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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]

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 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]

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 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 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 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 the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa–Holm equation on the real-line case. We show that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces B p , r s (R) with s > 1 2 and 1 ≤ p , r < ∞ , which improves the previous works Himonas et al. (2007) [23] , Li et al. (2020) [32] and Li et al. (2021) [31]. Furthermore, we present a strengthening of our previous work in Li et al. (2020) [32] and prove that the data-to-solution map for the Camassa–Holm equation is nowhere uniformly continuous in B p , r s (R) with s > max ⁡ { 1 + 1 / p , 3 / 2 } and (p , r) ∈ [ 1 , ∞ ] × [ 1 , ∞). The method applies also to the b-family of equations which contain the Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large deviation principle of the non-local fractional stochastic reaction-diffusion equation with a polynomial drift of arbitrary degree driven by multiplicative noise defined on unbounded domains. We first prove the strong convergence of the solutions of a control equation with respect to the weak topology of controls, and then show the convergence in distribution of the solutions of the stochastic equation when the noise intensity approaches zero. We finally establish the large deviations of the stochastic equation by the weak convergence method. The main difficulty of the paper is caused by the non-compactness of Sobolev embeddings on unbounded domains, and the idea of uniform tail-ends estimates is employed to circumvent the obstacle in order to obtain the tightness of distribution laws of the stochastic equation and the precompactness of the control equation. [ABSTRACT FROM AUTHOR]

We investigate the long time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper [ arXiv:2108.03650 ], which gives the soliton resolution for the defocusing mKdV equation in the central asymptotic sector { (x , t) : | ξ | < 6 } with ξ : = x / t. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the soliton-less regions { (x , t) : | ξ | > 6 , | ξ | = O (1) } for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics is in compatible with the "background solution" and the error terms are derived via rigorous analysis. [ABSTRACT FROM AUTHOR]

We consider the Hamilton-Jacobi equation H (x , D u) + λ (x) u = c , x ∈ M , where M is a connected, closed and smooth Riemannian manifold. The functions H (x , p) and λ (x) are continuous. H (x , p) is convex, coercive with respect to p , and λ (x) changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao [11] under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem. To the best of our knowledge, it is the first detailed description of the large time behavior of the HJ equations with non-monotone dependence on the unknown function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the three dimensional parabolic-elliptic Keller-Segel equation with the large plane Couette-Poiseuille flow. Without advection, there exist solution with arbitrarily mass which blow up in finite time. Firstly, we introduce three dimensional plane Couette-Poiseuille flow and study the enhanced dissipation effect of such flows by resolvent estimate method. Next, we show that the enhanced dissipation effect of such flows can suppress blow-up of solution to three dimensional parabolic-elliptic Keller-Segel equation and establish global classical solution with large initial data. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new regularity criterion in terms of the magnetic field for the local-in-time smooth solution of the two-dimensional non-resistive MHD equations. In particular, our result settles an open problem remarked by Lei et al. (2010), [11]. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of ground state solutions for the following (p , q) -Choquard equation: − Δ p u − Δ q u + | u | p − 2 u + | u | q − 2 u = (I α ⁎ F (u)) f (u) in R N , where 2 ≤ p < q < N , Δ s is the s -Laplacian operator, with s ∈ { p , q } , I α is the Riesz potential of order α ∈ ((N − 2 q) + , N) , F ∈ C 1 (R , R) is a general nonlinearity of Berestycki-Lions type and F ′ = f. Furthermore, we analyze the regularity, symmetry and decay properties of these solutions. In particular, we extend the results in [33] to the (p , q) -Laplacian setting. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of highest, cusped, traveling wave solutions for the fractional KdV equations f t + f f x = | D | α f x for all α ∈ (− 1 , 0) and give their exact leading asymptotic behavior at zero. The proof combines careful asymptotic analysis and a computer-assisted approach. [ABSTRACT FROM AUTHOR]

In this paper, we use Aleksandrov's reflection principle to prove symmetry of solutions to F (∇ S n 2 u + u I) = f (u , u 2 + | ∇ S n u | 2 ) , where u is the support function of a convex body, and F is a function of principal radii. As a corollary, alongside [Ivaki, arXiv:2307.06252 ], we provide an alternative proof of the uniqueness of solution to the isotropic Gaussian-Minkowski problem. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity 0 < | v | < 1 and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the noncompact prescribed Chern scalar curvature problem which reduces to solve a Kazdan-Warner type equation on noncompact non-Kähler manifolds. By introducing an analytic condition on noncompact manifolds, we establish related existence results. As its another application, we further give a new geometric proof of a classical multiplicity theorem of Ni (1982) [28]. [ABSTRACT FROM AUTHOR]

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v = 2 arctan ⁡ u x along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u 0 ∈ H 1 (R) was proved. [ABSTRACT FROM AUTHOR]

