On smoothness of solutions of some elliptic functional-differential equations with degeneration

We consider second-order differential-difference equations in bounded domains in the case where several degenerate difference operators enter the equation. The degeneration leads to the fact that the multiplicity of the zero eigenvalue for the corresponding differential-difference operator becomes infinite. Regularity of generalized solutions for such equations is known to fail in the interior of the domain. However, we prove that the projections of solutions onto the orthogonal complement to the kernel of the “leading” difference operator remain regular in certain subdomains which form a decomposition of the original domain.     

Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions

We study the bifurcation of radially symmetric solutions of +f(u)=0 onn-balls, into asymmetric ones. We show that ifu satisfies homogeneous Neumann boundary conditions, the asymmetric components in the kernel of the linearized operators can have arbitrarily high dimension. For general boundary conditions, we prove some theorems which give bounds on the dimensions of the set of asymmetric solutions, and on the structure of the kernels of the linearized operators.Research supported in part by the NSF under Grant No. MCS-800 2337     

Behavior of solutions of semilinear elliptic equations in domains with complicated boundary

In the paper, boundary value problems in a bounded domain for semilinear elliptic equations are considered. It is claimed that, for small exponent of nonlinearity, the support of a bounded solution is concentrated near the part of the boundary of the domain where the boundary condition is inhomogeneous. Estimates for the size of a neighborhood containing the support of the solution are established. For a supercritical exponent of nonlinearity, the convergence of solutions to a limit solution in the unperforated domain is established for some family of perforated domains. The rate of convergence is polynomial, and it depends on the exponent of the nonlinearity and on the rate at which the sizes of the cavities decrease simultaneously with the growth of the number of cavities and does not depend on the boundary conditions at the boundaries of the cavities. No restrictions on the displacement of the cavities are imposed.     

Monotonicity,symmetry and antisymmetry of solutions of semilinear elliptic equations

This paper investigates certain properties of solutions of equations of the form Δu + ƒ(x, u, ▿ u) = 0 in a bounded domain Ω which is convex in the x₁ direction. Under various conditions on ƒ and on a solution u it is shown that u is increasing in x₁ in the «left halfå of Ω, or in all of Ω. Symmetry (in x₁) of some solutions is proved. Also antisymmetry results are obtained. The paper may be considered as an extension of [4].     

Existence,uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations

We study positive solutions of the Dirichlet problem: u(x)+f(u(x))=0,xD ⁿ ,u(x)=0,xD ⁿ , whereD ⁿ is ann-ball. We find necessary and sufficient conditions for solutions to be nondegenerate. We also give some new existence and uniqueness theorems.Research supported in part by NSF Contract Number MCS 80-02337     

Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on ℝ3

We study the positive solutions of a semilinear equation with a Coulomb potential on ?³. We give a new uniqueness theorem for the positive radial solutions of such an equation and we apply these results to the Thomas-Fermi-Dirac-von Weizsäcker equation without electrostatic repulsion.     

Classification of degenerate solutions of linear differential equations

An algorithm of obtaining partial solutions of linear differential equations in partial derivatives that admit certain nontrivial symmetry algebra but are nonintegrable by the standard methods is described. The notion of degenerate solution is introduced. Natural classification of solutions is suggested.     

A bound for global solutions of semilinear heat equations

We show that global positive solutions of the initial-boundary value problem foru _t =u+u ^p are bounded, provided thatp>1 is subcritical. Our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity.Partially supported by the Sakkokai Foundation     

Local behavior of solutions of some elliptic equations

We study the local behavior of solutions of some nonlinear elliptic equations. These equations are of interest in differential geometry and mathematical physics.     

On a class of solutions of nonlinear Boltzmann equations

In a recent paper by Krook and Wu, the nonlinear Boltzmann equation for an infinite, spatially homogeneous, isotropic monoatomic gas of constant density and kinetic energy and with an elastic differential cross section that varies inversely as relative speed has been reduced to an infinite sequence of moment equations. The present note observes that the moment equations are successively integrable and shows that as time goes to infinity, the distribution tends to be Maxwellian.     

