Editorial backlog
- Kachalov V.I. Pseudoholomorphic functions and their application
Status: reviewing
Abstract. An analysis of asymptotic methods for solving singularly perturbed problems shows that the solutions obtained by means of these solutions depend in two ways on the small parameter: regularly and singularly. This dependence is particularly clearly demonstrated by the method of regularization of S.A.\,Lomov. Moreover, the regularized solu\-tions of singularly perturbed equations can converge in the usual sense. In this connection, it became necessary to study a special class of functions, pseudoholomorphic functions. This is a very important part of the analy\-sis, it is intended to substantiate the main points of the so-called analytic theory of singular perturbations. On the other hand, the relevance of the theory in question is also dictated by the fact that pseudoholomorphic functions, unlike holomorphic functions, are determined when the condi\-tions of the implicit function theorem are violated.
Date of submission: 16 May 2017 г.
- Kachalov V.I. On the holomorphic regularization of strongly nonlinear singularly perturbed problems
Status: reviewing
Abstract. The method of holomorphic regularization, which is a logical extension of the Lomov method, allows one to construct solutions of nonlinear singularly perturbed initial problems in the form of series converging in the usual sense in powers of a small parameter. The method itself is based on a generalization of the Poincare decomposition theorem: in the regular case, solutions depend holomorphically on a small parameter, in the singular case the first integrals inherit this dependence.
Date of submission: 29 May 2017 г.
- Zikkos E. A Taylor-Dirichlet series with no singularities on its abscissa of convergence
Status: reviewing
Abstract. In this paper it is proved that given any non-negative real number $d$,
there exists a Taylor-Dirichlet series of the form
\[
\sum_{n=1}^{\infty} \left(\sum_{k=0}^{\mu_n-1}c_{n,k}
z^k\right) e^{\lambda_n z},\quad c_{n,k}\in \mathbb{C}
\]
with no singularities on its abscissa of convergence, such that its associated multiplicity-sequence $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ has the following properties:
\noindent
(1) the terms of $\Lambda$ are positive real numbers and uniformly separated,
\noindent
$(\inf_{n\in\mathbb{N}}(\lambda_{n+1}-\lambda_n)>0)$,
\noindent
(2) $\Lambda$ has density equal to $d$, $\left(\lim_{t\to\infty}\frac{\sum_{\lambda_n\le t}\mu_n}{t}=d<\infty\right)$,
\noindent
(3) the multiplicities of the terms of $\Lambda$ are unbounded, $(\mu_n\not=O(1))$.
The proof is based on the fact that for this sequence $\Lambda$
its Krivosheev characteristic $S_{\Lambda}$ is negative.
We remark that when $\mu_n=1$ for all $n\in\mathbb{N}$ the result is false by a well known theorem of P\'{o}lya.
Date of submission: 30 May 2017 г.
- Vinnitskii B.V., Sharan V.L., Sheparovych I.B. About some interpolation problem in the class of functions of exponential type in a half-plane
Status: reviewing
Abstract. The conditions of solvability of the
interpolation problem $f(\lambda_{k} )=d_{k} $ are found in the
class of functions of exponential type. This results are applied
to research of some problem of the function's splitting.
Date of submission: 01 June 2017 г.
- Klimentov S.B. About Isomorphism of Some Integro-differential Operators
Status: accepted в т.0 №0
Abstract. In this work representations of <> for solutions of the linear general elliptic system of the first order in the unit circle are considered. The isomorphism of corresponding operators is established in Banach spaces $C^k_\alpha (\overline D) $ and $W^k_p (\overline D) $, $k\geq $1, $0 <\to alpha <$1, $p> $2. These results develop and supplement B.V. Boyarsky's works where representa\-tions of <> where obtained. Also this work supplements author’s results on representations of <> with more difficult operators.
Date of submission: 02 June 2017 г.
- Poluboyarova N.M. On the instability of extremals of the potential energy functional
Status: reviewing
Abstract. In this paper study has been done on problem of stability and instability of the potential energy functional. By stability we mean the sign-definiteness of the second variation. The expression for the second variation of the functional is calculated. With the capacitive method it is obtain to make the feature of instability extremals. Proved that stability parabolic extremals are planes. We have written the equation of extremals and the second variation of the functional for n-dimensional surfaces of revolution.
Date of submission: 18 June 2017 г.
