In several cases of nonlinear dispersive PDEs, the difference between the nonlinear and linear evolutions with the same initial data, i.e. the integral term in Duhamel's formula, exhibits improved regularity. This property is usually called nonlinear smoothing. We employ the method of infinite iterations of normal form reductions to obtain a very general theorem yielding existence of nonlinear smoothing

We consider nonlinear elliptic equations that are naturally obtained from the elliptic Schrödinger equation −Δu+Vu=0 in the setting of the calculus of variations, and obtain Lq-estimates for the gradient of weak solutions. In particular, we generalize a result of Shen (1995) [39] in the nonlinear setting by using a different approach. This allows us to consider discontinuous coefficients with a small

We propose a new approach to study the long time dynamics of the wave kinetic equation in the statistical description of acoustic turbulence. The approach is based on rewriting the discrete version of the wave kinetic equation in the form of a chemical reaction network, then employing techniques used to study the Global Attractor Conjecture to investigate the long time dynamics of the newly obtained

This paper deals with a class of nonlocal semilinear pseudo-parabolic equation with conical degenerationut−△But−△Bu=|u|p−1u−1|B|∫B|u|p−1udx1x1dx′, on a manifold with conical singularity, where △B is Fuchsian type Laplace operator with totally characteristic degeneracy on the boundary x1=0. By using the modified method of potential well with Galerkin approximation and concavity, the global existence

The theory of basic reproduction ratio R0 is established for a large class of almost periodic and time-delayed compartmental population models. We first present some dynamical properties for linear almost periodic functional differential systems. By using the product space and evolution semigroup approach, we then prove that R0−1 has the same sign as the exponential growth bound of an associated linear

Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly

In this paper, we study the wave propagation in a two-dimensional lattice dynamical system with global interaction, which arises in the study of a single species population with two age classes and a fixed maturation period living in a two-dimensional spatial unbounded environment. We consider the monostable case and show that for any given admissible speed, all the wave profiles propagating toward

In this article we obtain positive singular solutions of(1){−Δpu=|∇u|qinΩ,u=0on∂Ω, where Ω is a small C2 perturbation of the unit ball in RN. For (p−1)NN−1

In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and “Sobolev versus Hölder local minimizers” type of results. We prove that results similar to these

A class of nonlinear problems modeling thermoelastic extensible plates with rotational inertia and time-dependent delay in the internal feedback are considered. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution. The proof with or without rotational inertia, does not depend on the parameters γ1 and γ2 of time-dependent delay. Moreover

In this paper, we consider the following system{ut=Δu−χ∇⋅(u∇ω),vt=dvΔv−ξ∇⋅(v∇ω),mt=dmΔm+u−m,ωt=−(γ1u+m)ω, in two dimensional space with zero-flux boundary conditions. This model was proposed by Franssen et al. [7] to characterize the invasion and metastatic spread of cancer cells. We first establish the global existence of uniformly bounded global strong solutions. Then using the decay of ECM and the

In this paper, we apply the bifurcation theory to study the steady two-dimensional periodic equatorial water waves for the f-plane approximation. We consider the waves with vorticity which propagate with a specified fixed depth between the thermocline and the upper boundary of the centre layer. Moreover, we present a functional J and its first variation corresponds to the exact equations in the transformed

This paper is devoted to a class of integral type Brezis-Nirenbreg problems on the Heisenberg group. They are a class of nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities on the Heisenberg group, the nonexistence and existence results are obtained.

Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂M, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean ball, and consequently is achieved, if either (i) 9≤n≤11 and ∂M has a nonumbilic point;

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such asFi(x,Dui,D2ui)+fi(x,u1,…,un,Dui)=0,1≤i≤n, in a bounded domain Ω in RN with Dirichlet boundary condition u1=…,un=0 on ∂Ω. Here, fi's are nonincreasing with the radius r=|x|, and satisfy a cooperativity assumption. In addition, each fi is the sum of a locally Lipschitz with a nondecreasing

