We consider strictly log-concave measures, whose bounds degenerate at infinity. We prove that the optimal transport map from the Gaussian onto such a measure is locally Lipschitz, and that the eigenvalues of its Jacobian have controlled growth at infinity.

In this paper we study the global regularity for the solution to the Dirichlet problem of the equation of minimal graphs over a convex domain in hyperbolic spaces. We find that the global regularity depends only on the convexity of the domain but independent of its smoothness. Basing on the invariance of the problem under translation and rotation transforms, we construct the super-solution to the problem

Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measure-valued solutions to the two-dimensional isentropic compressible Euler equations, although they are energy admissible, can be discarded as unphysical, as they do not arise as vanishing viscosity limits. In fact

We consider nonlinear problems governed by the fractional p-Laplacian in presence of nonlocal Neumann boundary conditions and we show three different existence results: the first two theorems deal with a p-superlinear term, the last one with a source having p-linear growth. For the p-superlinear case we face two main difficulties. First: the p-superlinear term may not satisfy the Ambrosetti-Rabinowitz

The eigenvalues of the Laplace-Beltrami operator on the torus embedded in three-dimensional euclidean space are investigated. These eigenvalues are determined by the eigenvalues of Sturm-Liouville problems with separated boundary conditions. Eigenvalues are approximated by those of generalized eigenvalue problems involving tridiagonal matrices. A non-coexistence result is proved. The behavior of eigenvalues

An existence result for Neumann elliptic systems with singular, convective, sign-changing, arbitrarily growing reactions is established. Proofs are chiefly based on sub-super-solution and truncation techniques, nonlinear regularity theory, and fixed point arguments. As a consequence, infinitely many solutions are obtained through appropriate sequences of sub-super-solution pairs.

We consider here the magneto-hydrodynamics (MHD) equations on the whole space. For the 3D case, in the setting of the weighted L2 spaces we obtain a weak-strong uniqueness criterion provided that the velocity field and the magnetic field belong to a fairly general multipliers space. On the other hand, we study the local and global existence of weak suitable solutions for intermittent initial data,

We study the existence of sign changing solutions to the following problem(0.1){Δu+|u|p−1u=0inΩε;u=0on∂Ωε, where p=n+2n−2 is the critical Sobolev exponent and Ωε is a bounded smooth domain in Rn, n≥3, of the form Ωε=Ω\B(0,ε). Here Ω is a smooth bounded domain containing the origin 0 and B(0,ε) denotes the ball centered at the origin with radius ε>0. We construct a new type of sign-changing solutions

We consider Schrödinger operators on the line with potentials that are periodic with respect to the coordinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded in the right half-plane, then the high energy spectrum is real, and the corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann problems are

We investigate the spreading properties of a three-species competition-diffusion system, which is not order-preserving. We apply the Hamilton-Jacobi approach, due to Freidlin, Evans and Souganidis, to establish upper and lower estimates of spreading speed for the slowest species, which turn out to be dependent on the spreading speeds of the two faster species. The estimates we obtained are sharp in

A flower graph consists of a half line and N symmetric loops connected at a single vertex with N≥2 (it is called the tadpole graph if N=1). We consider positive single-lobe states on the flower graph in the framework of the cubic nonlinear Schrödinger equation. The main novelty of our paper is a rigorous application of the period function for second-order differential equations towards understanding

This paper focuses on a special two-dimensional (2D) Boussinesq system modeling buoyancy driven fluids. It governs the motion of the velocity and temperature perturbations near the hydrostatic balance. This is a partially dissipated system with the velocity involving only the vertical dissipation. We are able to establish the global stability and the large-time behavior of the solutions. In particular

In this article, we propose a general framework for the study of differential inclusions in the Wasserstein space of probability measures. Based on earlier geometric insights on the structure of continuity equations, we define solutions of differential inclusions as absolutely continuous curves whose driving velocity fields are measurable selections of multifunction taking their values in the space

We study solvability of some linear nonhomogeneous elliptic problems and establish that under reasonable technical conditions the convergence in L2(Rd) of their right sides implies the existence and the convergence in H4(Rd) of the solutions. The problems contain the squares of the sums of second order non-Fredholm differential operators and we use the methods of the spectral and scattering theory

Let n∈{2,3,4,…}, N∈{1,2,3,…} and p∈(1,2−1n]. Let β∈(1,∞) be such thatnpn−p<β′

Let A(s,t) be a continuous function with a positive lower bound m, and Ω be a bounded domain in RN. In this short note, we propose an iterative procedure for finding nonnegative solutions of the following Kirchhoff-Carrier type equations{−A(‖u‖p,‖∇u‖2)Δu=g(x,u)x∈Ω,u=0x∈∂Ω. The main advantage of our procedure is that the convergent proof of the iterative sequence depends only on comparison principle

