In this paper, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions' evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. After establishing the existence and uniqueness of solutions to the models that we

A priori estimates and existence of real-valued periodic solutions to the modified Benjamin-Ono equation with initial data in Hs for s>1/4 are proved locally in time. The approach relies on frequency dependent time localization, after which dispersive properties from Euclidean space are recovered. The same regularity results are proved for the cubic derivative nonlinear Schrödinger equation.

The paper concerns the mean hyperbolicity of the skew-product flows induced by the perturbation equations driven by varieties of non-periodic forcings. The weakly averaged horseshoe can be constructed as a mean hyperbolic invariant set in the Poincare section for high dimensional phase space. Due to the non-periodic property, the Poincare return map restricted to the weakly averaged horseshoe region

We consider the compressible inviscid model describing the time evolution of two fluids sharing the same velocity and enjoying the algebraic pressure closure. By employing the technique of convex integration, we prove the existence of infinitely many global-in-time weak solutions for any smooth initial data. We also show that for any piecewise constant initial densities, there exists suitable initial

We prove the existence of a bounded variation solution for a quasilinear elliptic problem involving the mean curvature operator and a sublinear nonlinearity. We obtain such a solution as the limit of the solutions of another quasilinear elliptic problem involving a parameter p>1 as p→1+. The analysis requires estimates independent on p, as well as a precise version of the weak Euler-Lagrange equation

We would like to study a weakly coupled system of semi-linear classical damped wave equations with moduli of continuity in nonlinearities whose powers belong to the critical curve in the p−q plane. The main goal of this paper is to find out sharp conditions of these moduli of continuity which classify between global (in time) existence of small data solutions and finite time blow-up of (even) small

Discrete breathers are spatially localized periodic solutions in nonlinear lattices. We prove the existence of odd symmetric, even symmetric, and multi-pulse discrete breathers in strong localization regime in one-dimensional infinite Fermi-Pasta-Ulam lattices with even interaction potentials. The multi-pulse discrete breather consists of an arbitrary number of the odd-like and/or even-like primary

We show the existence and uniqueness of local strong solutions of Keller-Segel system of parabolic-parabolic type for arbitrary initial data in the homogeneous Besov space which is scaling invariant. We also construct global strong solutions for small initial data, where the solutions belong to the Lorentz space in time direction. The proof is based on the maximal Lorentz regularity theorem of heat

In this paper, we consider the conditional regularity of weak solution to the 3D Navier–Stokes equations. More precisely, we prove that if one directional derivative of velocity, say ∂3u, satisfies ∂3u∈Lp0,1(0,T;Lq0(R3)) with 2p0+3q0=2 and 32

This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lamé operators of elasticity −Δ⁎+V in terms of suitable norms of the potential V. In particular, this allows to get sufficient conditions on the size of the potential such that the point spectrum of the perturbed operator remains empty. In three dimensions we show full

We discuss some of the geometric properties, such as the foliated Schwarz symmetry, the monotonicity along the axial and the affine-radial directions, of the first eigenfunctions of a Zaremba problem for the Laplace operator on annular domains. Together with the shape calculus, these fine geometric properties help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves

We study existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci's extremal operators in dimension two. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover for the Mλ,Λ− operator, we show the existence of a critical exponent and give bounds for it.

In this paper, we investigate well-posedness and well-posedness in the generalized sense for a new class of isoperimetric-type constrained variational control problems. For this purpose, in the first part of the current paper, we introduce new versions for the concepts of monotonicity, pseudomonotonicity and hemicontinuity associated with the considered curvilinear integral functional. Thereafter,

We consider a structural-acoustic wall problem in three dimensions, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and the acoustic medium is subject to boundary damping. For this model we study the existence of a continuous nonlinear semigroup associated with the model in the finite energy space. We show that strong/weak continuity of the semigroups depends on the support

Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou [16] proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the

