A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a distinction between the influence on the solution of the positive and the negative part of the right-hand side.

We prove the optimal W2,∞ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be C1 or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation(−Δ)σu=|x|αupinB1\{0} with an isolated singularity at the origin, where σ∈(0,1) and the punctured unit ball B1\{0}⊂Rn with n≥2. When −2σ<α<2σ and n+αn−2σ

In this article we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in R1. The analysis is performed in the weighted Sobolev spaces, H(a,b)s(I). Three different characterizations of H(a,b)s(I) are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded

The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., |ψ|p−2ψ) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite N-star graphs, the existence of standing waves bifurcating from the trivial solution

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C1/2-Hölder continuous. The coefficient of the time-dependent damping is given by μ(1+t)λ, (0<λ,0<μ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose

In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its

In this paper, we validate the boundary layer theory for 2D steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain {(X,Y)∈[0,L]×R+} under the assumption of a moving boundary at {Y=0}. The validity of boundary layer expansions and convergence rates are established in Sobolev sense. We extend the results for the case with the shear outer ideal MHD flows [3] to the case of the

We are concerned with the three dimensional Vlasov-Poisson plasma interacting with a like point charge in the case of infinite mass and velocities. We prove the global existence and uniqueness of the classical solution to the system by assuming that the initial density slightly decays in space variables and strongly decays in velocities. This result generalizes the previous one for the initial data

In this short note, we show the rigidity of a trace estimate for Steklov eigenvalues with respect to functions in our previous work (Shi and Yu (2016) [13]). Namely, we show that equality of the estimate holds if and only if the manifold is a direct product of a round ball and a closed manifold. The key ingredient in the proof is a splitting theorem for flat and totally geodesic Riemannian submersions

In this note, necessary and sufficient conditions for the existence of an entropy structure for certain classes of cross-diffusion systems with diffusion matrix A(u) are given, based on results from matrix factorization. The entropy structure is important in the analysis for such equations since A(u) is typically neither symmetric nor positive definite. In particular, the normal ellipticity of A(u)

We consider an initial-boundary value problem for reaction-diffusion equations coupled with the Keller-Segel system from the chemotaxis theory which describe a formation of colony patterns of bacteria Escherichia coli. The main goal of this work is to show that global-in-time solutions of this model converge towards stationary solutions depending on initial conditions.

In this paper, we consider inverse problems for the Dirac operator with complex-valued weight and the jump conditions inside the interval. We prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra.

In this paper we consider a family of active scalars with a velocity field given by u=Λ−1+α∇⊥θ, for α∈(0,1). This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to α=0. We consider the evolution of sharp fronts by studying families of almost-sharp fronts. These are smooth solutions with simple geometry in which

We consider the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps φ:Rn×Td→Rn×Td:φ(zθ)=(z+ϕ(z)+h(z,θ)+ϵf(z,θ)θ+ω), where ϕ:Rn→Rn is a homogeneous function of degree l with l≥2 and h=O(|z|l+1). We obtain the following results: (a) For n=1, l being odd and ϵ sufficiently small, parabolic invariant tori exist if ω satisfies the Brjuno-Rüssmann's non-resonant

In this paper, we prove the global well-posedness for the incompressible magnetohydrodynamics (MHD) equations in the three-dimensional unbounded domain Ω:=R+×R2. More precisely, we construct global small Sobolev regularity solutions with the initial data near 0 for the three-dimensional MHD equations in Ω. The key point of the proof is to find the suitable initial approximation function such that the

In this paper we study the time dependent Schrödinger equation with all possible self-adjoint singular interactions located at the origin, which include the δ and δ′-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Green's function and give a mathematical rigorous meaning to the corresponding

In this paper, we study existence times of strong solutions of the three-dimensional Navier-Stokes equations in time-varying analytic Gevrey classes based on Sobolev spaces Hs,s>12. This complements the seminal work of Foias and Temam (1989) [26] on H1 based Gevrey classes, thus enabling us to improve estimates of the analyticity radius of solutions for certain classes of initial data. The main thrust

