed harvesting season properly can prevent the species from extinction. Moreover, by comparing with the continuous harvesting model and combining the results in our previous paper and this article, we provide a complete understanding on the global dynamics for the periodic switching model. Some numerical simulations are also carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Ordinary and partial differential equations with operators containing the p -Laplace operator, the so called ϕ -Laplace operator, operators with variable exponents and the double phase operators have been studied thoroughly for at least the last 20 years and the amount of papers dealing with these subjects is huge. In this paper we deal with the problem of the existence of periodic solutions to some system of ODE's involving a fairly general operator generalizing in this form corresponding results for the areas of research just mentioned. In our results neither the function generating the differential operator nor the right hand side function in the boundary value problem considered are necessarily equal to the gradient of a function. Our main result is a here is a continuation theorem based on the Leray-Schauder degree theory which allows us to obtain the existence of solutions for the boundary value problem considered. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain Ω : = R × T 2 with T 2 = (R / Z) 2. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below. [ABSTRACT FROM AUTHOR]

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41] , and the algebraic multiplicity, χ , of Esquinas and López-Gómez [15,14,31] to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick–Pejsachowicz–Rabier degree, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F (λ , u) = 0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Finally, the unilateral theorems of [31,33] , as well as the refinement of Shi and Wang [53] , are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator. [ABSTRACT FROM AUTHOR]

In this paper we ascertain the singular point-wise behavior of the positive solutions of a semilinear elliptic boundary value problem (1) at the critical value of the parameter, λ , where it begins its metasolution regime. As the weight function m (x) changes sign in Ω, our result is a substantial extension of a previous, very recent, result of Li et al. [8] , where it was imposed the (very strong) condition that m ≥ 0 on a neighborhood of b − 1 ({ 0 }). In this paper, we are simply assuming that m (x 0) > 0 for some x 0 ∈ b − 1 ({ 0 }). • Theorem 1.1 proves that the behavior of the solutions proved by Li et al. [8] also occurs with much weaker hypotheses. • Theorem 3.1 is a substantial extension of Theorem 2.1 of López-Gómez and Sabina de Lis [12]. • Lemma 2.1 provides a useful estimate of eigenfunctions associated to an eigenvalue problem with sign changing weight. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reaction-diffusion SIS (susceptible-infected-susceptible) epidemic model governed by a saturated incidence function and linear birth-death growth in advective environments. The related model without birth and death case has been studied in our previous work [8]. In this paper, our main purpose is to investigate the combined effects of varying total population, saturation infection mechanism and spatial heterogeneity on the transmission dynamics and spatial distribution of disease. The extinction and persistence of the infectious disease in terms of the basic reproduction number are established. We discuss the global attractivity of the equilibria in two special cases and explore the asymptotic profiles of the endemic equilibrium with respect to the dispersal and advection rates. Compared with the results for the model without birth and death in [8] , our findings indicate that the linear birth-death growth can enhance the persistence of an infectious disease and may provide some prospective applications in disease prevention and control. [ABSTRACT FROM AUTHOR]

In this paper we focus on two aspects. First we study the initial-value problem of the Timoshenko-Fourier system with memory-type dissipation. By using the method of energy estimate combined with the technique of Fourier analysis, we obtain the decay estimates of solutions to the problem. Then we deal with the open problem proposed by N. Mori and S. Kawashima in the paper [18] entitled "Decay property of the Timoshenko-Cattaneo system" from the point view of energy estimate. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for the nonlinear Schrödinger equation − Δ u − λ u = f (u) in R N with an L 2 -constraint in the Sobolev subcritical case when f possesses several weaker L 2 -supercritical conditions and in the Sobolev critical case when f (u) = μ | u | q − 2 u + | u | 2 ⁎ − 2 u with μ > 0 and 2 < q < 2 ⁎ = 2 N N − 2 allowing to be L 2 -subcritical, L 2 -critical or L 2 -supercritical, where N ≥ 3 is the dimension, λ ∈ R and f ∈ C (R , R). By establishing several new critical point theorems on a manifold, we introduce a new variational approach which enables us to weaken the previous L 2 -supercritical conditions in the Sobolev subcritical case and also to present an alternative scheme for all 2 < q < 2 ⁎ , which is technically simpler compared to the Ghoussoub minimax principle [7] involving topological arguments, to construct bounded (PS) sequences on a manifold when the reaction performs as a mixed dispersion. In particular, the analysis developed in this paper also allows to control the energy level in the Sobolev critical case in a unified way whether N = 3 or N ≥ 4. Furthermore, this new approach can provide a new look at what were expected by Soave [13] and by Jeanjean-Le [10] and may it can be applied and modified to more related variational problems. [ABSTRACT FROM AUTHOR]

