For a centrally symmetric near-Hamiltonian system, we develop a method for computing all the coefficients in the expansions of three Melnikov functions near a double homoclinic loop. Moreover, we give a new estimation on the lower bound of H(2nˆ,5) for 11≤nˆ≤23, where H(2nˆ,5) is the maximal number of limit cycles for a kind of Liénard system, x˙=y,y˙=−g(x)+εf(x)y, with deg⁡g(x)=5 and deg⁡f(x)=2nˆ

The composites of materials with high contrasting properties is an interesting topic to study as it has applications. In this article, we wish to study problems in high oscillating domains, where the oscillatory part is made of two materials with high contrasting conductivities (or diffusivity). Thus the low contrast material acts as near insulation in-between the conducting materials. In the first

We study the Cauchy problem of the mass critical nonlinear Schrödinger equation with derivative with the 4π mass. One has the global well-posedness in H1 whenever “the mass is strictly less than 4π” or whenever “the mass is equal to 4π and the momentum is strictly less than zero”. In this paper, by the concentration compact principle as originally done by Kenig-Merle, we obtain the limiting profile

The mathematical model of the transport and diffusion of ions in biological channels is considered. It is described by the three-dimensional nonlinear evolution classical Poisson–Nernst–Planck (cPNP) system of partial differential equations with nonlinear coupled boundary conditions. In particular the Chang–Jaffé (CJ) conditions are given on the input and output of a channel. The Robin boundary conditions

We prove global existence of finite energy weak solutions to the quantum Navier-Stokes equations in the whole space with non trivial far-field condition in dimensions d=2,3. The vacuum regions are included in the weak formulation of the equations. Our method consists in an invading domains approach. More precisely, by using a suitable truncation argument we construct a sequence of approximate solutions

We consider Monge-Ampère equations with the right hand side function close to a positive constant and from a function class that is larger than any Hölder class and smaller than the Dini-continuous class. We establish an upper bound for the modulus of continuity of the solution's second derivatives. This bound depends exponentially on a quantity similar to but larger than the Dini semi-norm. We establish

In this paper, motivated by recent papers on the stabilization of evolution problems with nonlocal degenerate damping terms, we address an extensible beam model with degenerate nonlocal damping of Balakrishnan-Taylor type. We discuss initially on the well-posedness with respect to weak and regular solutions. Then we show for the first time how hard is to guarantee the stability of the energy solution

We study an optimal control problem consisting in minimizing the L∞ norm of a Borel measurable cost function, in finite time, and over all trajectories associated with a controlled dynamics which is reflected in a compact prox-regular set. The first part of the paper provides the viscosity characterization of the value function for uniformly continuous costs. The second part is concerned with linear

The purpose of this paper is to study the global wellposedness and large time behavior results of strong solutions for the compressible Oldroyd-B model derived by Barrett, Lu, Süli (Commun. Math. Sci., 15, 1265–1323, 2017). Exploiting the harmonic analysis tools (especially Littlewood-Paley theory), we first study the global well-posedness of the model with small initial data in spaces with low regularity

In this paper we consider in a bounded domain Ω⊂RN with smooth boundary an eigenvalue problem for the negative (p,q)-Laplacian with a Steklov-like boundary condition, where p,q∈(1,∞), p≠q, including the open case p∈(1,∞), q∈(1,2), p≠q. A full description of the set of eigenvalues of this problem is provided. Our results complement those previously obtained by Abreu-Madeira [1], Barbu-Moroşanu [2],

We consider a parabolic PDE with Dirichlet boundary condition and monotone operator A with non-standard growth controlled by an N-function depending on time and spatial variable. We do not assume continuity in time for the N-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic N-function, we

Stability and large-time behavior are essential properties of solutions to many partial differential equations (PDEs) and play crucial roles in many practical applications. When there is full Laplacian, many techniques such as the Fourier splitting method have been created to obtain the large-time decay rates. However, when a PDE is anisotropic and involves only partial dissipation, these methods no

In this paper, the existence, stability, and multiplicity of steady-state solutions and periodic solutions for a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition are investigated by using Lyapunov-Schmidt reduction. When the interior reaction term is weaker than the boundary reaction term, it is found that there is no Hopf bifurcation no matter how either of the

We consider a reaction-diffusion-advection problem in a perforated medium, with nonlinear reactions in the bulk and at the microscopic boundary, and slow diffusion scaling. The microstructure changes in time; the microstructural evolution is known a priori. The aim of the paper is the rigorous derivation of a homogenized model. We use appropriately scaled function spaces, which allow us to show compactness