The existence of stationary subsonic solutions and their stability for 3-D hydrodynamic model of unipolar semiconductors with the Ohmic contact boundary have been open for long time due to some technical reason, as we know. In this paper, we consider 3-D radial solutions to the system in a hollow ball, and prove that the 3-D radial subsonic stationary solutions uniquely exist and are asymptotically

The aim of this paper is to investigate the regime of high Mach number flows for compressible barotropic fluids of Korteweg type with density dependent viscosity. In particular we consider the models for isothermal capillary and quantum compressible fluids. For the capillary case we prove the existence of weak solutions and related properties for the system without pressure, and the convergence of

In this paper, we are concerned with the existence of local isometric embeddings into Euclidean space for analytic Riemannian metrics g, defined on a domain U⊂Rn, which are singular in the sense that the determinant of the metric tensor is allowed to vanish at an isolated point (say the origin). Specifically, we show that, under suitable technical assumptions, there exists a local analytic isometric

In this paper, we consider some hyperbolic variants of the mass conserving Allen–Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn–Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of solutions to the equation in a bounded interval of the real line with homogeneous

We study a forager-exploiter model with nonlinear diffusions{ut=∇⋅((u+1)m∇u)−∇⋅(u∇w),vt=∇⋅((v+1)l∇v)−∇⋅(v∇u),wt=Δw−(u+v)w−μw+r in a smooth bounded domain Ω∈Rn with homogeneous Neumann boundary conditions, where μ>0 and r is a given nonnegative function. We prove that, ifm≥1andl∈[1,∞)∩(n(n+2)2(n+1),∞), then the classical solution exists globally and remains bounded.

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential

We establish the existence of traveling waves for a Lotka-Volterra competition-diffusion model with a shifting habitat. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near ∞ and negative near −∞, and shifting rightward at a speed c. We show that under appropriate conditions, for the case that one species is competitively stronger near ∞ and the case that

A necessary and sufficient condition is given for quasi-homogeneous polynomial Hamiltonian systems having a center. Then it is shown that there exists a bound on the number of limit cycles bifurcating from the period annulus of quasi-homogeneous Hamiltonian systems at any order of Melnikov functions; and the explicit expression of this bound is given in terms of (n,k,s1,s2), where n is the degree of

We prove the mixed-norm Sobolev estimates for solutions to both divergence and non-divergence form time-dependent Stokes systems with unbounded measurable coefficients having small mean oscillations with respect to the spatial variable in small cylinders. As a special case, our results imply Caccioppoli type inequalities for the Stokes systems with variable coefficients. A new ϵ-regularity criterion

This paper solves the stability problem on a partially dissipated system of magnetohydrodynamic equations near a background magnetic field. Large-time behavior of the corresponding linearized system is also obtained. These results presented in this paper rigorously confirm a nonlinear phenomenon observed in physical experiments that the magnetic field actually stabilizes electrically conducting fluids

In this paper we study the motion of a surface gravity wave with viscosity. In particular we prove two well-posedness results. On the one hand, we establish the local solvability in Sobolev spaces for arbitrary dissipation. On the other hand, we establish the global well-posedness in Wiener spaces for a sufficiently large viscosity. These results are the first rigorous proofs of well-posedness for

In this paper we establish a new stability result for smooth volume preserving mean curvature flows in flat torus Tn in dimensions n=3,4. The result says roughly that if an initial set is near to a strictly stable set in Tn in H3-sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in W2,5-sense.

A classical problem in ergodic control theory consists in the study of the limit behaviour of λVλ(⋅) as λ↘0, when Vλ is the value function of a deterministic or stochastic control problem with discounted cost functional with infinite time horizon and discount factor λ. We study this problem for the lower value function Vλ of a stochastic differential game with recursive cost, i.e., the cost functional

In this paper, we focus on the standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation−Δu=λu+μ(|x|−α⁎|u|2)u+(|x|−β⁎|u|2)u,x∈RN under the normalized constraint∫RNu2=c>0, where N≥3, μ>0, α, β∈(0,N), and the frequency λ∈R is unknown and appears