In this work we consider an example of a linear time degenerate Schrödinger operator. We show that with the appropriate assumptions the operator satisfies a Kato smoothing effect. We also show that the solutions to the nonlinear initial value problems involving this operator and polynomial derivative nonlinearities are locally well-posed and their solutions also satisfy the same smoothing estimates

In this paper, we investigate the averaging principle for a stochastic coupled fast-slow atmosphere-ocean model. Precisely, we will first prove regularity estimates and tightness of the slow variable. Then, we prove the ergodicity of the fast variable with the frozen slow variable. In the end, we prove that the slow variable converges weakly and strongly to the solution of the averaged equation, when

In the case of symmetries with respect to a Coxeter group G⊂O(n) acting without non-zero fixed points on Rn, the stability of the solution of the Logarithmic Minkowski problem on Sn−1 is established.

In this paper, we consider positive supersolutions of the semilinear fourth-order problem{(−Δ)2u=ρ(x)f(u)inΩ,−Δu>0inΩ, where Ω is a domain in RN (bounded or not), f:Df=[0,af)→[0,∞) (00 for u>0 and ρ:Ω→R is a positive function. Using a maximum principle-based argument, we give explicit estimates on positive supersolutions that can easily be applied to obtain Liouville-type results for positive supersolutions

In the present paper we prove estimates on subsolutions of the equation −Av+c(x)v=0, x∈D, where D⊂Rd is a domain (i.e. an open and connected set) and A is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable

We establish a method of scaling spheres for the integral system{u(x)=∫Rn|y|avp(y)|x−y|n−αdy,x∈Rn,v(x)=∫Rn|y|buq(y)|x−y|n−βdy,x∈Rn, where 0<α,β−α, b>−β and p,q>0. By using this method, we obtain a Liouville theorem for nonnegative solutions when 0

This paper concerns the existence of bounded classical solutions to the attraction-repulsion chemotaxis system with logistic source(⋆){ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u),x∈Ω,t>0,0=Δv−βv+αu,x∈Ω,t>00=Δw−δw+γu,x∈Ω,t>0 in a smooth bounded domain Ω⊆RN(N≥1), subject to nonnegative initial data and homogeneous Neumann boundary conditions, where f(u)≤a−bur for all u≥0 with some a≥0,b>0 and r≥1. Here χ,α,ξ,β as well

It is rather well-known that hyperbolic operators have the shadowing property. In the setting of finite dimensional Banach spaces, having the shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al. constructed an operator with the shadowing property which is not hyperbolic, settling an open question. In the process, they introduced a class of operators which has come to be known

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in RN (N≥4). Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in BC(R,Ls,∞(RN)) when S(n)=n−12. Next for the case S(n)=0, we study the mild solution and point out it can become strong solution in BC(R,Ls,∞(RN)) if the

In this paper we consider the semilinear damped wave problem of the form{(α(t)ut)t−β(t)Δu+γ(t)ut+δ(t)u=β(t)f(u),x∈Ω,t>τ,u(x,t)=0,x∈∂Ω,t⩾τ,u(x,τ)=uτ(x),ut(x,τ)=vτ(x),x∈Ω, where Ω is a bounded smooth domain in RN, N⩾3, τ∈R, f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α,β,γ and δ are continuous real valued functions of a real

In this paper we investigate the Sturmian theory for general (possibly uncontrollable) linear Hamiltonian systems by means of the Lidskii angles, which are associated with a symplectic fundamental matrix of the system. In particular, under the Legendre condition we derive formulas for the multiplicities of the left and right proper focal points of a conjoined basis of the system, as well as the Sturmian

The integrable focusing nonlinear Schrödinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the space-time plane where solutions

Realizations of differential operators subject to differential boundary conditions on manifolds with conical singularities are shown to have a bounded H∞-calculus in appropriate Lp-Sobolev spaces provided suitable conditions of parameter-ellipticity are satisfied. Applications concern the Dirichlet and Neumann Laplacian and the porous medium equation.