We prove a Nekhoroshev type theorem for the nonlinear wave equation (NLW)(0.1)utt=uxx−mu−f(u) under Dirichlet boundary conditions where f is an analytic function. More precisely, we prove the subexponential long time stability result of the origin for equation (0.1) by using Birkhoff normal form technique for infinite dimensional Hamiltonian systems and the so-called tame property of the nonlinearity

In this paper, we develop a new method to estimate the smallest upper bound on the number of isolated zeros of Abelian integrals, and give an algebraic criterion for the case of Abelian integrals along energy level ovals of potential systems. As applications of our main result, we first obtain a criterion guaranteeing that two Abelian integrals have the Chebyshev property with accuracy one and give

In this work we consider an energy subcritical semi-linear wave equation (3

We study the one-dimensional diffusive generalized logistic problem with constant yield harvesting:{u″(x)+λg(u)−μ=0,−10. We assume that nonlinearity g satisfies g(0)=g(1)=0, g(u)>0 on (0,1), and g either is concave on (0,1) or (is concave-convex on (0,1) and satisfies a certain condition). We prove that, for any fixed μ>0, on the (λ,‖u‖∞)-plane, the bifurcation diagram consists

We consider the three–dimensional full compressible Navier–Stokes system with density–temperature–dependent viscosities in smooth bounded domains. For the case when the velocity u and absolute temperature θ admit the Dirichlet boundary condition, the strong solutions exist globally in time provided that ‖∇u0‖L22+‖∇θ0‖L22 is suitably small. Through some time–weighted a priori estimates, the main difficulties

We consider the model of a point-vortex under a periodic perturbation and give sufficient conditions for the existence of generalized quasi-periodic solutions with rotation number. The proof relies on Aubry-Mather theory to obtain the existence of a family of minimal orbits of the Poincaré map associated to the system.

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.

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this

The paper is concerned with the propagation of ion-acoustic shock waves in a collision dominated plasma whose equations of motion are described by the one-dimensional isothermal Navier-Stokes-Poisson system for ions with the electron density determined by the Boltzmann relation. The main results include three parts: (a) We establish the existence and uniqueness of a small-amplitude smooth traveling

In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in [7] by a geometric approach. More specifically, we prove that a heteroclinic orbit in the energy level 0 of a 2-dof Hamiltonian Hε is destroyed giving rise to heteroclinic connections between certain elements (at infinity)

We consider an evolving plane curve with two endpoints that can move freely on the x-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by the area-preserving curvature flow equation. The main result shows that any moving curve converges to a traveling wave if the moving curve starts from an

In the previous significant works by Alt et al. (1982) [2] and Alt et al. (1985) [6], the existence of the two-dimensional asymmetric incompressible jet flow and compressible subsonic jet flow were established. However, the uniqueness of the asymmetric jet flow has remained an unsolved open problem. In this paper, we will establish the uniqueness of asymmetric jet flow not only for incompressible case

The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudo-recurrence, Poisson stability) of solutions for semi-linear stochastic equationdx(t)=(Ax(t)+f(t,x(t)))dt+g(t

We present a weighted Lq(Lp)-theory (p,q∈(1,∞)) with Muckenhoupt weights for the equation∂tαu(t,x)=Δu(t,x)+f(t,x),t>0,x∈Rd. Here, α∈(0,2) and ∂tα is the Caputo fractional derivative of order α. In particular we prove that for any p,q∈(1,∞), w1(x)∈Ap and w2(t)∈Aq,∫0∞(∫Rd|uxx|pw1dx)q/pw2dt≤N∫0∞(∫Rd|f|pw1dx)q/pw2dt, where Ap is the class of Muckenhoupt Ap weights. Our approach is based on the sharp function

We consider an abstract class of differential inclusions, which covers differential-algebraic and non-autonomous problems as well as problems with delay. Under weak assumptions on the operators involved, we prove the well-posedness of those differential inclusions in a pure Hilbert space setting. Moreover, we study the causality of the associated solution operator. The theory is illustrated by an application

Given an isoparametric function f on the n-dimensional sphere, we consider the space of functions w∘f to reduce the Yamabe equation on the round sphere into a singular ODE on w in the interval [0,π], of the form w″+(h(r)/sin⁡r)w′+λ(|w|4/n−2w−w)=0 and boundary conditions w′(0)=0=w′(π), where h is a monotone function with exactly one zero on (0,π) and λ>0 is a constant. For any positive integer k we

We consider a general, nonlinear version of the bidomain system. Using the gradient structure of this system, but also the notion of j-subgradient, we prove wellposedness of the bidomain system in the energy space and provide first numerical experiments.

