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References 1. Cohen I., Nagel S. R., Scaling at the selective withdrawal transition through a tube suspended above the fluid surface, Phys. Rev. Lett. (2002) Vol. 88, 074501.

2. Jeong J.-T., Moffatt H. K., Free-surface cusps associated with flow at low Reynolds number, J.

Fluid Mech. (1992) Vol. 241, 1-22.

WAVE EQUATIONS WITH p(x, t)-LAPLACIAN: EXISTENCE AND BLOW-UP S. N. Antontsev CMAF, University of Lisbon, Portugal Let Rn be a bounded domain with Lipschitz-continuous boundary and QT = (0, T ].

We consider the problem utt = div a(x, t) |u|p(x,t)-2 u + ut + b(x, t)|u|(x,t)-2u, (x, t) QT, (1) u(x, 0) = u0(x), ut(x, 0) = u1(x), x, (2) u| = 0, T = (0, T ). (3) T Meleshko S. V. The exponents p(x, t), (x, t), the coefficients a = a(x, t), b = b(x, t) are given functions of their arguments and > 0. It is assumed that 0 a- a(x, t) a+, |at| Ca, (4) 1 < p- p(x, t) p+ <, (5) |pt| = -pt Cp, (6) 1 < - (x, t) +, 0 t C, (7) 0 b(x, t) b+, 0 bt, (8) 1,p(·,0) u0 L2() W, u1 L2(), f L2(QT ). (9) We discuss the questions of existence (local and global with respect to time) and blow up of energy solutions to the problem (1)–(3). The main attention is paid to the blow up effects caused by the variable nonlinearity of the equation under the study. The analysis is based on the methods developed in [1, 2, 3].

References 1. Antontsev S. N., Daz J. I., and Shmarev S., Energy Methods for Free Boundary Problems:Applications to Non-linear PDEs and Fluid Mechanics, Bikhuser, Boston, Progress in Nonlinear Differential Equations and Their Applications, Vol. 48, 2002.

2. Antontsev S. N., Shmarev S. Vanishing solutions of anisotropic equations with variable nonlinearity. Nonlinear Anal., Theory, Methods, Applications. 2010. Vol. 71. p. 371–391.

3. Antontsev S. N., Shmarev S. Blow-up of solutions to parabolic equations with nonstandard growth conditions. Journal of Computational and Applied Mathematics. 2010 (in press).

ON GROUP ANALYSIS OF STOCHASTIC DIFFERENTIAL EQUATIONS S. V. Meleshko School of Mathematics, Suranaree University of Technology Nakhon Ratchasima, Thailand Stochastic differential equations are often obtained by including random fluctuations in differential equations which have been deduced from phenomenological or physical view. For example, the motion of a small particle suspended in a moving liquid is described by the differential equation dx = b(t, x), dt where b(t, x) is the velocity of the fluid at the point x and at the time t. Let the function (t, x) represents the resistance caused by the viscosity of the liquid. If a particle is randomly bombarded by molecules of the fluid, then this can be modeled by the equation dX = b(t, X) + (t, X)Wt, (1) dt Panov E. Yu. where Wt denotes “white noise”. The second term on the right hand side represents the large number of collisions of the pollen grain with the molecules of the liquid. Formally, the white noise is written as Wt = dBt/dt and equation (1) is rewritten in the differential form dX = b(t, X) dt + (t, X) dBt, (2) Here Bt is a Brownian motion. Equation (2) is called a stochastic differential equation. Solution is a stochastic process. Equations of the type (2) have been used widely in other areas of the science.

In contrast to deterministic differential equations, only few attempts to apply group analysis to stochastic differential equations can be found in the literature. The presentation deals with applications of the group analysis method to stochastic differential equations. The presentation is organized as follows. Before defining an admitted symmetry for stochastic differential equations an introduction into stochastic differential equations is given. The introduction includes the discussion of a stochastic integration, a stochastic differential and a change of the variables (It formula) in stochastic differential equations. The It formula and the change of time in stochastic differential equations are the main tools of defining admitted transformations for stochastic differential equations. After introducing an admitted Lie group for SDEs and supporting material of the introduced definition, some examples of applications of the given definition are studied.

This research is supported by the Center of Excellence in Mathematics, the Commission on Higher Education, Thailand.