In this paper, we use a natural iteration technique to prove the existence of solutions to nonlinear Dirichlet problems. Among the examples included is the prescribed mean curvature equation. The nature of the technique allows applications to unbounded domains, as we show with domains of the form R n − 1 × (− d / 2 , d / 2). [ABSTRACT FROM AUTHOR]

In this paper, we investigate several properties of the positive solution to the nonlocal periodic-parabolic problem { u t = d [ ∫ Ω J (x − y) u (y , t) d y − u (x , t) ] + λ u − b (x) q (t) u p in Ω × [ 0 , T ] , u (x , 0) = u (x , T) in Ω , where Ω is a bounded smooth domain in R N (N ≥ 1), λ > 0 is a parameter, d > 0 and p > 1 are constants, q (t) is a positive and T -periodic continuous function, b (x) is a nonnegative continuous function, and is strictly positive only outside a domain Ω ¯ 0 ⊂ Ω. So the equation has a degeneracy over Ω 0. In such a case, it is known that there is a unique positive solution u λ if and only if λ is between the principle eigenvalues λ p (Ω) and λ p (Ω 0) of the associated nonlocal diffusion operator (over Ω and Ω 0 , respectively). We show that u λ is stable under perturbations of λ and b (x) in the equation, and when λ increases to λ p (Ω 0) , u λ converges to ∞ uniformly over Ω ¯ × [ 0 , T ] but a clear pattern is exhibited by u λ / ‖ u λ ‖ ∞ : u λ (x , t) ‖ u λ ‖ ∞ → { 0 for (x , t) ∈ (Ω ¯ ∖ Ω ¯ 0) × [ 0 , T ] , ϕ (x) for (x , t) ∈ Ω ¯ 0 × [ 0 , T ] , where ϕ (x) is the normalised positive eigenfunction associated to λ p (Ω 0). The proof relies on a general regularity result for the solutions of these types of nonlocal equations, which is of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study the propagation dynamics for a class of integro-difference equations with a shifting habitat. We first use the moving coordinates to transform such an equation to an integro-difference equation with a new kernel function containing the shifting speed c. In two directions of the spatial variable, the resulting equation has two limiting equations with spatial translation invariance. Under the hypothesis that each of these two limiting equations has both leftward and rightward spreading speeds, we establish the spreading properties of solutions and the existence of nontrivial forced waves for the original equation by appealing to the abstract theory of nonmonotone semiflows with asymptotic translation invariance. Further, we prove the stability and uniqueness of forced waves under appropriate conditions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stationary solutions to the one-dimensional full (non-isentropic) compressible Navier-Stokes-Korteweg equations with far field states and boundary data in the half line. When the fluids enter into and blow out the region through the boundary, respectively, the unique existence of the stationary solutions to the one-dimensional full compressible Navier-Stokes-Korteweg equations in the half line is shown provided that the boundary strength is small enough. Moreover, we also give the spatial decay rates of the stationary solutions. The main ingredient of the proof is the manifold theory and the center manifold theorem that take the accurate analysis of the cubic characteristic equations. [ABSTRACT FROM AUTHOR]

In this paper we consider determining a minimal surface embedded in a Riemannian manifold Σ × R. We show that if Σ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine Σ up to an isometry. [ABSTRACT FROM AUTHOR]

This paper introduces a new notion of dispersion for kinetic equations solely based on the conservation laws and independent of the specific type of interactions. We present new a-priori estimates for kinetic PDEs and improve the Bony-type functional approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals T λ f (x) = p.v. ∫ R n Ω (y) | y | n − λ f (x − y) d y for any f ∈ L q (R n) , 1 < q < ∞ and 0 < λ < n , which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation (0.1) ∂ t θ + u ⋅ ∇ θ + κ Λ 2 β θ = 0 , (x , t) ∈ R 2 × R + , κ > 0 , where u = − ∇ ⊥ Λ − 2 + 2 α θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ]. Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator T λ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [ b , T λ ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation. [ABSTRACT FROM AUTHOR]

This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established. [ABSTRACT FROM AUTHOR]

In this paper, we study the following integral equation u (x) = ∫ R n u p (y) (1 + | y | α) | x − y | n − α d y , u > 0 i n R n , where n ≥ 3 , α ∈ (0 , n) , and p > 0. When α = 2 , the equation is associated with the classical Matukuma equation − Δ u = u p (x) 1 + | x | 2 , u > 0 i n R n. We investigate qualitative properties of two kinds of positive solutions. One is the finite total mass solution, and the other is the finite energy solution. We investigate the qualitative properties of these solutions, including existence, radial symmetry, integrability and asymptotic behavior at infinity. Here we use the method of moving planes in integral forms introduced by Chen-Li-Ou to prove the radial symmetry, and develop a new comparing principle in integral forms to estimate the asymptotic rates. In addition, we consider the relation between finite total mass solutions and finite energy solutions, and in this process the Pohozaev identity in integral forms plays a key role. [ABSTRACT FROM AUTHOR]