We use a double mollification method to study a family of high order nonlinear partial differential equations with peakon weak solutions. This method gives the approximated N-peakon weak solutions which satisfy weak consistent property. By a limiting process, we prove the global existence of N-peakon weak solutions. Moreover, the traveling speeds of single peakon weak solutions (soliton waves) are

We consider the motion of a point mass in a one-dimensional viscous compressible barotropic fluid. The fluid–point mass system is governed by the barotropic compressible Navier–Stokes equations and Newton's equation of motion. Our main result concerns the long-time behavior of the fluid and the point mass. Pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium

We study the Cauchy problem of the incompressible damped wave type magnetohydrodynamic system in RN (N≥2). The purpose of this paper is to show the global well-posedness and a singular limit of the problem in Fourier–Sobolev spaces. For the proof of the results, we use the Lp-Lq type estimates for the fundamental solutions of the damped wave equation and end-point maximal regularity for the inhomogeneous

We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. This technique is useful to study lower bounds for the local cyclicity of centers. We denote by M(n) the

In this paper, we construct two classes of planar polynomial Hamiltonian systems having a center at the origin, and obtain the lower bounds for the number of critical periods for these systems. For polynomial potential systems of degree n, we provide a lower bound of n−2 for the number of critical periods, and for polynomial systems of degree n, we acquire a lower bound of n2/2+n−5/2 when n is odd

We define new matrix fourth Painlevé hierarchies and a new matrix second Painlevé hierarchy. For our matrix fourth Painlevé hierarchies, properties such as auto-Bäcklund transformations, Bäcklund transformations and special integrals are derived, and known relations between fourth Painlevé transformations are extended to the matrix ODE hierarchies. The matrix ODE hierarchies presented here are derived

This paper investigates the dynamics of a general reaction-diffusion-advection equation with nonlocal delay effect and Dirichlet boundary condition. The existence and stability of positive spatially nonhomogeneous steady state solution are shown. By analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized equation, the existence of Hopf bifurcation is

This article begins with a full description of the quadratic planar vector fields which display two centers. We follow the method proposed by Chengzhi Li and provide more detailed analysis of the different types of double centers using the classification: Hamiltonian, reversible, Lotka-Volterra, Q4, currently used for centers of quadratic planar vector fields. We also describe completely the different

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities:y(t)≤E(t,τ)‖yτ‖+∫τtK1(t,s)‖ys‖ds+∫t∞K2(t,s)‖ys‖ds+ρ,t≥τ≥0. As a simple example of application, the retarded scalar functional differential equation x˙=−a(t)x+B(t,xt) is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is

We study an ill-posed forward-backward parabolic equation with techniques of nonstandard analysis. Equations of this form arise in many applications, ranging from the description of phase transitions, to the dynamics of aggregating populations, and to image enhancing. The equation is ill-posed in the sense that it only has measure-valued solutions that in general are not unique. By using grid functions

We are concerned with asymptotic behaviours of solutions for linear wave equations with frictional damping only on Wentzell boundary, but without any interior damping. Making some elaborate and subtle analysis of an associated auxiliary system, we obtain an ideal estimate of the resolvent of the generator of the system along the imaginary axis. This enables us to prove that the energies of the system

In this paper we consider the Cauchy problem of a model of the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity. We show that there is a unique global smooth solution to this system for arbitrarily large initial data in Sobolev spaces. Our key argument is the De Giorgi-Nash-Moser estimates for the vorticity equation.

In this paper, we derive the a priori second order estimate for admissible solutions in Γk+1 cone of complex Hessian equations with both sides of the equation depending on the gradient on compact Hermitian manifolds.

