36 PD07 Abstracts

36 PD07 Abstracts CP1 CP1 Energy Functional of the Dbvp and the First Eigen- The Boundary Value Problems with Strong Singu- value of the Laplacian larity of Solution. n n Let B1 be a ball of radius r1 in S (H ) with r1 <π We consider the boundary value problems with strong sin- n for S . Let B0 be a smaller ball of radius r0 such that gularity. Depending on the coefficients and right-hands B0 ⊂ B1. Let u be a solution of the problem −Δu =1in sides of the equation and of the boundary conditions, the Ω:=B1 \ B0 vanishing on the boundary. It is shown that solution can be infinite at some points (the case of strong the energy functional E(Ω) is minimal if and only if the singularity, u ∈ H1(Ω)). For these problems we offer to de- balls are concentric. It is also shown that first Dirichlet fine the solution as Rν -generalized one. Such a new concept eigenvalue of the Laplacian on Ω is maximal if and only if of solution led to distinction of two classes of the boundary the balls are concentric. value problems: the problems with coordinated and non- coordinated degeneration of initial date; and also made Anisa Chorwadwala it possible to study the existence and uniqueness of solu- The Institute of Mathematical Sciences tions as well as coercivity and differential properties in the [email protected] weighted Sobolev spaces. Victor Rukavishnikov CP1 Russian Academy of Scinces Singularities, Not Existing Solutions, Mac and [email protected] PDE The non existing solution of Dirichlet problem for simple CP1 bounded domain is analyzed. If the domain is a ring for Uniqueness and Asymptotic Stability for the Ra- Laplace equation then the solution exist for finite inner and dial Solutions of Semilinear Elliptic Equations outer radius but it can have singularity in the solution for small inner radius. The solution does not exist for zero in- Less is known of the uniqueness for the radial solutions u = n ner radius. To obtain the physically acceptable solution the u(r) of the problem Δu = f(u+)=0inR (n>2),u(ρ)= method of additional conditions (MAC) is applied. Then 0,u(0) = 0, besides the cases where limr→∞u(r)=0;and the bounded solution can be obtained. for the cases based only on the evolution of the functions d f(t) f(t) and dt t . This paper proves uniqueness for the prob- Igor Neygebauer lem Da + f(u+)=0(r>0),u(ρ)=0,u(0) = 0 based Associate Professor 1 on the assumption that f ∈ C ([0, ∞)) and that ρ satisfies National University of Rwanda a boundedness condition. Furthermore, we prove asymp- [email protected] totic stability for Da + φ(r)f(u+) = 0 based only on the evolution of u(r) and u − φ(r)f(u). CP1 Sikiru Adigun Sanni A Theory of Infinite-Dimensional Evans Function University of Uyo, Uyo, Nigeria and the Associated Bundle. [email protected] The purpose of this talk is to introduce a topological frame- work to treat elliptic eigenvalue problems in a channel or CP2 bounded domain in higher dimension. The main objects Ultra-Fast Soliton Propagation in Nonlinear Opti- are generalizations of the Evans function and the bundle cal Media construction which are originally used in the stability analysis of traveling wave in R. Motivated by the recent interest in optical materials with embedded nanostructures (so-called metamaterials), Shunsaku Nii we study stability of ultra-fast pulses governed by the Kyushu University Maxwell-Duffing model. A solitary wave is obtained as a [email protected] simulated annealing solution for the two-point boundary- value problem of the 6 by 6 system of ODEs for the trav- Jian Deng eling wave. Stability of solitary waves is demonstrated Central University of Finance and Economics through a series of DNS as well as through a numerical [email protected] linearized stability analysis. Yevgeniy Frenkel, Victor Roytburd CP1 Department of Mathematical Sciences Singular Limit Problem for Some Elliptic Systems RPI [email protected], [email protected] For the sharp interface problem arising in the singular limit of some elliptic systems, we prove the existence and the nondegeneracy of solutions whose interface is a distorted CP2 circle in a two-dimensional bounded domain without any Evaluation of the Finite Impulse in Plastic-Elastic assumption on the symmetry of the domain. Wave Propagation Yoshihito Oshita We consider a wave picture in semi-infinite bar in case of Department of Mathematics plastic-elastic loading and elastic unloading. The stress in Okayama University the endpoint of the bar represents the finite impulse. The [email protected] boundary between loading and unloading zones is called the unloading wave. In case under consideration the initial PD07 Abstracts 37 part of the unloading wave is the strong wave, but a jump Goethert Pkwy of strains decreases. We obtain the condition for impulse [email protected] amplitude, such that this jump equals zero within finite time. After this jump has equaled zero the unloading wave becomes the weak wave. We obtain the form of this wave CP2 analytically. Numerical Simulation on Optimal Quantum Sys- tem Control Alexander A. Kosmodemyanskiy