In the present paper we initiate the variational analysis of a super sinh-Gordon system on compact surfaces, yielding the first example of non-trivial solution of min-max type. The proof is based on a linking argument jointly with a suitably defined Nehari manifold and a careful analysis of Palais-Smale sequences. We complement this study with a multiplicity result exploiting the symmetry of the problem

We show that the viscous resistive magnetohydrodynamics system with Hall effect is locally well-posed in Hs(R3)×Hs+1−ε(R3) with s>12 and any small enough ε>0 such that s−ε>12. This space is to date the largest local well-posedness space in the class of Sobolev spaces for the system. It is also optimal according to the predominant scalings of the two equations in the system.

In this paper, we discuss the following chemotaxis-Stokes system with nonlinear diffusion and rotation{nt+u⋅∇n=Δnm−∇⋅(nS(x,n,c)⋅∇c),x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 with m>0, where Ω is a bounded domain in R3, S(x,n,c) is a chemotactic sensitivity tensor satisfying S∈C2(Ω¯×[0,∞)2;R3×3) and |S(x,n,c)|≤(1+n)−αS0(c) for all (x,n,c)∈Ω×[0,∞)2 with α≥0 and some nondecreasing

We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a W1,1 convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and

We are concerned with two kinds of singular limits of the Cauchy problem to the two-layer rotating shallow water equations as the Rossby number and the Froude number tend to zero. First we construct the uniform estimates for the strong solutions to the system under the condition that the Froude number is small enough. Different from the previously studied cases, the large operator of this model is

Motivated by various physical, cellular and ecological applications, there has been a recent resurgence of interest in studying the boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface, by using “bulk-surface models” (involving volumetric densities and surface densities) or by using models involving dynamical boundary conditions for volumetric densities. The surface

We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in Lq with q>n. Our approach is based on the scaling and comparison arguments of [13], and we show that all our

In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these

This paper analyses the existence and properties of solutions of the extended play-type model which was proposed in [16] to incorporate hysteresis in unsaturated flow through porous media. The model, when regularised, reduces to a nonlinear degenerate parabolic equation coupled to an ordinary differential equation. This PDE–ODE coupled system, also studied in the fields of bio-film and cellular biology

Motivated by applications in relativistic fluid dynamics, this paper studies a general class of linear systems of partial differential equations in which a second-order symmetric hyperbolic part and a first-order symmetric hyperbolic part interact dissipatively. It shows that corresponding algebraic properties of the two operators induce decay in natural Sobolev spaces, and discusses examples which

This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with

Consider the time decay rates of the smooth solution to the compressible MHD system on T3. It is shown that if the density ρ is uniformly bounded from up and below, and the L6 norm of the magnetic H is uniformly bounded, then all derivatives of the smooth solution to the compressible MHD system decay exponentially fast to the equilibrium. Furthermore, we construct a global smooth solution obeying these

We present an Lq(Lp)-theory for the equation∂tαu=ϕ(Δ)u+f,t>0,x∈Rd;u(0,⋅)=u0. Here p,q>1, α∈(0,1), ∂tα is the Caputo fractional derivative of order α, and ϕ is a Bernstein function satisfying the following: ∃δ0∈(0,1] and c>0 such that(0.1)c(Rr)δ0≤ϕ(R)ϕ(r),0

We analyze the one dimensional Cucker–Smale (in short CS) model with a weak singular communication weight ψ(x)=|x|−β with β∈(0,1). We first establish a global-in-time existence of measure-valued solutions to the kinetic CS equation. For this, we use a proper change of variable to reformulate the particle CS model as a first-order particle system and provide the uniform-in-time stability for that particle

The Keller-Segel-Stokes type system with indirect signal production, as given by{nt+u⋅∇n=Δn−∇⋅(nS(n)∇v),x∈Ω,t>0,vt+u⋅∇v=Δv−v+w,x∈Ω,t>0,wt+u⋅∇w=Δw−w+n,x∈Ω,t>0,ut+∇P=Δu+n∇ϕ,∇⋅u=0,x∈Ω,t>0, is considered in a bounded domain Ω⊂RN, N∈{2,3}, with smooth boundary, where ϕ∈C2(Ω‾) and S∈C2([0,∞)). Under the assumption that there exist CS>0 and α≥0 such that|S(n)|≤CS(n+1)−αfor all n≥0, it is shown that for all

We consider the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility in a bounded domain Ω⊂Rd, d≤3. We first prove the existence of maximal strong solutions in weighted (in time) Lp spaces. Then we establish further regularity properties of the solution through maximal regularity theory. Finally, we revisit the separation property in an appendix.

