Limiting smoothness of the solution of a nonstationary problem with one or two obstacles

The regularity of the solution of a nonstationary problem with an obstable for various forms of parabolic operators has been thoroughly investigated. Under the condition of sufficient smoothness of the data of the problem, one proves that the solutionW _q ^2,1 (Q) belongs to the Sobolev space In the present paper one establishes that the limiting possible smoothness of the solution of a nonstationary problem with one or two obstacles is the boundedness of the second derivatives of the solution with respect to the spatial variables and of the first derivatives with respect to t. One assumes that the operator is linear and the functions defining the obstacles have the minimal possible smoothness.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 149–157, 1984.     

Asymptotic behavior of a differential operator with discontinuities at two points

In this paper, we study a Sturm–Liouville operator with eigenparameter‐dependent boundary conditions and transmission conditions at two interior points. By establishing a new operator A associated with the problem, we prove that the operator A is self‐adjoint in an appropriate space H, discuss completeness of its eigenfunctions in H, and obtain its Green function. Copyright © 2010 John Wiley & Sons, Ltd.     

Asymptotic properties of a solution of a boundary-value problem

The asymptotic behavior of the solution of a boundary-value problem for the equation u_txx+ u_x =f when the time tends to infinity is investigated. It is proved that the time mean of the solution tends to a stationary solution everywhere except in a boundary region at the left end of the interval.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 273–284, September, 1970.     

On one nonlinear problem of sequential approach of a controlled object to two evading points

We consider a problem in which a pursuer described by a nonlinear third-order system aims to sequentially approach two points moving along straight lines in a minimal time. The evaders aim to increase the approach time as much as possible by choosing their directions of motion.     

Asymptotic solution of a Coulomb multichannel scattering problem with a nonadiabatic channel coupling

Theoretical and Mathematical Physics - We develop a well-posed multiparticle Coulomb scattering problem based on asymptotic solutions of a Coulomb multichannel scattering problem constructed in the...     

Hyperbolic four-manifolds with one cusp

We introduce an algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds with one cusp. More generally, we show that the number of k-cusped hyperbolic four-manifolds with volume ? V grows like C ^{V ln V} for any fixed k. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.     

On a singular perturbation problem with two second-order turning points

In this paper, we study the singular perturbation problem

where 0<ε1 is a small positive parameter, p(x) and q(x) are sufficiently smooth and strictly positive functions. The main feature of this equation is that there are two second-order turning points in the interval (0,1). Based on the rigorous results on singular perturbation problems with one second-order turning point in our previous work, we obtain a uniform asymptotic approximation for the general solution of the above equation by means of a matching technique.     

Asymptotic stability of one prey and two predators model with two functional responses

We formulate a mathematical model to study the complex dynamical behavior of a three dimensional model consisting of one prey and two predators involving Beddington–DeAngelis and Crowley–Martin functional responses. The existence and stability conditions of the equilibrium points are analyzed. The global asymptotic stability of the interior equilibrium point, if exists, is proved by considering Lyapunov function. Several numerical simulations are performed to illustrate the theoretical analysis. The multiple states of stability are observed in one example whereas another example exhibits the global stability of interior equilibrium point.

     

Asymptotic form and infinite product representation of solution of a second order initial value problem with a complex parameter and a finite number of turning points

The paper studies the differential equation * $y'' + (\rho ^2 \varphi ^2 (x) - q(x))y = 0$ on the interval I = [0, 1], containing a finite number of zeros 0 < x ₁ < x ₂ < ... < x _m < 1 of ? ², i.e. so-called turning points. Using asymptotic estimates from [6] for appropriate fundamental systems of solutions of (*) as |ρ| → ∞, it is proved that, if there is an asymptotic solution of the initial value problem generated by (*) in the interval [0, x ₁), then the asymptotic solutions in the remaining intervals can be obtained recursively. Furthermore, an infinite product representation of solutions of (*) is studied. The representations are useful in the study of inverse spectral problems for such equations.     

Asymptotic expansion of the solution of a boundary value problem

We study, in the rectangle Ω=(0,a)× (0,b), the Dirichlet boundary value problem for the elliptic partial differential equation

Asymptotic solution of a quasistatic thermoelasticity problem for a slender rod

An asymptotic expansion is constructed for solving a quasistatic thermo-elasticity problem for a slender cylindrical rod in the presence of mass forces and non-linear heat sources. The algorithm for constructing the asymptotic form, based on the method of boundary functions, is fairly simple and convenient for carrying out numerical calculations. A deduction is made on the basis of the asymptotic form constructed on how to select correctly a simplified one-dimensional model so as to obtain a better approximation for the solution of the initial two-dimensional problem. An existence theorem for the solution is proved under certain conditions.     

Asymptotic behavior of the least energy solution of a problem with competing powers

We consider the problem ε²Δu−uq+up=0 in Ω, u>0 in Ω, u=0 on ∂Ω. Here Ω is a smooth bounded domain in RN, if N?3 and ε is a small positive parameter. We study the asymptotic behavior of the least energy solution as ε goes to zero in the case . We show that the limiting behavior is dominated by the singular solution ΔG−Gq=0 in Ω\{P}, G=0 on ∂Ω. The reduced energy is of nonlocal type.     

Construction of Siegel cusp forms of genus two in terms of cusp forms of genus one

One proves that the convolution of a modular form of semiintegral weight and genus one with the theta-series of an indefinite quadratic form of signature (3.2) is a Siegel modular form of genus two of an even weight. One establishes the relationship between the zeta-functions of these modular forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 93, pp. 186–191, 1980.     

Existence of a smooth solution of one boundary-value problem

We study a periodic boundary-value problem for the quasilinear equationu _tt–u_xx=F[u, u_t], u(0, t)=u(, t)=0,u(x, t+2)=u(x, t). We establish conditions that guarantee the validity of the uniqueness theorem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1717–1719, December, 1995.     

Asymptotic solution method for a quasilinear minimum force control problem

For a transient process in a quasilinear system, we consider an optimization problem of finding a (multi-dimensional) control with minimum intensity. We suggest an algorithm for constructing asymptotic approximations to the solution of this problem. The main advantage of the algorithm is that an optimal control problem for a linear system is solved instead of the original essentially nonlinear problem.     

Exact solution of one boundary-value problem

We study the boundary-value perlodic problem u _tt −u _xx =F(x, t), u(0, t)=u(π, t)=0, u(x, t+T)=u(x, t), (x, t) ∈ R ². By using the Vejvoda-Shtedry operator, we determine a solution of this problem. Ternopol Pedagogical Institute, Temopol. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 998–1001, July, 1997.     

Smooth solution of one boundary-value problem

We study the boundary value problem for the quasilinear equation u _u ? u_xx=F[u, u_t], u(x, 0)= u(x, π)=0, u(x+w, t)=u(x, t), x ε ®, t ε [0, π], and establish conditions under which a theorem on the uniqueness of a smooth solution is true.     

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED PROBLEM FOR A NONLINEAR SINGULAR EQUATION

The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.     

Asymptotic representation of a solution to a singular perturbation linear time-optimal problem

A time-optimal control problem is considered for a linear system with fast and slow variables and smooth geometric constraints on the control. An asymptotic expansion of the optimal time up to the second order of smallness is constructed and validated.