Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions

The Cauchy problem for the damped Boussinesq equation with small initial data is considered in two space dimensions. Existence and uniqueness of its classical solution is proved and the solution is constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly and a uniform in space estimate of the residual term is given

Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation

For the damped Boussinesq equation utt−2butxx=−αuxxxx+uxx+β(u2)xx,x∈(0,π),t>0;α,b=const>0,β=const∈R1, the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtained in the explicit form and the question of the blow up of the solution in a certain case is examined. The possibility of passing to the limit b→+0 in the constructed solution is investigated

A method for assessing the stress-strain state of reinforced concrete structures

The article analyses the modern methods of estimation of stress-strain state of reinforced concrete structures. The result of the analysis is a new method for estimating the stress-strain state of reinforced concrete structures. The method is based on extracting a small sample of concrete from the array. The article describes the method of execution of works, the method of calculating the stresses. Previously, the method was investigated under laboratory conditions. The results are presented in graphs and tables. The research was conducted to assess the stress state of existing concrete structures. As the objects of research, two industrial buildings of 1933 and 1941 construction years were taken. An assessment of a stress state of a panel residential building was held. The measurement results were analyzed. The method for determining the stresses in reinforced concrete buildings and structures is recommended

Применение эхоконтрастных препаратов в клинике и перспективы синхронизации УЗИ, КТ- и МРТ-изображений (собственный опыт и обзор литературы)

Radioology plays a primary role in the diagnosis of various cancer diseases. Continuous development and improvement of ultrasound method compels us pay more attention to emerging new technologies, in order to timely implement them in clinical practice. In the present article we have tried to highlight the main aspects and possible applications echo contrast agents, as well as the ability to synchronize ultrasound, CT and MRI images. In the current article are described classification, physical principles and the basic types of echo contrast agents (ECA), the methodology of performing ultrasound study with contrast agents, the most important advantages and disadvantages of this technique and our own clinical observations. Especially promising is the synchronization of CT and MRI images with contrast ultrasound images when there is an opportunity to take advantage of CEUS and avoid the adverse effects of the additional radiation exposure, and introduction of iodine and gadolinium-containing agents. Undoubtedly, the use of ehokontrastirovaniya opens new horizons in ultrasound diagnostics, allowing to increase its effectiveness and informativity, providing a largely unique diagnostic information.Лучевая диагностика играет важную роль в первичной диагностике различных онкологических заболеваний. Постоянное развитие и совершенствование ультразвукового метода заставляет нас уделять все больше внимания появляющимся новым технологиям, чтобы своевременно внедрять их в клиническую практику. В предлагаемой работе мы постарались осветить основные аспекты и возможности применения эхоконтрастных препаратов, а также возможность синхронизации УЗИ, КТ- и МРТ-изображений. Представлены классификация, физические принципы и основные типы эхоконтрастных препаратов, подробно описана методология выполнения УЗИ с контрастированием, приведены наиболее значимые преимущества и недостатки этой методики и собственные клинические наблюдения. Особенно перспективным представляется синхронизация КТ- и МРТ-изображений с контрастными ультразвуковыми изображениями, когда имеется возможность использовать преимущества эхоконтрастирования и избежать неблагоприятного влияния дополнительной лучевой нагрузки, введения йод- и гадолинийсодержащих препаратов. Несомненно, использование эхоконтрастирования открывает новые горизонты в ультразвуковой диагностике, позволяя повысить ее эффективность и информативность, предоставляя во многом уникальную диагностическую информацию

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

AbstractRayleigh functions σl(ν) are defined as series in inverse powers of the Bessel function zeros λν,n≠0,σl(ν)=∑n=1∞1λν,n2l, where l=1,2,…; ν is the index of the Bessel function Jν(x) and n=1,2,… is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index,Rl(m)=∑k=−∞∞σl(|m−k|)σl(|k|)for l=1,2,…;m=0,±1,±2,…, are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699–725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424], where the properties of R1(m) were investigated. In the present work a general representation of Rl(m) in terms of σl(ν) is deduced. On the basis of this a representation for the function R2(m) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R2(m) as |m|→∞. Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded domains with periodicity conditions in one coordinate. As an example of application of Rl(m) a forced Boussinesq equationutt−2bΔut=−αΔ2u+Δu+βΔ(u2)+f with α,b=const>0 and β=const∈R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R1(m) and R2(m) which are responsible for the nonlinear smoothing effect

Time estimates for the Cauchy problem for a third-order hyperbolic equation

A classical solution is considered for the Cauchy problem: (utt−Δu)t+utt−αΔu=f(x,t), x∈ℝ3, t>0; u(x,0)=f0(x), ut(x,0)=f1(x), and utt(x)=f2(x), x∈ℝ3, where α=const, 0<α<1. The above equation governs the propagation of time-dependent acoustic waves in a relaxing medium. A classical solution of this problem is obtained in the form of convolutions of the right-hand side and the initial data with the fundamental solution of the equation. Sharp time estimates are deduced for the solution in question which show polynomial growth for small times and exponential decay for large time when f=0. They also show the time evolution of the solution when f≠0

Long-time asymptotics for the damped Boussinesq equation in a disk

For the damped Boussinesq equation the first initial-boundary value problem is considered in a unit disk. Its strong solution is constructed in the form of a series in the small parameter present in the initial conditions. The global-in-time solvability follows from the construction. The first-order long-time asymptotics is calculated with the uniform in space estimate of the remainder

On the Kuramoto-Sivashinsky equation in a disk

We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and its asymptotics with respect to the nonlinearity constant. The method can work for other dissipative parabolic equations with dispersion