In this paper we study the initial boundary value problem for the system ut−Δum=−div(uq∇v),vt−Δv+v=u. This problem is the so-called Keller-Segel model with nonlinear diffusion. Our investigation reveals that nonlinear diffusion can prevent overcrowding. To be precise, we show that solutions are bounded as long as m>q>0, thereby substantially generalizing the known results in this area. Furthermore

We study the decay property for symmetric hyperbolic systems with memory-type diffusion. Under the structural condition (called Craftsmanship condition) we prove that the system is uniformly dissipative and the solutions satisfy the corresponding decay property. Our proof is based on a technical energy method in the Fourier space which makes use of the properties of strongly positive definite kernels

In this paper, we first establish the existence of uniform random attractor for 2D stochastic Navier-Stokes equation in H with deterministic non-autonomous external force being normal in Lloc2(R;V′), which is the measurable minimal compact set and uniformly attracts bounded random set in H in the sense of pullback. We also show that uniform random attractor with respect to the deterministic non-autonomous

We investigate the Cauchy problem for a quasilinear equation with transport rough input of the form du−∂i(aij(u)∂ju)dt=dXti(x)∂iut, u0∈L2 on the torus Td, where X is two-step enhancement of a family of coefficients (Xti(x))i=1,…d, akin to a geometric rough path with Hölder regularity α>1/3. Using energy estimates, we provide sufficient conditions that guarantee existence in any dimension, and uniqueness

This paper is concerned with a class of semilinear elliptic equations with a potential function−Δu=λ|x|−αu−|x|σupinΩ∖{0}, where λ,σ∈R, α>0, p>1, and Ω⊂RN (N≥3) is a bounded smooth domain with 0∈Ω. We establish the existence, nonexistence and asymptotic behavior of positive solutions when the potential function |x|−α has strong singularity at the origin. When the potential function |x|−α has weak singularity

A well-known diffuse interface model consists of the Navier-Stokes equations for the average velocity, nonlinearly coupled with a convective Cahn-Hilliard type equation for the order (phase) parameter. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a stochastic version of this model forced by

In this paper, we investigate the following Schrödinger equation{−Δu−μ|x|2u=g(u)inRN∖{0},u∈H1(RN), where N≥3, μ<(N−2)24, 1|x|2 is called the Hardy potential (the inverse-square potential) and g satisfies the Berestycki-Lions type condition. If 0<μ<(N−2)24, combining variational methods with analytical skills, we show that the above problem has a positive and radial ground state solution. At the same

In this paper, we define the second generalized Yamabe invariant on manifolds with boundary. We prove some of its properties and study when the invariant is attained by some metric. In another direction, by using the conformal mean curvature flow, we prove a version of the conformal Schwarz lemma for manifolds with boundary.

It was proved that both the normal hyperbolicity and invariant manifold for a uniformly hyperbolic compact invariant manifold and the invariant manifold for a uniformly hyperbolic noncompact invariant manifold are persistent under small perturbation. In this paper, we weaken the uniform normal hyperbolicity to the nonuniform one and prove that both the nonuniform normal hyperbolicity and invariant

We study positive solutions of the Yamabe equation with isolated singularity and prove the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.

This paper is concerned with the stationary problem for a diffusive Lotka-Volterra predator-prey-mutualist model with a protection zone under homogeneous Neumann boundary conditions. Compared with the case where the mutualist is absent in [12], this paper aims to reveal the effects of mutualism coefficients α and β on the existence, number and stability of steady states. It turns out that when α is

We study the large time behavior of solutions v:Ω×(0,∞)→R of the PDE ∂t(|v|p−2v)=Δpv. We show that e(λp/(p−1))tv(x,t) converges to an extremal of a Poincaré inequality on Ω with optimal constant λp, as t→∞. We also prove that the large time values of solutions approximate the extremals of a corresponding “dual” Poincaré inequality on Ω. Moreover, our theory allows us to deduce the large time asymptotics

We present a general formulation of the averaging method in the setting of a semilinear equation Lx=εN(x,ε), being L a linear Fredholm mapping of index zero. Our general approach provides new results even in the classical periodic framework. Among the applications we obtained there are: a partial answer to an open problem related to the Liebau phenomenon, the multiplicity of periodic solutions for

In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21], to prove that there exists a family of cubic centers, labeled by CD3112 in [25], with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we

In this work we apply the so called Unfolding Operator Method to analyze the asymptotic behavior of the solutions of the p-Laplacian equation with Neumann boundary condition in a bounded thin domain of the type Rε={(x,y)∈R2:x∈(0,1) and 01 representing respectively weak, resonant and high oscillations at the top boundary. In the three cases we deduce the homogenized limit and obtain correctors.