PARABOLIC H-MEASURES AND THE STRONG PRE-COMPACTNESS PROPERTY E. Yu. Panov Novgorod State University, Veliky Novgorod In a domain Rn we consider the equation div(x, u) - D2 · B(x, u) = 0, (1) where D2 · B(x, u) = x xjbij(x, u), u = u(x) (we use the conventional rule of summation over i repeated indexes), B(x, u) = {bij(x, u)}n is a symmetric matrix such that bij(x, u) i,j=L2 (, C(R)), i, j = 1,..., n, and x, u1, u2 R sign (u1 - u2)(B(x, u1) - B(x, u2)) 0. We loc also assume that B(x, u) is degenerated on a linear subspace X Rn: X (B(x, u)-B(x, 0)) = 0. Concerning the convective terms, we suppose that (x, u) = (1(x, u),..., n(x, u)) L2 (, C(R, Rn)).

loc On the base of the localization principles for parabolic H-measures we establish the following strong pre-compactness property for sequences uk(x) of approximate solutions of (1), which are supposed to be merely measurable functions satisfying the requirement that the sequences of distributions div (x, sa,b(uk(x)))-D2·B(x, sa,b(uk(x))) are pre-compact in the anisotropic Sobolev -1,-space Wd,loc () for some d > 1 and each a, b R, a < b, where sa,b(u) = max(a, min(u, b)) are cut-off functions.

Theorem 1. Suppose that for almost all x for all X, X such that = 0, = · · the function (x, ), B(x, ) are not constant on non-degenerate intervals. Then a) there exists a measurable function u(x) R {±} such that, after extraction of a subsequence ur, r N, sa,b(ur) sa,b(u) as r in L1 () a, b R, a < b.

loc b) If, in addition, m(uk(x))dx CK for each compact set K, where m(u) is a positive K super-linear function, then u(x) L1 () and ur u in L1 () as r.

loc loc Theorem 1 allows to establish the existence of entropy solutions to the Cauchy problem for evolutionary parabolic equations of the kind (1), see the details in [1]. In the particular case Shmarev S. I., Diaz J. I. B = B(u) the assertion of Theorem 1 was proved in [2] under the weaker non-degeneracy condition:

for a.e. x for all Rn, = 0 the functions · (x, ), B() · are not constant simultaneously on non-degenerate intervals. Here no parabolicity assumptions on the existence of the subspace X are required.

The work was carried out under financial support of the Russian Foundation for Basic Research (project 09-01-00490-a).

References 1. Panov E. Yu. Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property. J. of Mathematical Sciences. 2009. V. 159. N 2. P. 180–228.

2. Holden H., Karlsen K. H., Mitrovic D., Panov E., Yu. Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function. Acta Mathematica Scientia. 2009. V. 29B. N6. P. 1573–1612.

HOMOGENIZATION OF TIME HARMONIC MAXWELL EQUATIONS AND THE FREQUENCY DISPERSION EFFECT V. V. Shelukhin Lavrentyev Institute of Hydrodynamics, Novosibirsk We perform homogenization of the time-harmonic Maxwell equations in order to determine an effective dielectric permittivity h and an effective electric conductivity h. We prove that h and h depend on the frequency ; this phenomenon is known as the frequency dispersion effect.

Moreover, the macroscopic Maxwell equations also depend on ; they are different for small and large values of.

References 1. Shelukhin V. V, Terentev S. A. Frequency dispersion of dielectric permittivity and electric conductivity of rocks via two-scale homogenization of the Maxwell equations. Progress in electromagnetic research B. 2009. V. 14. P. 175–202.

LAGRANGIAN COORDINATES IN PARABOLIC EQUATIONS NOT IN DIVERGENCE FORM: APPLICATION TO FREE BOUNDARY PROBLEMS IN CLIMATOLOGY S. I. Shmarev1, J. I. DazUniversity of Oviedo, Oviedo, Spain University Computense, Madrid, Spain We study the dynamics and regularity of the level sets in solutions of the semilinear parabolic equation ut - u a H(u - µ) in Q = (0, T ], (1) Алехно А. Г., Севрук А. Б. where Rn is a ring–shaped domain, a and µ are given constants, H(·) is the Heaviside maximal monotone graph: H(s) = 1 if s > 0, H(0) = [0, 1], H(s) = 0 if s < 0. Such equations arise in climatology: equation (1) is a simplified version of the celebrated energy balance model proposed by M. Budyko in 1969 [1]. We show that under certain conditions on the initial data the level sets µ = {(x, t) : u(x, t) = µ} are n–dimensional hypersurfaces in the (x, t)–space even in the case when meas {u0(x) = µ} = 0. We show that the dynamics of µ is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This equation expresses the velocity of advancement of the level surface µ through the spatial derivatives of the solution u.

Similar results are obtained for the energy balance model proposed and justified by P. Stone in 1972:

ut - p u + f a H(u - µ), p (1, ), (2) where p u denotes the p-Laplace operator.

The study is based on the introduction of a local set of Lagrangian coordinates which render stationary the thought free boundary: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows one to watch the trajectory of each of the fluid particles. The results are published in the papers [2, 3].

References 1. Budyko M. The efects of solar radiation variations on the climate of the earth, Tellus, 1969.

V. 21. P. 611–619.

2. Diaz J. I., Shmarev S. I. Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology. Arch. Rational Mech. Anal. 2009. V. 194. P. 75–103.