In this paper we explicitly calculate the control sets associated with a linear control system on the two dimensional solvable Lie group. We show that a linear control system of such kind admits exactly one control set or infinite control sets depending on some algebraic conditions.

The paper concerns regularities of solutions of elliptic equations in non-divergence form with directional homogenization. The main interest is to investigate the differences of regularities of datums that are needed and of solutions that are obtained between directions with homogenization and without homogenization. We establish uniform interior Lp (1

In this paper we study the global bifurcation from a critical orbit of a strongly indefinite functional. This problem arises in considering the bifurcations of solutions to non-cooperative semi-linear elliptic systems with Neumann boundary conditions. We define an index of an orbit in terms of an equvariant degree and establish its relationship with the index of a critical point of the mapping restricted

In this paper we study the asymptotic behaviors of global solutions to the fully parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇v)+ru−μu1+σ, vt=Δv−v+uγ, subject to the homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn (n≥2), where parameters μ,σ,γ>0, r∈R, and the nonlinearity D,S∈C2([0,∞)) are supposed to generalize the prototypesD(u)≥a0(u+1)−α,0≤S(u)≤b0u(u+1)β−1 with a0,b0>0

This paper is concerned with the asymptotic stability of the initial-boundary value problem of a singular PDE-ODE hybrid chemotaxis system in the half space R+=[0,∞). We show that when the non-zero flux boundary condition at x=0 is prescribed and the initial data are suitably chosen, the solution of the initial-boundary value problem converges, as time tend to infinity, to a shifted traveling wavefront

In this paper, we consider the boundary value problem for the one-dimensional non-isentropic Euler-Poisson system. The physical subsonic inflow and outflow conditions are prescribed at boundaries. We prove the existence of multi-transonic solutions, i.e., the existence of subsonic-sonic-supersonic-shock-subsonic flows in fact nozzles. Furthermore, the monotonicity between the location of the transonic

In this paper, we establish the existence, uniqueness and attraction properties of an invariant measure for the real Ginzburg-Landau equation in the presence of a degenerate stochastic forcing acting only in four directions. The main challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. To achieve this, we propose a condition which only

Second order partial differential equations which describe spherical surfaces (ss) or pseudospherical surfaces (pss) are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature K=1 or K=−1, respectively, and they can be seen as the compatibility condition of an associated su(2)-valued or sl(2,R)-valued linear problem, also referred to as a zero curvature

A family of Camassa-Holm type equations with a linear term and cubic and quartic nonlinearities is considered. Local well-posedness results are established via Kato's approach. Conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is a time-dependent, monotonically decreasing function of time, and bounded from above by the Sobolev norm

This paper is devoted to the investigation of generalized Abel equation x˙=S(x,t)=∑i=0mai(t)xi, where ai∈C∞([0,1]). A solution x(t) is called a periodic solution if x(0)=x(1). In order to estimate the number of isolated periodic solutions of the equation, we propose a hypothesis (H) which is only concerned with S(x,t) on m straight lines: There exist m real numbers λ1<⋯<λm such that either (−1)i⋅S(λi

We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak

We study arbitrary cubic perturbations of the symmetric 8-loop Hamiltonian, which are linear with respect to the small parameter. It is shown that when the first 4 coefficients in the expansion of the displacement functions corresponding to both period annuli inside the loop vanish, the system becomes integrable with the following three strata in the center manifold: Hamiltonian, reversible in y and

In this paper, we explore the three-dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions. We extend previous results on the field and

We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-α model, stochastic power law fluids, the stochastic

We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type Lu:=−r−θ(rα|u′(r)|βu′(r))′, where θ,β≥0 and α>0, are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the p-Laplacian and k-Hessian operators in the radial case. Our results involve