We study the fourth order Schrödinger operator H=(−Δ)2+V for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the L1→L∞ dispersive bounds. In all cases, we show that the natural |t|−34 decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct

This paper is concerned with analysis of the global behavior for a type of reaction-diffusion system incorporating the pathogens' incubation period, which describes the infection of bacteria in a population. By using the operator semigroup theory and dynamical system approaches, the global dynamics with respect to the homogeneous Dirichlet problem are investigated. Firstly, the positivity, boundedness

The quasilinear differential systemx′=ax+b|y|p⁎−2y+k(t,x,y),y′=c|x|p−2x+dy+l(t,x,y) is considered, where a, b, c and d are real constants with b2+c2>0, p and p⁎ are positive numbers with (1/p)+(1/p⁎)=1, and k and l are continuous for t≥t0 and small x2+y2. When p=2, this system is reduced to the linear perturbed system. It is shown that the behavior of solutions near the origin (0,0) is very similar

Global existence and boundedness of classical solutions of the chemotaxis–consumption systemnt=Δn−∇⋅(n∇c),0=Δc−nc, under no-flux boundary conditions for n and Robin-type boundary conditions∂νc=(γ−c)g for c (with γ>0 and C1+β(∂Ω)∋g>0 for some β∈(0,1)) are established in bounded domains Ω⊂RN, N≥1. Under a smallness condition on γ, moreover, we show convergence to the stationary solution.

The two-phase Hele-Shaw problem when two immiscible fluids have different viscosities is considered. A method of solution using the Schwarz function approach is discussed. Examples of different scenarios of the evolution of an elliptical interface in Hele-Shaw cells with both constant and a uniformly changing distance between the plates are shown. In these examples, the sinks and sources are not only

We consider a two dimensional molecular beam epitaxy equation with biharmonic dissipation. We give an improved gradient bound, which removes the logarithmic dependence on the dissipation coefficient previously shown in [1].

In this paper, we consider a class of Hessian quotient equations in Euclidean space. Under some sufficient conditions, we obtain an existence result using standard degree theory based on a priori estimates for solutions to Hessian quotient equations.

In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter–Saxton equation (1.1), and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking

In this document we discuss the long time behaviour for the homogeneous Landau-Fermi-Dirac equation in the hard potential case. Uniform in time estimates for statistical moments and Sobolev regularity are presented and used to prove exponential relaxation of non degenerate distributions to the Fermi-Dirac statistics. All these results are valid for rather general initial datum. An important feature

Spreading speed of spatio-temporal nonlinear dynamical system can sometimes be determined either by its corresponding linear system with an explicit speed formula, or by the complicated nonlinear system itself with the existence of a pushed wavefront. In this paper, the spreading speed (the minimal speed of wavefronts) for a Lotka-Volterra competition model in spatially periodic habitats is investigated

We study the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equationiqt(x,t)+qxx(x,t)+2q2(x,t)q¯(−x,t)=0 with a step-like initial data: q(x,0)=q0(x), where q0(x)=o(1) as x→−∞ and q0(x)=A+o(1) as x→∞, with an arbitrary positive constant A>0. The main aim is to study the long-time behavior of the solution of this problem. We show that the asymptotics has qualitatively different

This is the continuation of our previous work [10], where we introduced and studied some nonlinear integral equations on bounded domains that are related to the sharp Hardy-Littlewood-Sobolev inequality. In this paper, we introduce some nonlinear integral equations on bounded domains that are related to the sharp reversed Hardy-Littlewood-Sobolev inequality. These are integral equations with nonlinear

The aim of this paper is to provide sharp regularity estimates for locally bounded solutions of the degenerate doubly nonlinear parabolic equationut−div(m|u|m−1|Du|p−2Du)=f, where m>1, p>2 and f∈Lq,r. More precisely, we show that solutions are locally of class C0,β, where β depends explicitly only on the optimal Hölder exponent for solutions of the homogeneous case, the integrability of f in space

Using the Euler-Jacobi formula there is a relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the polynomial differential systems x˙=P(x,y,z), y˙=Q(x,y,z), z˙=R(x,y,z) with degrees of P, Q and R equal to two when these systems have the maximum

We combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions. We analyze the behavior of facets at the parts of the boundary where these conditions are imposed. We devote particular attention to the radially symmetric data. We observe that the boundary layer detachment actually can happen at concave parts of the boundary.

We consider a chemotaxis-Navier-Stokes system modelling cellular swimming in fluid drops where an exchange of oxygen between the drop and its environment is taken into account. This phenomenon results in an inhomogeneous Robin-type boundary condition. Moreover, the system is studied without the logistic growth of the bacteria population. We prove that in one or two dimensions, the system has a unique

In this paper we consider the linear quasi-periodic systemx˙=(A+ϵP(t))x,x∈Rd, where A is a d×d constant matrix of elliptic type and has multiple eigenvalues, P(t) is analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-degeneracy condition and 0≤β(α)

In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying

This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system on the real line. We prove the existence of some entire solutions that are asymptotic, as t→−∞, to a traveling wave consisting of a single species. Depending on system parameters, these entire solutions evolve into two ore more stacked invasion waves as t→+∞. Our results cover both the weak