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation ∂ t u = Δ V u + H (u) on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As its application, on the one hand, we obtain new and improved estimates for logarithmic type equation and Yamabe type equation. On the other hand, we establish some sharp estimates for them under non-negative Bakry-Émery curvature condition. As their natural corollary, we also obtain some sharp Harnack inequalities and Liouville type theorems. [ABSTRACT FROM AUTHOR]

In this paper, we justify the hydrodynamic limit of the self-organized kinetic (SOK) model with density-dependent parameters. Amic Frouvelle [10] derived this SOK model and formally deduced its hydrodynamic limit to get a self-organized hydrodynamic (SOH) model. Firstly, the local existence of the classical solution of the SOH equations is proved. Then we use Hilbert expansion method and generalized collision invariants (GCIs) hypothesis to construct the solutions of the SOK equations. Finally, we verify that the equilibrium related to the solution of the SOH equations is the limit of solutions of the SOK equations as ϵ → 0. [ABSTRACT FROM AUTHOR]

Let Δ d be the discrete Laplacian defined on Z d by setting Δ d f (n →) = ∑ j = 1 d − [ f (n → + e → j) + f (n → − e → j) − 2 f (n →) ] , n → ∈ Z d , where { e → j : j = 1 , ... , d } is the standard basis for R d. In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation { u t + Δ d u = f , t ∈ [ 0 , T) u (0 , ⋅) = 0 , where T ∈ (0 , ∞). [ABSTRACT FROM AUTHOR]

This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion on the periodic domain. The stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit large-time decay rate of the solutions. [ABSTRACT FROM AUTHOR]

This paper considers a kind of coupled FitzHugh-Nagumo (FHN) equations, combined the classical FHN equations with the mechanics equation. The traveling pulses in coupled FHN equations in the presence of doubly-diffusive effect and local time delay are investigated. The singular orbits are constructed by analyzing limit dynamics of the equations in the traveling wave framework. Particularly, the full system involves three time scales, making it more challenging to seek for the invariant manifold. In order to establish the traveling pulses for the full system, the main analysis relies on the geometric singular perturbation theory and Exchange Lemma. [ABSTRACT FROM AUTHOR]

The Landau–Lifshitz–Baryakhtar (LLBar) equation is a generalisation of the Landau–Lifshitz–Gilbert and the Landau–Lifshitz–Bloch equations which takes into account contributions from nonlocal damping and is valid at moderate temperature below the Curie temperature. Therefore, it is used to explain some discrepancies between the experimental observations and the known theories in various problems on magnonics and magnetic domain-wall dynamics. In this paper, the existence and uniqueness of global weak, strong, and regular solutions to LLBar equation are proven. Hölder continuity of the solution is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the Cauchy problem for the Fornberg–Whitham equation is not locally well-posed in B p , r s (R) with (s , p , r) ∈ (1 , 1 + 1 p) × [ 2 , ∞) × [ 1 , ∞ ] or (s , p , r) ∈ { 1 } × [ 2 , ∞) × [ 1 , 2 ] by showing norm inflation phenomena of the solution for some special initial data. [ABSTRACT FROM AUTHOR]

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h , it has been shown that for any M > 1 there exists s M ∈ R such that the equation x ″ + f (x) x ′ + h (t , x) = s has two positive T -periodic solutions u 1 (⋅ ; s) and u 2 (⋅ ; s) satisfying min ⁡ { u 1 (t ; s) : t ∈ [ 0 , T ] } > M and min ⁡ { u 2 (t ; s) : t ∈ [ 0 , T ] } < 1 / M for every s < s M. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T -periodic solutions of indefinite differential equations. • Establish a refinement result of the Ambrosetti-Prodi type in the absence of anti-coercivity condition. • Investigate the dynamic behaviors of positive periodic solutions to a generalized Liénard equation. • Provide a new approach to study the existence of positive periodic solutions for indefinite differential equations. • Find a relationship between the strict lower functions and Leray-Schauder degree. [ABSTRACT FROM AUTHOR]

This paper is concerned with the qualitative properties of the positive ground state solutions to the nonlocal Choquard type equation on a ball B R. First, we prove the radial symmetry of all the positive ground state solutions by using Talenti's inequality. Next we develop the Newton's Theorem and then resort to the contraction mapping principle to establish the uniqueness of the positive ground state solution. Finally, by constructing cut-off functions and applying energy comparison method, we show the convergence of the unique positive ground state solutions as the radius R → ∞. Our results extend and complement the existing ones in the literature. [ABSTRACT FROM AUTHOR]

Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. In this paper, we study three types of averaging principle for stochastic complex Ginzburg-Landau equations. Firstly, we prove that the solution of the original equation converges to that of the averaged equation on finite intervals as the time scale ε goes to zero when the initial data are the same. Secondly, we show that there exists a unique recurrent solution (in particular, periodic, almost periodic, almost automorphic, etc.) to the original equation in a neighborhood of the stationary solution of the averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as ε goes to zero. [ABSTRACT FROM AUTHOR]

For the famous Camassa-Holm equation, the well-posedness in C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p ≤ 2 and the ill-posedness in C ([ 0 , T ] ; B ∞ , 1 1 (R)) had been studied in [15,16,23]. That is to say, it left an open problem in the critical case C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 2 < p < + ∞ proposed by Danchin in [15,16]. In this paper, we solve this problem and obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p < + ∞. It is worth mentioning that our method is suitable for many Camassa-Holm type equations, such as the Novikov equation and the two-component Camassa-Holm system, and can also improve their index of the local well-posedness. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the two-dimensional Euler-Boussinesq equations with temperature-dependent thermal diffusivity. More precisely, we prove the global-in-time existence and uniqueness for this system under Yudovich-type initial data. In addition, the global propagation of striated regularity is also considered. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of global solutions in H 3 (R) ∩ H 2 , 1 (R) to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons. A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation dynamics of nonlocal dispersal monostable equations in time-space periodic habitats. We first show that such an equation admits a single spreading speed in every direction under certain conditions and then give several spreading properties in terms of spreading speeds such as asymptotic spreading ray speeds and asymptotic spreading sets. Furthermore, we consider the dependence of the spreading speed on the dispersal rate and reaction term and prove that taking the temporal average or spatial average can decrease the spreading speed. Finally, we employ the viscosity vanishing method to establish the existence of time-space periodic traveling fronts with the critical speed in every direction under the partially temporally homogeneous case and partially nearly flat case, which solves partially the open problem raised by Rawal, Shen, and Zhang (Discrete Contin. Dyn. Syst. 35 (2015) 1609–1640). [ABSTRACT FROM AUTHOR]

In this paper we are concerned with Kirchhoff equations with strong damping and quasi-periodic external forcing on arbitrary flat tori. Such models arise from nonlinear forced vibrations of multidimensional bodies in which the dependence of the tension on the deformation cannot be neglected. We establish the existence of quasi-periodic traveling waves whose frequency is consistent with that of the forcing term. Due to strong damping, we don't impose any non-resonance conditions. Our arguments provide a general framework for studying singular perturbation problems with strong damping. [ABSTRACT FROM AUTHOR]

This paper deals with the following radial Caffarelli-Kohn-Nirenberg-type inequality, ∫ R N | x | α | Δ u | 2 d x ≥ S r a d (N , α) ( ∫ R N | x | l | u | p α ⁎ d x) 2 p α ⁎ , u ∈ C c ∞ (R N) , where N ≥ 3 , 2 < α < N , l = 4 (α − 2) (N − 2) N − α − α and p α ⁎ = 2 (N + l) N − 4 + α. Then we consider the related Euler-Lagrange equation: Δ (| x | α Δ u) = | x | l u p α ⁎ − 1 , u > 0 in R N. For α ≠ 0 or l ≠ 0 , it is known the solutions of above equation are invariant for dilations λ N − 4 + α 2 u (λ x) but not for translations. However we show that if α is an even integer, there exist new solutions to the linearized problem, related to the radial solution that "replace" the ones due to the translations invariance. As an application, we turn back to investigate the remainder terms of the above inequality. As well as some fourth-order Liouville-type equations with singular data. [ABSTRACT FROM AUTHOR]

This paper deals with the study of the existence and multiplicity of solutions for the class of nonlocal problems that have arised in recent developments in the mathematical physics of string theory and cosmology given by (P) { − Δ e − c Δ u + u = λ P (x) (u + f (x , u)) , in R N lim | x | → ∞ ⁡ u (x) = 0 , u ∈ H c , ∞ (R N) , where N ≥ 3 , c > 0 , λ > 0 , P : R N → R is a positive continuous function, f : R N × R → R is C 1 -function, e − c Δ is defined via a power series and H c , ∞ (R N) is a Hilbert space as introduced in [11]. The main tools used here are: the Minimax Theorems and a bifurcation result via variational methods due to Rabinowitz. [ABSTRACT FROM AUTHOR]

In this paper we use variational methods to establish the existence of heteroclinic solution for the prescribed mean curvature equation of the form − ( q ′ 1 + (q ′) 2 ) ′ + a (t) V ′ (q) = 0 in R , where V is a double-well potential with minima at t = ± α and a ∈ L ∞ (R) is an even non-negative function with 0 < a 0 : = inf t ≥ M ⁡ a (t) for some M > 0. [ABSTRACT FROM AUTHOR]