The Bogdanov-Takens bifurcation of the SIRS epidemic model with nonlinear incidence rate was studied by Ruan and Wang (2003) [11] , Tang et al. (2008) [13] and Lu et al. (2019) [9] in recent years. The results in the mentioned papers showed that the SIRS epidemic model with nonlinear incidence rate k I 2 / (1 + ω I 2) can undergo a Bogdanov-Takens bifurcation of codimension two. In this paper we study the SIRS epidemic model with nonlinear incidence rate k I p / (1 + ω I q) for general p and q. The bifurcation analysis indicates that there is a saddle-node or a cusp of codimension two for various parameter values and the model can undergo a saddle-node bifurcation or a Bogdanov-Takens bifurcation of codimension two if suitable bifurcation parameters are selected. It means that there are some SIRS epidemic models which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of Bogdanov-Takens bifurcation is at most two. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case. It is difficult to estimate the upper and lower bounds of the holes due to the coupling of electrons and holes and the degeneracy of electrons at the boundary. Thus, we use the topological degree method to prove the well-posedness of solution. We prove the ill-posedness of subsonic solution under some conditions by direct mathematical analysis and contradiction method. The ill-posedness property shows significant difference to the unipolar model. Another highlight of this paper is the application of specific energy method to obtain the uniqueness of solution in two cases. One case is the relaxation time τ = ∞ , namely, the pure Euler-Poisson case; the other case is j τ ≪ 1 , which means that, when the current flow is sufficiently small and the relaxation time is sufficiently large both can satisfy. [ABSTRACT FROM AUTHOR]

This paper presents Part I of a two-part series on studying the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. Part I establishes a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems, while Part II will focus on analyzing their non-autonomous periodic counterparts. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this paper introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation methods exhibit good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our methods in several application examples from biology and ecology. [ABSTRACT FROM AUTHOR]

In this paper we discuss the perturbation of a generalized Liénard system, defined in two regions. For x ≤ 0 the system is regular, while for x > 0 the system is singularly perturbed. An essential difference with other papers is that a degenerate canard point appears which requires a blow-up analysis. We prove that under generic conditions the perturbed system has at most one small-amplitude limit cycle and at most N canard limit cycles, where N is the number of local extrema of the function F (x) in the singular part of the Liénard system. For critical values of the control parameters the mechanism of a canard explosion is discussed. As an application we prove that the upper bound for the number of limit cycles in a Gause predator-prey system with a general monotonic functional response function with cut-off is two times the number of maxima of the natural prey growth function after the cut-off. [ABSTRACT FROM AUTHOR]

This paper investigates the expansions of Melnikov functions near a homoclinic loop with a nilpotent cusp of order m. It presents a methodology for calculating all coefficients in these expansions, which can be employed to study the problem of limit cycle bifurcation. As an application, by utilizing the obtained results, the paper rigorously establishes that a polynomial Liénard system of degree n + 1 has at least n + [ n 4 ] limit cycles near the homoclinic loop with a nilpotent cusp of order one. This work not only updates and generalizes existing results, but also provides a rigorous application of the obtained findings in the context of limit cycle bifurcation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate two important problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincaré-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincaré-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Liénard systems. With proper perturbations, explicit center conditions are derived for switching Liénard systems at a nilpotent center. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center. [ABSTRACT FROM AUTHOR]