MIIT (Moscow Institute of Railways) Amazing quantum world is the most frontier realm. Among Comp. Math Department this, quantum system control, with its supprisingly devel- [email protected] oping and rapidly growing, is extensively demanded area. It attracts numerous attention of worldwide researchers and scientists. Fortunately, laboratory simulation provides CP2 the essential step to realize quantum control in advanced A Variational Approach to Inverse Recovery of Co- laser technology. With this motivation, the novel point of efficient Functions in the Acoustic Wave Equation this work focus on exploring substantial stage in the direc- tion of real quantum control via computational experiment. We present a full waveform, fully nonlinear minimization approach for the reflection seismology problem of recover- Quan-Fang Wang ing the subsurface density ρ, sound speed κ, and absorption Mechanics and Automation Engineering γ from direct measurements of pressure over a part of the The Chinese University of Hong Kong water surface. Our approach consists of transforming the [email protected] underlying hyperbolic partial differential equation 1 1 CP3 (ptt + γ(x)pt) −∇. ∇p = F (x, t) ρ(x)κ2(x) ρ(x) The Stability and Stabilization of Solutions of the Nonlinear Differential Equations And Their Appli- to the self-adjoint elliptic equation cations 2 −λ −∇.(A(x)∇u)+(λ B(x)+λC(x))u = −e (λR(x)+S(Thex)) presentation consists of the three parts. In the first part we give a review of the works of the author devoted ˆ + F (x, λ) to Liaponov stability of the motion. Given new criterion’s of stability of nonlinear differential and difference equa- via the finite Laplace transform and minimizing a functions in Banach spaces, of stability of solution of nonlinear tional that is believed to possess a unique stationary point. systems of differential equations with lateness, nonlinear The minimization procedure is based on a conjugate gra- systems of differential equations with a small parameter at- dient descent algorithm using Sobolev gradients. tached to derivative, nonlinear systems of partial differen- Ian W. Knowles tial equations. Stability of solution of nonlinear systems of University of Alabama at Birmingham differential equations with discontinuous right-hand sides Department of Mathematics is investigated also. We investigate the Turing stability [email protected] of solution of partial differential equations. Criterion’s of Liapunov stability are applied to regular case and to all possible critical cases simultaneously. In the second part Mandar S. Kulkarni of the presentation we give some new criterion’s of stabi- Department of Mathematics lization of solutions of differential equations. In the third University of Alabama at Birmingham part of the presentation we give applications of these crite- [email protected] rions to some tasks of physics, ecology and economics. In particular, we offer new criterion of stability of Hotelling- CP2 Scellam models in ecology and economics, dissipative and Schrodinger systems. New criterion of stability of Kol- Discrete Nonlinear Waves As Solutions of Linear mogorov model in ecology are given too. Wave Equation Ilya V. Boykov A new method is described to compute thin pulse solu- Professor tions to the linear wave equation, discretized on a uniform [email protected] lattice. Unlike conventional Eulerian numerical solutions, these waves remain narrow, several grid cells (h) in width, and do not spread due to discretization error even over CP3 indefinite distances. These “pulses” represent a traveling The Extremal Solutions of Nonlinear Singular wave front as a codimension 1 surface in the limit h-¿0. The Boundary Value pulse is essentially a nonlinear solitary wave that “lives” on the lattice. It accurately represents the total amplitude of We consider the nonlinear boundary value problems for an arriving pulse at each lattice point in one or three di- partial differential equations, that are not satisfy the co- mensions, as well as the arrival time. Multiple arrival times ercivity condition and for which there are not the global can also be computed. solvability results. By using the method of the insertion of spurious control function the given problem is regularized. Subhashini Chitta For each value of control functions the obtained problem UTSI has at least one global solution. We form the discrepancy [email protected] functional, then we prove some theorems for existence of optimal control. The obtained solutions are regularizators John Steinhoff UTSI 38 PD07 Abstracts for original problem. ear Viscoelasticity Victor Ivanenko In this paper it will be studied the boundary integral Institute for Applied and System Analysis NAS of formulation of the initial-boundary value problem of a Ukraine Volterra type integro-differential equation [email protected] t ∂u(x, t) (t − s)α−1 − μΔu(x, t) − Δu(x, s)ds =0, Pavlo Kasyanov ∂t 0 Γ(α) Kyiv National Taras Shevchenko University, Ukraine in Ω u = f, [email protected] on ∂Ω u(x, 0) = 0.

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