Conditions for the Existence of Solutions of a Periodic Boundary-Value Problem for an Inhomogeneous Linear Hyperbolic Equation of the Second Order. I

We consider the periodic boundary-value problem u _tt − u _xx = g(x, t), u(0, t) = u(π, t) = 0, u(x, t + ω) = u(x, t). By representing a solution of this problem in the form u(x, t) = u ⁰(x, t) + ũ(x, t), where u ⁰(x, t) is a solution of the corresponding homogeneous problem and ũ(x, t) is the exact solution of the inhomogeneous equation such that ũ(x, t + ω) u x = ũ(x, t), we obtain conditions for the solvability of the inhomogeneous periodic boundary-value problem for certain values of the period ω. We show that the relation obtained for a solution includes known results established earlier. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 912–921, July, 2005.     

The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.     

The solvability of a boundary-value periodic problem

In the space of functions B _a³⁺={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T₃/2)=g(x, −t)}, we establish that if the condition aT ₃ (2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u _u −a ² u _xx =g(x, t), u(0, t)=u(π, t)=0, u(x, t+T ₃ )=u(x, t), ℝ², is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator. Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997 Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997     

Local solvability of the cauchy problem of a fifth-order nonlinear dispersive equation

The solvability of the fifth-order nonlinear dispersive equation δtu＋au （δxu）^2＋βδx^3u＋γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5（R） and s〉1/4,then the Cauchy problem has a unique solution in C（[-T,T],H^5（R）） for some T〉0.     

Asymptotic behavior of the solution to the two-dimensional stationary problem of flow past a body far from it

In the exterior domain Ω⊂ℝ² we consider the two-dimensional Navier-stokes system Δu-▽p=(u,▽)u, div u=0 whose solution possesses a finite Dirichlet integral and satisfies the condition lim_|x|→∞ u(x)=(1, 0). For this solution, we establish the estimate |u(x)−(1, 0)|≤c|x| ^−α, where α>1/4. This estimate implies an asymptotic expression for the solution indicating the presence of a track behind the body. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 246–253, February, 1999.     

Distortion-suppression optimization for a class of nonlinear optical systems with feedback

We consider the initial boundary-value problem for the quasilinear diffusion equation ∂ _t u+u − DΔu = K(x, y)(1 + γ(u + ϕ(x, y))) describing the dynamics of optical systems with controlled feedback wave intensity modulation K(x, y) in the presence of incoming-wave phase perturbations ϕ(x, y). The control problem for the parameter K(x, y) is formulated with the objective of smoothing out the spatial nonhomogeneities of the total output phase u(x, y, T) + ϕ(x, y). We prove existence and uniqueness theorems for the generalized solutions of the direct and conjugate problems, solvability theorems for the optimization problems, and Frechet-differentiability of the objective functional. A formula for the functional gradient is derived and the efficiency of the gradient projection method is demonstrated numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 20, pp. 80–99, 2005.     

The autotopism group of ap-primitive semifield plane of orderq 4

Let κ be a semifield plane of order q⁴ with kernel K≅GF(q²), where q=p^r, p is prime. Previously, Johnson determined the form of p-primitive semifield planes of order q⁴, q=p^r, and Cordero specified the form of autotopisms and proved the solvability of an autotopism group for the particular case q=p. The goal of the present article is to give an explication of the form of autotopisms and prove the solvability of an autotopism group in the general case. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 334–344, May–June, 1996.     

On the behavior of free boundaries near the boundary of the domain

Let u be a solution to the obstacle problem in a domain Ω⊂ℝ ⁿ . In this paper, the behavior of the free boundary in a neighborhood of ϖΩ is studied. It is proved that under some conditions the free boundary touches ϖΩ at contact points. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 5–19. Translated by T. N. Rozhkovskaya.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.     

The first boundary-value problem for a singular nonlinear ordinary differential equation of fourth order

The solvability of the boundary-value problem

Existence of a global solution of the Whitham equations

The Cauchy problem for Whitham equations with monotonic analytic initial data is studied. If the initial data f(u) satisfies the condition f^(2N+1)(u)<0 for all u∈ℝ except a number of isolated points, then the genus of the solution of the Whitham equations is at most equal to N, where 1≤N∈ℕ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 58–71, January, 1999.     

A periodic problem for the inhomogeneous equation of string oscillations

We study a periodic problem for the equation u _tt−u_xx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ² and establish conditions of the existence and uniqueness of the classical solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 558–565, April, 1997.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

On the behavior of solutions of a semilinear parabolic or elliptic equation satisfying a nonlinear boundary condition in a cylindrical domain

One considers a semilinear parabolic equation u _t = Lu − a(x)f(u) or an elliptic equation u _tt + Lu − a(x)f(u) = 0 in a semi-infinite cylinder Ω × ℝ₊ with the nonlinear boundary condition , where L is a uniformly elliptic divergent operator in a bounded domain Ω ∈ ℝⁿ; a(x) and b(x) are nonnegative measurable functions in Ω. One studies the asymptotic behavior of solutions of such boundary-value problems for t → ∞. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 368–389, 2007.     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

On a smooth solution of a boundary-value problem

We study a periodic boundary-value problem for the quasilinear equation u _tt −u _xx =F[u, u _t , u _x ], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998.     

Existence of the Vejvoda-Shtedry spaces

We investigate the linear periodic problem u _tt −u _xx =F(x, t), u(x+2π, t)=u(x, t+T)=u(x, t), ∈ ℝ², and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry spaces. Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, February, 1997.     

On the embedding problem with non-Abelian kernel of orderp 4. IV

In the present paper, the embedding problem is considered for number fields with p-groups whose kernel is either of two groups with two generators α and β and with the following relations: (1) α^ρ=1, α^ρ=1, [α,β,β]=1, [α,β,α,α]=1, or (2) α^ρ=[α, β α], β^ρ=1, [α,β,β]=1. It is shown that for the solvability of the original embedding problem it is necessary and sufficient to have the solvability of the associated Abelian and local problems for all completions of the base fields. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 120–126. Translated by V. V. Ishkhanov.     

Parallelepipeds circumscribed about a convex body in 3-space

It is proved that each convex body K ⊂ ℝ³ of volume V(K) is contained in a parallelepiped of volume . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 79–87.     

Stochastic equations for fluctuating potentials in donor-acceptor systems with low compensation

To solve the stochastic equations for fluctuating potentials in donor-acceptor systems with low compensations K, an asymptotic perturbation theory is elaborated and a correction to the Fermi level of impurity electrons is calculated: δμ(K) = c₁K + c₂K^3/2 + c₃K² log(1/K) + c₄K². The constants c_i are an asymptotic series in the powers of α = 0.23 with coefficients depending on Λ, where 1/Λ is the mean size of the local two-acceptor complex. The constant c₁ is proportional to Λ. Thus, despite the low compensation of two-acceptor local complexes, their structure determines the leading term of the asymptotic behavior of δμ(K). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 2, pp. 295–307, August, 1997.