Multiple solutions to Hessian equations

This paper concerns with the multiple solutions of Hessian equations ?? _k (??(D ² u))?=?f (x, u) in a (k ? 1)-convex domain ${\Omega\subset \mathbb{R}^{n}}$ . Using the methods of degree theory and a priori estimates we prove the existence of two or more solutions to the Hessian equations.     

On the Explicit Determination of Certain Solutions of Periodic Differential Equations of Higher Order

We consider the class of differential equations $ y^{(k)}+\Sigma_{k- 2}^{\nu=1}A_{\nu}y^{(\nu)}+A_0(z)y=0\ {\rm where}\ A_{1},\dots,A_{k- 2}$ are constants, k ≥ 3 and where A₀(z) is a non-constant periodic entire function, which is a rational function of e ^z. In this paper we develop a method that enables us to decide if this equation can have solutions with few zeros, and we also present the construction of these solutions.     

Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

We study a nonlinear equation in the half-space {x ₁ > 0} with a Hardy potential, specifically

$$ - \Delta u - \frac{\mu }{{x_1^2}}u + {u^p} = 0in\mathbb{R}_ + ^n,$$

where p > 1 and ?∞ < μ < 1/4. The admissible boundary behavior of the positive solutions is either O(x ₁ ^?2/(p?1)) as x ₁ → 0, or is determined by the solutions of the linear problem \( - \Delta h - \frac{\mu }{{x_1^2}}h = 0\). In the first part we study in full detail the separable solutions of the linear equations for the whole range of μ. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like O(x ₁ ^?2/(p?1)) near the origin and which, away from the origin, have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like O(x ₁ ^?2/(p?1)) at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.

     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ε-approximations

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (?⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω??³. Bibliography: 25 titles.     

A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x ^1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.     

On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations

Here we prove Hölder regularity for bounded weak solutions of nonlinear parabolic equations with measurable coefficients. The prototype of this class of equations isu _t =Div(|u|^β|Du| ^p?2 Du)p>1, β>1?p     

Uniqueness of conservative solutions for nonlinear wave equations via characteristics

For some classes of one-dimensional nonlinear wave equations, solutions are Hölder continuous and the ODEs for characteristics admit multiple solutions. Introducing an additional conservation equation and a suitable set of transformed variables, one obtains a new ODE whose right hand side is either Lipschitz continuous or has directionally bounded variation. In this way, a unique characteristic can be singled out through each initial point. This approach yields the uniqueness of conservative solutions to various equations, including the Camassa-Holm and the variational wave equation u_tt ? c(u)(c(u)u_x )_x = 0, for general initial data in H¹(R).     

On the navier-stokes equation with the additional conditionu 1 1 =u 3=0

We study the Navier-Stokes equation with the additional conditionu ₁ ¹ =u ³=0. In certain cases, solutions are represented in a closed form. In other cases, the investigated system reduces to simpler systems of partial differential equations. We study the symmetry properties of these systems and construct classes of their particular solutions.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

On sharp bounds for marginal densities of product measures

We develop a new approach for counting integral solutions of the system of equations associated to rational lines on cubic hypersurface. As a consequence, we deduce the density result for rational lines on the cubic hypersurface defined by c1z₁³ + ··· + c_sz_s³ = 0 as soon as s≥21.     

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry

In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate t ^?2λ , where λ = 1, 2, . . . is the angular momentum. Our technique is to use Chandrasekhar’s separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as t ^?2λ . For the second set, in general, the solutions tend to some explicit profile at the rate t ^?2λ . The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green’s functions for time independent Schrödinger equations associated with these wave equations.     

A class of large solutions to the 3D incompressible MHD and Euler equations with damping

In this paper, we derive the global existence of smooth solutions of the 3 D incompressible Euler equations with damping for a class of laxge initial data, whose Sobolev norms H~s can be arbitrarily large for any s ≥ 0. The approach is through studying the quantity representing the difference between the vorticity and velocity. And also, we construct a family of large solutions for MHD equations with damping.     

A remark on the Cauchy problem of 1D compressible Navier-Stokes equations with density-dependent viscosity coefficients

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefcients.By assumingρ0∈L1（R）,we will prove the existence of weak solutions to the Cauchy problems forθ〉0.This will improve results in Jiu and Xin’s paper（Kinet.Relat.Models,1（2）：313–330（2008））in whichθ〉12is required.In addition,We will study the large time asymptotic behavior of such weak solutions.     

Boundedness of solutions of difference equations and application to numerical solution of Volterra integral equations of the second kind

We study the difference equations obtained when some numerical methods for Volterra integral equations of the second kind are applied to the linear test problem y(t) = 1 + ∝₀^t (λ + μt + vs) y(s) ds, t ⩾ 0, with fixed stepsize h. The resulting difference equations are of Poincaré type and we formulate a criterion for boundedness of solutions of these equations if the associated characteristic polynomial is a simple von Neumann polynomial. This result is then used in stability analysis of reducible quadrature methods for Volterra integral equations.     

An application of the maximum principle to describe the layer behavior of large solutions and related problems

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u ^p(x), ??u = ?m(x)u?+?a(x)u ^p(x) where a(x)??? a ₀ >?0, p(x)??? 1 in ??, and ??u = e ^p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? _i :?= ??? _c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u ^p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.     

Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

The present paper deals with the study of semilinear and non-homogeneous Schrödinger equations on a manifold with conical singularity. We provide a suitable constant by Sobolev embedding constant and for p ∈ (2, 2?) with respect to non-homogeneous term g(x) ∈ L ₂ ^n/2 (B), which helps to find multiple solutions of our problem. More precisely, we prove the existence of two solutions to the problem 1.1 with negative and positive energy in cone Sobolev space H _2,0 ^1,n/2 (B). Finally, we consider p = 2 and we prove the existence and uniqueness of Fuchsian-Poisson problem.     

Conformally flat Lorentzian hypersurfaces in ℝ<Stack><Subscript>1</Subscript><Superscript>4</Superscript></Stack> with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x: M ₁ ³ → ? ₁ ⁴ is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of ? ₁ ⁴ . Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.     

A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation: IV

We continue the study of invertible formal transformations of two-dimensional autonomous systems of differential equations with zero approximation represented by homogeneous polynomials of degree 2 and with perturbations in the form of power series without terms of order < 3. In the regular case, we consider systems that have the canonical form (αx ₁ ² ? sgnα x ₂ ² , x ₁ x ₂) with α ≠ 0 as the zero approximation. For such systems, we obtain resonance equations in closed form and use them to prove the theorem on the formal equivalence of systems and establish a generalized normal form to which any original system can be reduced by an invertible change of variables.     

Discrete nonlinear hyperbolic equations. Classification of integrable cases

We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ?². The fields are associated with the vertices and an equation of the form Q(x ₁, x ₂, x ₃, x ₄) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ? ^N . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.     

Local solvability of the k-Hessian equations

We give a classification of second-order polynomial solutions for the homogeneous k-Hessian equation σ_k[u] = 0. There are only two classes of polynomial solutions: One is convex polynomial; another one must not be(k + 1)-convex, and in the second case, the k-Hessian equations are uniformly elliptic with respect to that solution. Based on this classification, we obtain the existence of C∞local solution for nonhomogeneous term f without sign assumptions.