A classical regularity result is that non-negative solutions to the Dirichlet problem Δu=f in a bounded domain Ω, where f∈Lq(Ω), q>n2, satisfy ‖u‖L∞(Ω)≤C‖f‖Lq(Ω). We extend this result in three ways: we replace the Laplacian with a degenerate elliptic operator; we show that we can take the data f in an Orlicz space LA(Ω) that lies strictly between Ln2(Ω) and Lq(Ω), q>n2; and we show that we can replace

In (Euzébio et al., 2016 [10]; Chen and Tang, 2020 [8]), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x˙=F(x)−y,y˙=g(x)−α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system

In this paper, we are concerned with almost periodic sequences. We study various types of almost periodic sequences arising in the literature and adapt their notions to the parametric framework. Then, similarly to the continuous case, we aim to prove variational principles. These variational principles are used to obtain some structural results and some theorems of existence in a Hilbert framework

We provide an extension of the Hartman–Knobloch theorem for periodic solutions of vector differential systems to a general class of ϕ-Laplacian differential operators. Our main tool is a variant of the Manásevich–Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. Our results concern the case of convex bound sets for which we show some

We consider a nonautonomous systemx˙=A(t)x+f(t,x,y),y˙=g(t,y) and give conditions under which there is a transformation of the form H(t,x,y)=(x+h(t,x,y),y) taking its solutions onto the solutions of the partially linearized systemx˙=A(t)x,y˙=g(t,y). Shi and Xiong [28] proved a special case where g(t,y) was a linear function of y and x˙=A(t)x had an exponential dichotomy. Our assumptions on A and f

We show that the geometric deformation of shearing yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in an unbounded strip. The proof is based on the Hardy inequality due to the shearing established in [2] and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

In this paper, we study a class of explicit positive solutions to G-heat equations by solving second order nonlinear ordinary differential equations. Based on the positive solutions, we give the sharp order of G-capacity cσ([−ε,ε]) when ε→0.

In this paper, by establishing the localized Lp-Lq estimate and Sobolev estimates for parabolic partial differential equations with a singular first order term and a Lipschitz first order term, a new Zvonkin-type transformation is given for stochastic differential equations with singular and Lipschitz drifts. The associated Krylov's estimate is established. As applications, dimension-free Harnack inequalities

In this paper we give a multi-scale analysis proof of the power-law localization for random operators on Zd for arbitrary d≥1.

It was proved by Karch and Pilarczyk that Landau solutions are asymptotically stable under any L2-perturbation. In our earlier work with L. Li, we have classified all (−1)-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability

In this paper we introduce and study several mean forms of sensitive tuples. It is shown that the topological or measure-theoretical entropy tuples are correspondingly mean sensitive tuples under certain conditions (minimal in the topological setting or ergodic in the measure-theoretical setting). Characterizations of the question when every non-diagonal tuple is mean sensitive are presented. Among

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle S1,ut=uxx+f(t,u,ux),t>0,x∈S1=R/2πZ, where f is independent of t or T-periodic in t. Assume that the equation admits a compact global attractor. It is proved that, any non-wandering point is a limit point of the system (that is, it is a point in some ω-limit set). More precisely, in the

We study the simplest parabolic-elliptic model of chemotaxis in space dimensions N≥3, and show the optimal conditions on the initial data for the finite time blow-up and the global existence of solutions in terms of stationary solutions. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution.