We study the coupled Hartree system{−Δu+V1(x)u=α1(|x|−4⁎u2)u+β(|x|−4⁎v2)uinRN,−Δv+V2(x)v=α2(|x|−4⁎v2)v+β(|x|−4⁎u2)vinRN, where N≥5, β>max⁡{α1,α2}≥min⁡{α1,α2}>0, and V1,V2∈LN/2(RN)∩Lloc∞(RN) are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with V1=V2=0 we employ moving sphere arguments in integral form to classify positive solutions

In this article we will develop some techniques aimed at the strong couplings in two-dimensional wave-Klein-Gordon system. We distinguish the roles of different type of decay factors and develop a method which permits us to “exchange” one type of decay into the other. Then a global existence result of a model problem is established. We also give a sketch of the Klein-Gordon-Zakharov model system and

We consider the Hénon equation(Pα)−Δu=|x|α|u|p−1uinBN,u=0on∂BN, where BN⊂RN is the open unit ball centered at the origin, N≥3, p>1 and α>0 is a parameter. We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation−Δw=|w|p−1winB2,w=0on∂B2, where B2⊂R2 is the open unit ball, is the limit problem of (Pα), as α→∞, in the framework of radial solutions. We exploit this fact to prove

We consider a class of logarithmic Keller-Segel type systems modeling the spatio-temporal behavior of either chemotactic cells or criminal activities in spatial dimensions two and higher. Under certain assumptions on parameter values and given functions, the existence of classical solutions is established globally in time, provided that initial data are sufficiently regular. In particular, we enlarge

We use Lie brackets of unbounded vector fields to consider a dissipative relation that generalizes the differential inequality which defines classic control Lyapunov functions. Under minimal regularity assumptions, we employ locally semiconcave solutions of this extended relation, called in the following degree-k control Lyapunov functions, in order to design degree-k Lyapunov feedbacks, i.e. particular

It has been shown by Chen that if all masses are in the same proper vector subspace of Rd(d≥2) on one of the free boundaries, local minimizers connecting the two free boundaries have no collision on that boundary. However, not much is known for other types of boundary configurations. In this paper, we consider minimizers connecting two free boundaries and study possible quadruple collisions under the

In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p>2){nt+u⋅∇n=∇⋅(|∇n|p−2∇n)−χ∇⋅(n∇c),ct+u⋅∇c−Δc=−cn,ut+∇π=Δu+n∇φ,divu=0 in a bounded domain Ω of R3 with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem

We correct an error in our paper “Inviscid limit of the compressible Navier–Stokes equations for asymptotically isothermal gases”, which was made at four distinct points in the document. The main result therein (Theorem 1.4) is correct as stated, with the only exception that the requirement α>1/2 must be imposed in point (3) of Definition 1.1 of the paper.

In this paper, we are interested in providing lower estimations for the maximum number of limit cycles H(n) that planar piecewise linear differential systems with two zones separated by the curve y=xn can have, where n is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that H(2)≥4, H(3)≥8, H(n)≥7

In this paper, we study the planar Lp-Minkowski problem(0.1)uθθ+u=fup−1,θ∈S1 for all p∈R, which was introduced by Lutwak [21]. A detailed exploration of (0.1) on solvability will be presented. More precisely, we will prove that for p∈(0,2), there exists a positive function f∈Cα(S1),α∈(0,1) such that (0.1) admits a nonnegative solution vanishes somewhere on S1. In case p∈(−1,0], a surprising a-priori

In this paper we study regularity of partial differential equations with polynomial coefficients in non isotropic Beurling spaces of ultradifferentiable functions of global type. We study the action of transformations of Gabor and Wigner type in such spaces and we prove that a suitable representation of Wigner type allows to prove regularity for classes of operators that do not have classical hypoellipticity

This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has a countably infinite state space. After establishing the existence and uniqueness of a global weak solution via the martingale approach under very mild conditions, the paper next proves the strong Feller property for regime-switching stochastic damping Hamiltonian systems

In this paper, we are concerned with the global existence and time decay rates of the solutions to Boltzmann equation for hard potential in the critical Chemin-Lerner type space. The evolution of the energy in negative Besov space is derived to be preserved. Our results are based on the modified trilinear estimates and some new observations. We use the Lyapunov-type inequality of the energy functionals