非线性发展方程的丰富的Jacobi椭圆函数解

通过把十二个Jacobi椭圆函数分类成四组，提出了新的广泛的Jacobi椭圆函数展开法，利用这一方法求得了非线性发展方程的丰富的Jacobi椭圆函数双周期解.当模数m→0或1时，这些解退化为相应的三角函数解或孤立波解和冲击波解. 关键词： 非线性发展方程 Jacobi椭圆函数 双周期解 行波解     

非线性波动方程的Jacobi椭圆函数包络周期解

应用Jacobi椭圆函数展开法求得了一类非线性波方程的包络周期解,而且用这种方法得到的周期解在一定条件下可以退化为包络冲击波解或包络孤立波解 关键词： Jacobi椭圆函数 非线性方程 包络周期解 包络孤立波解     

A class of elliptic solutions to the star-triangle relation

The star-triangle relation for the N-state IRF model is investigated. A class of elliptic solutions is found. The corresponding lattice model splits into mutually non-interacting diagonal lattices with the interactions between nearest-neighbour spins.     

一类非线性微分方程的近似解析解

从理论上研究了一类广义扩散方程的求解问题. 利用相似变换和解析拆分技巧给出了求解该类非线性微分方程近似解的一种有效方法, 方程的解可以表示为一个收敛的幂级数. 近似解结果和数值结果非常符合，证明了所提出的方法的准确性和可靠性, 该方法可以用于解决其他科学和工程技术问题. 关键词： 广义扩散方程 非线性边界值问题 解析拆分 近似解析解     

A class of solutions of Einstein-Maxwell equations

A metric defined by $\text{ds}{}^2\text{= [}\text{(p}{}^2\text{+ q}{}^2\text{)}\text{P}\text{] dp}{}^2\text{+ [}\text{P}\text{(p}{}^2\text{+ q}{}^2\text{)}\text{](dτ + q}{}^2\text{dσ)}{}^2\text{+ [}\text{(p}{}^2\text{+q}{}^2\text{)}\text{Q}\text{] dq}{}^2\text{? [}\text{Q}\text{(p}{}^2\text{+ q}{}^2\text{)}\text{](dτ ? p}{}^2\text{dσ)}{}^2$, with P = P(p), Q = Q(q), is studied; the first sections investigate its connections and curvature; the metric is of type D, with Einstein tensor of the electromagnetic algebraic type. Metrics with R = const are characterized by $\text{P}$ and $\text{Q}$ being polynomials of 4th order. In Section 5, by applying Rainich-Wheeler procedure, the electromagnetic field associated with the studied metric is constructed. Section 6 describes change-of-scale transformations of the derived solution of Einstein-Maxwell equations with λ; Sections 7 and 8 study geodesics and trajectories of charged test particles in the field of this solution; with H-J equation separable, the integration process reduces to quadratures. Section 9 gives a summary of basic results, Sections 10 and 11 investigate contractions of general solution with 6 continuous and 1 discrete parameter to the generalized NUT, anti-NUT and Bertotti-Robinson solutions. Section 12 specializes our general solution to the combined NUT and Kerr-Newman solution. Section 13 investigates a complex extension and the double Kerr-Schild form of our solution of Einstein-Maxwell equations with λ. Finally, Section 14 investigates the special-relativistic limit of the discussed solutions: a construction of a topology of flat space-time is proposed in such a manner, that in a sense it represents a “riemannian sheet” of the analytic structure of the electromagnetic field of the Kerr-Newman solution. Concluding remarks which indicate a further generalization of the present results, derived together with Demiañski, close this paper.     

On the solutions in elliptic functions of integrable nonlinear equations

A new solution in elliptic functions for the KdV equation is constructed using the method proposed by Belokolos and the author.     

A class of nondispersive evolution equations with solitary wave solutions

We present a class of nonlinear evolution equations possessing stable solitary wave solutions with a sech² profile. These equations are related to the Korteweg-de Vries (KdV) and regularised longwave (RLW) equations, but, unlike the latter, are dispersion free in the linear limit.     

Behavior of bounded solutions of quasilinear elliptic equations on Riemannian manifolds

Bounded solutions of the equation Δ _p u = c(x)|u| ^p−2 u are studied.