In this paper we are interested in the inverse inclusion problem in the plane. We derived Hölder stability estimates for the inversion using a general single boundary measurement, and under the assumption that the inclusion has a circular shape. The Hölder power in the stability estimates only depends on the position of the target inclusion and shows that the identification is better when the inclusion

The vanishing viscosity limit for the multi-dimensional compressible Navier-Stokes equations to the corresponding Euler equations is a difficult and challenging problem in the mathematics. Recently, L.Li, D.Wang and Y.Wang [17] verified that the solutions for the 2D compressible isentropic Navier-Stokes equations converge to the planar rarefaction wave solution for the corresponding 2D Euler equations

We consider the non-isentropic compressible Navier-Stokes equations with hyperbolic heat conduction and a law for the stress tensor which is modified correspondingly by Maxwell's law. These two relaxations, turning the whole system into a hyperbolic one, are not only treated simultaneously, but are also considered in a version having Galilean invariance. For this more complicated relaxed system, the

In this paper, we investigate multiple existence of homoclinic solutions for a periodically perturbed N-dimensional autonomous differential equation with a degenerate homoclinic solution of degeneracy degree d. Known results were obtained with a functional perturbation, which is regarded as an infinite-dimensional parameter. In this paper we consider a single parameter perturbation, a special form

We give unified proofs of the well-known Hölder regularity of locally bounded solutions to parabolic equations with structures modeled after the parabolic p-Laplace equation and the porous medium equation. This approach is based on expansion of positivity. The proofs make no use of logarithmic energy estimates.

We study the problem of stabilization for a class of evolution systems with fractional-damping. After writing the equations as an augmented system we prove in this article first that the problem is well posed. Second, using the LaSalle's invariance principle we show that the system is strongly stable. Then, based on a resolvent approach we show a luck of uniform stabilization. Next, using multiplier

In this paper, we build up an output feedback law to stabilize a sampled-data controlled heat equation (with a potential) in a bounded domain Ω. The feedback law abides the following rules: First, we divide equally the time interval [0,+∞) into infinitely many disjoint time periods, and divide each time period into three disjoint subintervals. Second, for each time period, we observe a solution over

In this paper, we construct the global large solution to the three-dimensional incompressible Hall-MHD equations with a class of initial data. Here the “large solution” means that the L∞ norm can be arbitrarily large initially.

It is well-known that the 3D incompressible Euler equations admit some local-in-time solutions for which the vorticity is piecewise smooth and discontinuous across a smooth time-dependent hypersurface which evolves with the flow. In this paper we prove that such a solution can be obtained as zero-viscosity limit of strong solutions to the Navier-Stokes equations whose vorticity has sharp variations

In this paper we consider the Hénon problem in the unit disc with Dirichlet boundary conditions. We study the asymptotic profile of least energy and nodal least energy radial solutions and then deduce the exact computation of their Morse index for large values of the exponent p. As a consequence of this computation a multiplicity result for positive and nodal solutions is obtained.

In this paper we extend the classical “decomposing+embedding” method for systems of delay differential equations. Our extension has two advantages: (1) enhancing the range of applicability of the classical method and (2) providing delay dependent criteria of global attraction. The leading ideas are, on the one hand, the extension of the notion of dominance introduced first for difference equations

A Sobolev type embedding for radially symmetric functions on the unit ball B in Rn, n≥3, into the variable exponent Lebesgue space L2⋆+|x|α(B), 2⋆=2n/(n−2), α>0, is known due to J.M. do Ó, B. Ruf, and P. Ubilla, namely, the inequalitysup⁡{∫B|u(x)|2⋆+|x|αdx:u∈H0,rad1(B),‖∇u‖L2(B)=1}<+∞ holds. In this work, we generalize the above inequality for higher order Sobolev spaces of radially symmetric functions

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality LAu=−div[A(x,u,∇u)]≥(Iα⁎up)uq in Ω, where Ω⊂RN,N≥1, is an open set. Here Iα stands for the Riesz potential of order α∈(0,N), p>0 and q∈R. For a large class of operators LA (which includes the m-Laplace and the m-mean curvature operator) we obtain optimal ranges of exponents p,q and α for which