We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter ε entering an Allen-Cahn-type energy

We prove the global well-posedness of the strong solution to the two-dimensional inhomogeneous incompressible primitive equations for initial data with small H12-norm, which also satisfies a natural compatibility condition. A logarithmic type Sobolev embedding inequality for the anisotropic Lx∞Lz2(Ω) norm is established to obtain the global in time a priori H1(Ω)∩W1,6(Ω) estimate of the density, which

An approximation of a Generalized Langevin equations with small mass and randomly fast oscillation is derived. By some bounded estimates and a diffusion approximation the limiting equation is shown to be stochastic partial differential equations (SPDEs) driven by white noise.

This paper is concerned with an initial and boundary value problem of the compressible Navier-Stokes equations for one-dimensional viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients. In the case when the viscosity μ(θ)=θα and the heat-conductivity κ(θ)=θβ with α,β∈[0,∞), we prove the global-in-time existence of strong solutions under some assumptions

In this paper, we consider an eigenvalue problem defined on a two-patch domain. Our first aim is to show that the two-patch eigenvalue problem is equivalent to the eigenvalue problem of a measure differential equation defined on an one-patch domain. Our second aim is to study the existence of principal eigenvalue of the measure differential equation, and we will prove the principal eigenvalue is continuously

In previous articles, the authors provided sufficient conditions for global solvability of linear differential operators defined on a n-dimensional manifold, with real principal part, under the hypothesis of the existence of one hyperbolic singular point. In this work, for n=2, we extend the result for non-degenerate singular semi-hyperbolic point.

In this paper, we investigate steady inviscid compressible flows with radial symmetry in an annulus. The major concerns are transonic flows with or without shocks. One of the main motivations is to elucidate the role played by the angular velocity in the structure of steady inviscid compressible flows. We give a complete classification of flow patterns in terms of boundary conditions at the inner and

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general assumptions and based on variational tools we prove the existence of at least one nontrivial weak solution for such a problem. This is the first work dealing with a Finsler

In this paper, we introduce traveling waves to one-dimensional Cauchy problems (CP) for scalar parabolic-hyperbolic conservation laws. The equation is regarded as a linear combination of the scalar hyperbolic conservation laws and the porous medium type equations. Thus, this equation has both properties of hyperbolic equations and those of parabolic equations. Accordingly, it is difficult to investigate

We construct modal outgoing Green's kernels for the simplified Galbrun's equation under spherical symmetry, in the context of helioseismology. The coefficients of the equation are C2 functions representing the solar interior model S, complemented with an isothermal atmospheric model. We solve the equation in vectorial spherical harmonics basis to obtain modal equations for the different components

We study the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity:utt−uxx+μtut=tα|u|p,t>t0,x∈R, where μ>0, p>1 and α>−2. Here either t0=0 (singular problem) or t0>0 (regular problem). We show that this model may be interpreted as a semilinear wave equation with borderline dissipation: the existence of global small data solutions depends not only on the power p, but also

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results

The aim of this article is to study a Cahn-Hilliard model for a multicomponent mixture with cross-diffusion effects, degenerate mobility and where only one of the species does separate from the others. We define a notion of weak solution adapted to possible degeneracies and our main result is (global in time) existence. In order to overcome the lack of a-priori estimates, our proof uses the formal

First, we provide a necessary and sufficient condition of the existence of viscosity solutions of the nonlinear first order PDEF(x,u,Du)=0,x∈M, under which we prove the compactness of the set of all viscosity solutions. Here, F(x,u,p) satisfies Tonelli conditions with respect to the argument p and −λ≤∂F∂u<0 for some λ>0, and M is a compact manifold without boundary. Second, we study the long time behavior

In the standard theory of delay equations, the fundamental solution does not ‘live’ in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we

The Landau-Lifshitz-Bloch-Maxwell equations with polarization describe the evolution of the mean fields in continuous ferromagnetics. In this paper, we firstly use the energy method to prove the existence of weak solution to the Landau-Lifshitz-Bloch-Maxwell equations with polarization for the viscosity problem in two dimensions. Then we prove that the estimates are uniformed in ϵ for solutions to

An important question of ongoing interest for linear time-delay systems is to provide conditions on its parameters guaranteeing exponential stability of solutions. Recent works have explored spectral techniques to show that, for some low-order delay-differential equations of retarded type, spectral values of maximal multiplicity are dominant, and hence determine the asymptotic behavior of the system