We study the long-time behavior of the solution to a type of dissipative wave equation, where the operator in the equation is time-dependent and the solution is defined on a metric measure space (X,m) satisfying appropriate conditions. The operator is assumed to be self-adjoint and is related to a time-dependent Dirichlet form. We link hyperbolic PDEs with the firmly established theories for parabolic

This paper deals with the quasilinear chemotaxis-haptotaxis model of cancer invasion{ut=∇⋅(D(u)∇u)+∇⋅(S1(u)∇v)+∇⋅(S2(u)∇w)+f(u,w),x∈Ω,t>0,τvt=Δv−v+g1(w)g2(u),x∈Ω,t>0,wt=−vw,x∈Ω,t>0 in a bounded smooth domain Ω⊂RN(N≥1) with zero-flux boundary conditions, where τ∈{0,1}, the functions D(u),S1(u),S2(u)∈C2([0,∞)), f(u,w)∈C1([0,∞)2),g1(w),g2(u)∈C1([0,∞)) fulfill D(u)≥CD(u+1)−α,S1(u)≤χu(u+1)β−1,S2(u)≤ξu(u+1)γ−1

We study convergence of variational solutions of the nonlinear eigenvalue problem−Δu=λ|u|p−2u,u∈H01(Ω), as p↓2 or as p↑2, where Ω is a bounded domain in RN with smooth boundary. It turns out that if λ is not an eigenvalue of −Δ then the solutions either blow up or vanish according to p↓2 or p↑2, while if λ is an eigenvalue of −Δ then the solutions converge to the associated eigenspace.

We consider a delay differential equation for tick population with diapause, derived from an age-structured population model, with two time lags due to normal and diapause mediated development. We derive threshold conditions for the global asymptotic stability of biologically important equilibria, and give a general geometric criterion for the appearance of Hopf bifurcations in the delay differential

This paper first investigates all possible phase portraits of a class of planar quintic vector field given byx˙=(1+2bx2+2dx4)y,y˙=−2(2ax2+by2+3cx4+2dx2y2)x,a,b,c,d∈R. Then, we study the bifurcation problem of its small perturbed vector field with polynomial perturbations of arbitrary degree n,n∈N by the corresponding Abelian integral, and prove that the lower bound for the maximal number of limit cycles

Population mobility in an SIRS epidemic model is considered via graph Laplacian diffusion. We show the existence and uniqueness of solutions to the SIRS model defined on weighed graph. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1. By constructing Lyapunov function, we show that the endemic

In this paper, we study the existence of mild periodic solutions of abstract semilinear equations in a setting that includes several other types of equations such as delay differential equations, first-order hyperbolic partial differential equations, and reaction-diffusion equations. Under different assumptions on the linear operator and the nonhomogeneous function, sufficient conditions are derived

In this paper, we study the quasi-shadowing property for quasi-partially hyperbolic δ-pseudo-orbit made by quasi-partially hyperbolic strings. We also investigate the corresponding Lp, limit and asymptotic quasi-shadowing properties. As an application, we prove that the set of points with periodic center leaves is dense in the non-wondering set for some partially hyperbolic flows.

We study the well-posedness of Cauchy problems on the upper half space R+n+1 associated to higher order systems ∂tu=(−1)m+1divmA∇mu with bounded measurable and uniformly elliptic coefficients. We address initial data lying in Lp (12, what is also new for the case m>1.

We consider a multi-agent system whose dynamics is governed by a gradient flow on the unit sphere associated with the interaction potential between positions of all agents measured by a weighted distance |xi−xj|p+2 for any p≠0. In this paper, we employ both attractive and repulsive couplings to study the asymptotic behavior of the system accompanied by both p>0 (positive range) and p<0 (negative range)

This paper is concerned with a haptotactic cross-diffusion system as the model for oncolytic virotherapy, which describes the influence of the extracellular matrix (ECM) taxis over the tumor-oncolytic virus interaction in the form of haptotaxis of both cancer cells and oncolytic virus toward higher ECM densities. The main results assert the global boundedness of solutions to an associated spatially

Consider a non-doubling manifold with ends M=Rn♯Rm where Rn=Rn×Sm−n for m>n≥3. We say that an operator L has a generalised Poisson kernel if L generates a semigroup e−tL whose kernel pt(x,y) has an upper bound similar to the kernel of e−tΔ where Δ is the Laplace-Beltrami operator on M. An example for operators with generalised Gaussian bounds is the Schrödinger operator L=Δ+V where V is an arbitrary