We study a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effects. This model also takes into account some significant mechanisms such as active transport and nonlocal interactions of Oono's type. The system under investigation couples the Navier–Stokes equations for the fluid velocity, a convective Cahn–Hilliard equation with physically relevant

We study the convergence of the method of reflections for the Stokes equations in domains perforated by countably many spherical particles with boundary conditions typical for the suspension of rigid particles. We prove that a relaxed version of the method is always convergent in H˙1 under a mild separation condition on the particles. Moreover, we prove optimal convergence rates of the method in W˙1

In this paper, we study the stability problem of a star-shaped network of elastic strings with a local Kelvin-Voigt damping. Under the assumption that the damping coefficients have some singularities near the transmission point, we prove that the semigroup corresponding to the system is polynomially stable and the decay rates depend on the speed of the degeneracy. This result improves the decay rate

For initial data in Sobolev spaces Hs(T), 12

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation{−Δu−γ|x|2u=1|x|s|u|ps−2u in RN∖{0},u≥0, where N≥3, s∈[0,2), ps=2(N−s)N−2 and γ∈(−∞,(N−2)24). We extend results of E.N. Dancer, F. Gladiali, M. Grossi (2017) [12] where only the case s=0 is considered. The results specially rely on a careful analysis of the kernel of the linearized operator. Moreover, thanks

We study the existence and non-existence of classical solutions for inequalities of type±Δmu≥(Ψ(|x|)⁎up)uq in RN(N≥1). Here, Δm (m≥1) is the polyharmonic operator, p,q>0 and ⁎ denotes the convolution operator, where Ψ>0 is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to

We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For

In this paper, we construct a nontrivial solitary wave solution to Chern-Simons-Higgs system for non-self-dual case and verify its convergence of non-relativistic limit to a minimal mass solution to the non-relativistic Jackiw-Pi model. We also provide with an explicit rate and high regularity of the convergence, which is naturally obtained in process of construction.

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on T1×S3. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the

The global equi-continuity estimate on Lp-viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of Lp-viscosity solutions is established via an approximation of given data. The local Hölder continuity estimate on the space derivative of Lp-viscosity solutions is shown when the obstacles belong to C1

In this paper we obtain higher order asymptotic expansions of solutions to the Cauchy problem of the linear damped wave equation in Rnutt−Δu+ut=0,u(0,x)=u0(x),ut(0,x)=u1(x), where n∈N and u0, u1∈L1(Rn)∩L2(Rn). Established hyperbolic effects seem to be new in the sense that the order of obtained expansions depends on the spatial dimension.

In this work I show how a diffusion-advection equation in three space-dimensions may have its advection term weakly limited to a velocity field localized to a moving curve. This is accomplished through the technique of concentrated capacity, and the form of the limit along with small time existence of solutions is determined. This problem is motivated by mathematical biology and the study of proteins

Let n≥2 and Ω be a bounded Lipschitz domain of Rn. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second-order elliptic equations of divergence form with real-valued, bounded, measurable coefficients on Ω. More precisely, let p∈(n/(n−1),∞). Using a real-variable argument, the authors obtain two necessary and sufficient

In this paper we study the existence of a positive solution for a nonlinear Dirichlet problem driven by the p-Laplacian and a reaction which has the competing effects of a singular term and of a convection (gradient dependent) term. Using the “frozen” variable method and eventually the Leray-Schauder Alternative Theorem, we show that the problem has a positive smooth solution. We mention that on the

The paper is concerned with two-person zero-sum mean-field linear-quadratic stochastic differential games over finite horizons. By a Hilbert space method, a necessary condition and a sufficient condition are derived for the existence of an open-loop saddle point. It is shown that under the sufficient condition, the associated two Riccati equations admit unique strongly regular solutions, in terms of

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient K(x) behaves like |x|−b for 0

We consider Dirac operators with dislocation potentials on the line. The dislocation potential is a periodic potential for x<0 and the same potential but shifted by t∈R for x>0. Its spectrum has an absolutely continuous part (the union of bands separated by gaps) plus at most two eigenvalues in each gap. Its resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface. We prove that

We consider smooth systems limiting as ϵ→0 to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with sufficiently small but non-zero ϵ, using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation

We analyze a quasi-static Biot system of poroelasticity for both compressible and incompressible constituents. The main feature of this model is a nonlinear coupling of pressure and dilation through the system's permeability tensor. Such a model has been analyzed previously from the point of view of constructing weak solutions through a fully discretized approach. In this treatment, we consider simplified