In the previous paper Liu and Pu (2019) [17] , we proved the nonlinear Schrödinger (NLS) approximation for the Euler-Poisson system for a hot ion-acoustic plasma, where the appearance of resonances and the loss of derivatives of quadratic terms are the main difficulties. Note that when the ion-acoustic plasma is hot, the Euler-Poisson system is Friedrich symmetrizable, and the linear term can provide a derivative to compensate the loss of derivative induced by quadratic terms after diagonalizing the linearized system. When the ion-acoustic plasma is cold, as considered in the present paper, the situation is very different from that in the previous paper. The Euler-Poisson system becomes a pressureless system, so the linear operator has no regularity, and the quadratic terms still lose a derivative in the diagonalized system. This fact makes it more difficult to prove the NLS approximation of Euler-Poisson system for a cold ion-acoustic plasma. In this paper, we take advantage of the special structure of the pressureless Euler-Poisson system and the normal-form transformation to deal with the difficulties caused by resonances, especially the difficulties caused by derivative loss, in order to prove the NLS approximation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation dynamics of the time periodic Lotka-Volterra competition systems with nonlocal dispersal in a shifting habitat. We first obtain three types of time-periodic forced waves connecting the extinction state to the co-existence state, itself and the semi-trivial state, which describe the conversion from the state of two aboriginal co-existent competing species, two invading alien competitors, and a saturated aboriginal competitor with another invading alien competitor to the extinction state, respectively. This provides a comprehensive explanation of the point-wise extinction dynamics of these two competing species under such a time-periodic worsening habitat. Then, we establish the spreading properties of the associated Cauchy problem depending on the range of the shifting speed. More specifically, we give a complete description on the threshold values for the extinction as well as persistence (by moving with asymptotic speed). Our results reveal the possibility that a competitively weaker species with a much faster spreading speed can drive a competitively stronger species with a slower spreading speed to extinction. The discussion in this paper applies to both cases of weak competition and strong-weak competition. In particular, we need to point out that some combined effects of nonlocal dispersal, two-variable coupling and time-periodic shifting heterogeneity in this system pose extra difficulties in mathematical treatment, which are dealt with by introducing new approaches. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic dynamics for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole of size proportional to a positive parameter ε. In other words, we prove that the family of attractors behaves continuously as ε → 0. We will also provide the convergence rates in terms of the parameter. [ABSTRACT FROM AUTHOR]

This paper studies the zero-viscosity limit for the Navier-Stokes equations with rotation, an additive white noise and nonslip boundary condition in dimension three. By establishing the well-posedness of the limit problems and constructing approximate solutions, we obtain the convergence in the mean square for the Oseen type equation with nonlinear source term, and the convergence in probability of the martingale solution of the stochastic Navier-Stokes equations with rotation. [ABSTRACT FROM AUTHOR]

A modified Poisson–Nernst–Planck system in a bounded domain with mixed Dirichlet–Neumann boundary conditions is analyzed. It describes the concentrations of ions immersed in a polar solvent and the correlated electric potential due to the ion–solvent interaction. The concentrations solve cross-diffusion equations, which are thermodynamically consistent. The considered mixture is saturated, meaning that the sum of the ion and solvent concentrations is constant. The correlated electric potential depends nonlocally on the electric potential and solves the fourth-order Poisson–Fermi equation. The existence of global bounded weak solutions is proved by using the boundedness-by-entropy method. The novelty of the paper is the proof of the weak–strong uniqueness property. In contrast to the existence proof, we include the solvent concentration in the cross-diffusion system, leading to a diffusion matrix with nontrivial kernel. Then the proof is based on the relative entropy method for the extended cross-diffusion system and the positive definiteness of a related diffusion matrix on a subspace. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schrödinger equation in R × R 2 with the nonlinearity η | u | 2 where η ∈ C ∖ { 0 } and low regularity initial data. If s < − 1 / 4 , the ill-posedness result in the Sobolev space H s (R 2) is known. We will prove the well-posedness in H s (R 2) for − 1 / 2 < s < − 1 / 4 by assuming some angular regularity on initial data. The key tools are the modified Fourier restriction norm and the convolution estimate on thickened hypersurfaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the pathwise dynamics of stochastic p -Laplacian lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. When the stochastic system is driven by a linear multiplicative noise, we prove the convergence of solutions and the upper semicontinuity of random attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero. [ABSTRACT FROM AUTHOR]

The maximum number of centers is an open problem proposed by Gasull for planar polynomial differential systems of degree n with n ≥ 4. In this paper we study the problem for planar polynomial Kolmogorov differential systems of degree n , prove that the maximum number of centers is exactly seven for planar quartic polynomial Kolmogorov differential systems, and give the upper and lower bound for the maximum number of centers that the Kolmogorov differential systems of degree n can have. [ABSTRACT FROM AUTHOR]