3. Diaz J. I., Shmarev S. I. Lagrangian approach to the study of level sets II: A quasilinear equation in climatology. J. Math. Anal. Appl., 2009. V. 352. P. 475–495.

ОДНОРОДНАЯ КРАЕВАЯ ЗАДАЧА РИМАНА СО СЧЕТНЫМ МНОЖЕСТВОМ РАЗРЫВОВ ПЕРВОГО РОДА ЕЕ КОЭФФИЦИЕНТОВ А. Г. Алехно, А. Б. Севрук Белорусский государственный университет, Минск Пусть D комплексная плоскость, разрезанная вдоль луча L = [0, ). Решена однородная краевая задача Римана, состоящая в нахождении всех ограниченных аналитических в D функций, непрерывные граничные значения которых на берегах разреза по L удовлетворяют соотношению +(t) = G(t) -(t), t L. (1) Заданный коэффициент G(t) задачи подчинен условию G(t) = exp 2i (t) t(t) - -1 (t) t(t) +, (2) (t) Hµ, () = > 0, > 1, () = > 0, где [a] означает целую часть числа a, а (t) уточненный порядок. Из условия (2) следует, что аргумент коэффициента G(t) задачи имеет счетное множество точек разрыва первого рода tn, в которых (t)t(t) + 1/2 = n, n N, и arg G(t) испытывает скачок n = 2-1.

Алимжанов Е. С. Следуя методу Н.В. Говорова [1], исследование задачи сводится к изучению асимптотики при z канонической функции задачи Римана, которая вводится по формуле z ln G(t) dt X(z) = exp.

2i t (t - z) Лемма. Если µ > (2 - 1)(2 + 1)-1, то каноническая функция является ограниченным решением задачи Римана (1), (2).

Вообще говоря, задача (1), (2) имеет бесконечное множество ограниченных решений вида (z) = X(z)F (z), где F (z) целые функции, множество решений которых подчинено требованиям, аналогичным установленным в [1] для задачи Римана с бесконечным индексом степенного порядка.

Серьезные трудности представляет вопрос об отыскании условий существования единственного ограниченного решения задачи.

Теорема. Однородная краевая задача Римана (1), (2) имеет единственное ограниченное линейно независимое решение (z) = CX(z), где C произвольная комплексная постоянная, если выполнено одно из условий 1) 0 < < 1/2, µ > ;

2) (t) > 0, (t) > 0.

Список литературы 1. Говоров Н. В. Краевая задача Римана с бесконечным индексом. М.: Наука, 1986.

О ЗАДАЧЕ С ОДНОСТОРОННИМИ ОГРАНИЧЕНИЯМИ ТЕОРИИ ПОЛУПРОВОДНИКОВ Е. С. Алимжанов Казахский национальный университет имени аль-Фараби, Алматы, Казахстан Система уравнении из теории полупроводников, описывающая процесс (p - n) – перехода в диоде, в стационарном случае приводит к задаче с односторонними ограничениями для уравнения Пуассона [1] - u(x) = f(x) в D = {x : (u) < u(x) < (u)}, (1) где искомой функцией является электростатический потенциал u(x), а f(x) – распределение зарядов. Данная задача рассматривается внутри области Rn, которая представляет собою полупроводник. Здесь функции–препятствия (u) и (u), так называемые квазипотенциалы Ферми, неявно зависят от искомой функции и делят область на три части:

= {x : u = } и = {x : u = } области с p- и n- проводимостью соответственно, и D область, где происходит (p - n) – переход. Эта задача со свободными границами, так как границы между описанными областями являются неизвестными. Для однозначного определения функции u(x) в (1) нужно задать условия на остальной части границы области.

Багдерина Ю. Ю. Данная задача с различными условиями на границе была исследована в работе [2], где она была рассмотрена в вариационной постановке и решена с применением методов теории вариационных неравенств (см., напр. [3]).

В работе задача (1) рассмотрена в постановке в виде квазивариационного неравенства K (u) = {v H 1() : (u) v (u) п.в. в } u · (v - u) dx f(v - u) dx + (V1 - U1)h1 + (V2 - U2)h2, v K (u), где K (u) – замкнутое выпуклое подмножество пространства H 1() = {v H1() : v = i Vi- неизвестные константы, i = 1, 2}, а i свободные границы. Также заданы нелокальные граничные условия на и, однородные условия Неймана на остальной части границы. Доказано, что при выполнении определенных условий на данные задачи решение существует и единственно.

Список литературы 1. Markowich, P. A. The Stationary Semiconductor Device Equations. Springer-Verlag / Wien, 1986. – 193 p.

2. Rodrigues, J. F. On some quasi-variational inequality arising in semiconductor theory // Universidad Computense de Madrid, vol. 5, num. 1; 1992. – P. 137–151.

3. Байокки К., Капелло А. Вариационные и квазивариационные неравенства. М.: Наука, 1988.

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