Motivated by the propagation of nonlinear sound waves through relaxing hereditary media, we study a nonlocal third-order Jordan–Moore–Gibson–Thompson acoustic wave equation. Under the assumption that the relaxation kernel decays exponentially, we prove local well-posedness in unbounded two- and three-dimensional domains. In addition, we show that the solution of the three-dimensional model exists globally

Firstly we use the Lie derivatives of spinors with respect to Killing vector fields and conformal Killing vector fields to obtain the generalized commuting vector field set adapted to the Dirac operator. By the conservation law of the charge current and total angular momentum, as well as the weighted Sobolev estimates, it gives the optimal decay for the linear massless spinor fields and more decay

We prove that for almost every initial data (u0,u1)∈Hs×Hs−1 with s>p−3p−1 there exists a global weak solution to the supercritical semilinear wave equation ∂t2u−Δu+|u|p−1u=0 where p>5, in both R3 and T3. This improves in a probabilistic framework the classical result of Strauss [20] who proved global existence of weak solutions associated to H1×L2 initial data. The proof relies on techniques introduced

In this paper on hyperbolic systems of conservation laws in one space dimension, we give a complete picture of stability for all solutions to the Riemann problem which contain only extremal shocks. We study stability of the Riemann problem amongst a large class of solutions. We show stability among the family of solutions with shocks from any family. We assume solutions verify at least one entropy

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from [1], we prove that if one component of the velocity remains small enough in a sub-space of H˙12 “almost” scaling invariant, then the 3D Navier-Stokes equations are globally wellposed

We prove that polyharmonic maps of arbitrary order from complete nonparabolic Riemannian manifolds to arbitrary Riemannian manifolds must be harmonic if certain smallness and integrability conditions hold.

We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces, Bp,qs, where the parameters p,q and s considered here are such that the local existence and uniqueness result holds.

We give a lower estimate of the number of anti-periodic solutions of implicit differential inclusions on tori. Our approach is based on the application of the topological essential fixed point theory, jointly with the Nielsen theory for multivalued admissible maps. Since one of the conditions is rather technical (zero topological dimension of a fixed point set), some simple illustrative examples to

We prove existence of the largest and the smallest entropy solutions to the Cauchy problem for a nonlinear degenerate anisotropic parabolic equation. Applying this result, we establish the comparison principle in the case when at least one of the initial functions is periodic. In the case when initial function vanishes at infinity (in the sense of strong average) we prove the long time decay of an

We investigate different and new thermoelastic Timoshenko systems with or without history, and with Fourier or Cattaneo law for heat conduction, with respect to (non-)exponential stability. Results are obtained that shed a new light on the role of history terms and that of the heat conduction law. Improvements of previous results of earlier work [12] are presented, clarifying open questions, as well

In the article, we prove the large data scattering for two models, i.e. the defocusing quintic nonlinear Schrödinger equation on R2 × T and the defocusing cubic nonlinear Schrödinger equation on R3 × T. Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper

It is known that the Melnikov function method is equivalent to the averaging method for studying the number of limit cycles of planar analytic (or C∞) near-Hamiltonian differential systems. In this paper, we study piecewise smooth near-integrable systems and establish the Melnikov function method and the averaging method for finding limit cycles. We also show the equivalence of the two methods even

We construct explicitly potentials, Darboux matrix functions and corresponding solutions of Dirac, dynamical Dirac and Dirac–Weyl systems using generalised Bäcklund-Darboux transformation (GBDT) in the important case of nontrivial initial systems. In this way, we construct explicit solutions of systems with non-vanishing at infinity potentials, including steplike and power-law growth potentials. Thus