This paper concerns the structural stability of subsonic steady-states to the bipolar Euler-Poisson equations under small perturbation of doping profiles. Here, the electron density is imposed with degenerate sonic boundary and considered in interiorly subsonic case, while the hole density is considered in fully subsonic case. Unlike the unipolar model, we show that the structural stability in bipolar model holds regardless of the type of doping profile and propose a new version of comparison principle, which captures the intrinsic negative correlation between electrons and holes. To overcome the difficulty caused by the degenerate effect at the sonic boundary, we introduce two different weight functions to handle the singularity near both sides of endpoints separately. The approaches adopted to prove the structural stability include the local singularity analysis, the monotonicity method, the continuation argument and the squeezing skill. Several numerical simulations are performed which support our theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the coupled fourth-order parabolic equations, subject to Dirichlet or Navier boundary conditions. We combine the generalized potential well method, the concave method, with the auxiliary methods to classify the initial energy about the existence of blow-up solutions and global solutions. Moreover, the decay estimate of global solutions and the bounds of blow-up time and rate are determined, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the long time dynamical behavior of the solutions for d -dimensional nonlinear Schödinger equation with general nonlinearity by using Birkhoff normal form technique and the so-called tame property in Gevrey space and modified Sobolev space. [ABSTRACT FROM AUTHOR]

In this paper we study the harmonic map heat flow on the euclidean space R d and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B ˙ p , ∞ d p (R d) where d < p < ∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖ ∇ u (t) ‖ L ∞ (R d) ≤ C t − 1 2 . Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying in R n × (0 , T) (n ≥ 1) the Cauchy problem for a semilinear heat equation with source term f (x) non-rarefied at infinity, subjected to zero initial data. To such a problem, the novelty of this work lies in: (a) similar to no source case, non-rarefaction of f at infinity brings about its classical solution blowing up in a finite time T ⁎ > 0 ; (b) we derive the optimal upper and lower bound on the life span T ⁎. This may throw some light on studies of life span. [ABSTRACT FROM AUTHOR]

We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurrence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of concentrated traveling-rotating helical vortex for the 3D incompressible Euler equations in an infinite pipe. The solutions are obtained by maximization of the energy over the set of rearrangments of a fixed bounded function with compact support, and tends asymptotically to singular helical vortex filament evolving by the binormal curvature flow. We also show nonlinear stability of maximizers under L p perturbation for p ≥ 2. [ABSTRACT FROM AUTHOR]

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the one-dimensional Euler-Poisson (or hydrodynamic) model for semiconductors, where the energy equation is replaced by a pressure-density relation. First, the existence of global entropy solutions is proved by using the vanishing artificial viscosity method, where, a special flux approximate is introduced to ensure the uniform boundedness of the electric field E and the a-priori L ∞ estimate, 0 < 2 δ ≤ ρ ε , δ ≤ M (t) , | u ε , δ | ≤ M (t) , where M (t) could tend to infinity as the time t tends to infinity, on the viscosity-flux approximate solutions (ρ ε , δ , u ε , δ) ; Second, the compensated compactness theory is applied to prove the pointwise convergence of (ρ ε , δ , u ε , δ) as ε , δ go to zero, and that the limit (ρ (x , t) , u (x , t)) is a global entropy solution; Third, a technique, to apply the maximum principle to the combination of the Riemann invariants and ∫ − ∞ x ρ ε , δ (x , t) − 2 δ d x , deduces the uniform L ∞ estimate, 0 < 2 δ ≤ ρ ε , δ ≤ M , | u ε , δ | ≤ M , independent of the time t and ε , δ ; Finally, as a by-product, the known compactness framework [14,29] is applied to show the relaxation limit, as the relation time τ and ε , δ go to zero, for general pressure P (ρ). [ABSTRACT FROM AUTHOR]

The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to H δ (R 2) × H s (R 2) for δ ∈ [ s − 1 , s + 1 ] if s ∈ R , s > 2 , for δ ∈ (1 , s + 1 ] if s ∈ (0 , 2 ] and for δ ∈ [ 0 , 1 ] if s = 0. In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in H m (R 2) , m ∈ N and in H ˙ m − 1 (R 2) with m ∈ N , m ≥ 2 , respectively. [ABSTRACT FROM AUTHOR]

This paper presents the global stability and the large-time behavior of solutions to the compressible Oldroyd-B model in a periodic domain. Moreover, we show that the H 3 -norm of the solutions decays exponentially in time. It is worth pointing out that the coupling parameter ω is not necessary to be small enough in our result. [ABSTRACT FROM AUTHOR]

This paper aims to study a semilinear transfer equation with a delay term defined over a non-cylindrical domain. We prove the existence and regularity of weak solutions as well as the existence, regularity and finiteness of the fractal dimension of pullback attractors on tempered universes that depend on a non-increasing function. We address the problem of estimating the fractal dimension of pullback attractors, under current techniques, which consists of readjusting the proof presented in [9, Lemma 1.3] and extending it to families of normed spaces parameterized in time, see Theorem 4.11. [ABSTRACT FROM AUTHOR]

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-Bénard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (2012) [44]. [ABSTRACT FROM AUTHOR]

In this paper we study the unique weighted measurable shadowing property for weighted pseudo-orbits of random systems on Banach spaces with the property that linear part of random system admits a tempered exponential dichotomy. We also prove for linear random systems that the tempered exponential dichotomy is necessary for the unique weighted measurable shadowing property to hold. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions of the p -Laplacian equation with mixed homogeneous Neumann-Dirichlet boundary conditions. It is posed in a two-dimensional rough thin domain with two different composites periodically distributed. Each composite has its own periodicity and roughness order. Here, we obtain distinct homogenized limit equations which will depend on the relationship among the roughness and thickness orders of each one. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a more general class of equations called functional Volterra integral equations involving measures, which encompass many types of equations such as functional Volterra equations, functional Volterra equations with impulses, functional Volterra delta integral equations on time scales, functional fractional differential equations with and without impulses, among others. Also we present some important results such as: local existence, uniqueness and prolongation of solutions, which play an important role for further investigations. We provide many examples and applications. [ABSTRACT FROM AUTHOR]

In this paper we deal with the following boundary value problem { − Δ p u + g (u) | ∇ u | p = h (u) f in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, in a domain Ω ⊂ R N (N ≥ 2) , where 1 ≤ p < N , g is a positive and continuous function on [ 0 , ∞) , and h is a continuous function on [ 0 , ∞) (possibly blowing up at the origin). We show how the presence of regularizing terms h and g allows to prove existence of finite energy solutions for nonnegative data f only belonging to L 1 (Ω). [ABSTRACT FROM AUTHOR]

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with singular sensitivity and Lotka-Volterra competition kinetics (0.1) { u t = Δ u − χ 1 ∇ ⋅ (u w ∇ w) + μ 1 u (1 − u − a 1 v) , t > 0 , x ∈ Ω , v t = Δ v − χ 2 ∇ ⋅ (v w ∇ w) + μ 2 v (1 − v − a 2 u) , t > 0 , x ∈ Ω , w t = Δ w − w + u + v , t > 0 , x ∈ Ω , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , t > 0 , x ∈ ∂ Ω , u (0 , x) = u 0 (x) , v (0 , x) = v 0 (x) , w (0 , x) = w 0 (x) , x ∈ Ω , where Ω ⊂ R N (N ≥ 2) is a bounded smooth convex domain, and the parameters χ 1 , χ 2 , μ 1 , μ 2 , a 1 and a 2 are positive constants. It is shown that the system possesses globally bounded classical solutions under the following conditions χ 1 , χ 2 ∈ (0 , 1 2) for N = 2 , 3 or χ 1 , χ 2 ∈ (0 , N − 2 N − 1) for N ≥ 4. Moreover, if min ⁡ { μ 1 , μ 2 } > max ⁡ { χ 1 , χ 2 } 2 4 , we obtain the uniformly lower bound for w. Finally, when χ 1 , χ 2 are suitably small, it is proved that if 0 < a 1 , a 2 < 1 , then the solution (u , v , w) converges to (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , 2 − a 1 − a 2 1 − a 1 a 2 ) in L ∞ norm as t → ∞ ; if 0 < a 2 < 1 ≤ a 1 , then the solution (u , v , w) converges to (0 , 1 , 1) in L ∞ norm as t → ∞. [ABSTRACT FROM AUTHOR]

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation ∂ t u + (− Δ) 2 u = ∇ ⋅ F (∇ u) , where F satisfies a cubic growth conditions. We establish existence and uniqueness of the solution for small initial data in local BMO spaces. In the cubic case F (ξ) = ± | ξ | 2 ξ , we also examine the large time behaviour and stability of global solutions for arbitrary and small initial data in VMO, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in detail in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T , we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M C H T and find explicitly what operators T have the property that M C H T = C. [ABSTRACT FROM AUTHOR]

In this paper we deal with an eigenvalue problem in an interface elliptic equation. We characterize the set of principal eigenvalues as a level set of a concave and regular function. As application, we study a problem arising in population dynamics. In these problems each species lives in a subdomain, and they interact in a common border, which acts as a geographical barrier; but unlike previous results, we consider the case of different growth rates in each subdomain. [ABSTRACT FROM AUTHOR]