- Berdellima A. ON A CONJECTURE OF KHABIBULLIN ABOUT A PAIR OF INTEGRAL INEQUALITIES
Status: accepted в т.9 №2
Abstract. It is known that in general Khabibullin’s conjecture is not true. Sharipov [8]
constructed a counterexample when $n = 2$ and $\alpha = 2$. In this paper we develop a method
of how to construct a counterexample for the more general case $n > 2$ and $\alpha > 1/2$.
Date of submission: 24 June 2017 г.
- Kopachevsky N.D., Tsvetkov D.O. Малые движения идеальной стратифицированной жидкости со свободной поверхностью, полностью покрытой крошеным льдом
Status: reviewing
Abstract. Let a rigid immovable vessel be partially filled with an ideal incompressible stratified fluid. We assume that in an
equilibrium state the density of a fluid is a function of the vertical variable $x_3,$ i.e., $\rho_0=\rho_0(x_3).$ In this case the
gravitational field with constant acceleration $\vec g=-g\vec e_3$ acts on the fluid, here $g>0$ and $\vec e_3$ is unit vector of the vertical
axis $Ox_3,$ which is directed opposite to $\vec g.$ Let $\Omega$ be the domain filled with a fluid in equilibrium state, $S$ be rigid wall of
the vessel adherent to the fluid, $\Gamma$ be a free surface completely covered with a crumbled ice.
The initial boundary value problem is reduced to the Cauchy problem
\begin{equation*}
\begin{split}
&\mathcal A \frac{d^2x}{dt^2} + \mathcal C x = f(t),
\quad x(0)=x^0, \quad x^{'}(0)=x^1, \\
&0<< \mathcal A= \mathcal A^{*} \in \mathcal L(\mathcal H),
\quad
0 \leq \mathcal C = \mathcal C^{*} \in \mathcal L(\mathcal H).
\end{split}
\end{equation*}
in some Hilbert space $\mathcal H$. The theorem on strong solvability of initial boundary value problem is proved.
Date of submission: 29 June 2017 г.
- Baskakov A.G., Uskova N.B. Linear differential operator with an involution as generator of group of operators
Status: accepted в т.0 №0
Abstract. We consider mixed problem for first order differential equation with involution.
By the method of similar operators, he differential operator, which defined by this differential
equation, is transformed in orthogonal direct sum of operators. By the main theorem we construct group of operators and
we describe the weak solutions of this problem. We use this theorem for Fourier method.
Date of submission: 29 June 2017 г.
- Garayev M., Guediri H., Sadrawi H. New Characterizations of Bloch spaces, Bers-type and Zygmund-type spaces and
Related Questions
Status: reviewing
Abstract. We give in terms of Berezin symbols new characterizations of\ the Bloch spaces
$\mathcal{B}$ and $\mathcal{B}_{0},$ Bers-type and the Zygmund-type spaces of
analytic functions on the unit disc $\mathbb{D}$ of the complex plane
$\mathbb{C}.$ Moreover, we discuss some properties of Toeplitz operators on
the Bergman space $L_{a}^{2}(\mathbb{D}).$ A new characterization of\ some
function space with variable exponents is also given.
Date of submission: 29 June 2017 г.
- Andriyan S.M., Kroyan A.K., Khachatryan K.A. On Solvability of a Class of Nonlinear Integral Equations in $p$ -adic String Theory
Status: accepted в т.0 №0
Abstract. In this paper a class of nonlinear integral equations, which has direct application in the $ p $ -adic string theory, is studied. The existence of a nontrivial continuous odd and bounded solution on the whole axis is proved. With some additional conditions, the uniqueness of the constructed solution in the certain class of continuous functions is established as well.
Date of submission: 15 July 2017 г.
- Muravnik A.B. On Qualitative Properties of Solutions of Quasilinear Parabolic
Equations Admitting Degenerations at Infinity
Status: reviewing
Abstract. We consider the Cauchy problem для for
quasilinear parabolic equations of the kind
$\rho(x)u_t=\Delta u + g(u)|\nabla u|^2,$ where the positive coefficient $\rho$
admits a degeneration at infinity, while the coefficient $g$ either is a continuous function
or admits power singularities such that the power does not exceed one.
The long-time
behavior of (classical) solutions of the specified problem is
investigated.
Date of submission: 21 July 2017 г.
- Ehrgashev T.G. Third Double-Layer Potential for a Generalized Bi-Axially
Symmetric
Helmholtz Equation
Status: accepted в т.0 №0
Abstract. The double-layer potential plays an
important role in solving boundary value problems for elliptic
equations, and in the study of which for a certain
equation, the properties of the fundamental solutions of the given
equation are used. All the fundamental solutions of the
generalized bi-axially symmetric Helmholtz equation were known,
and only for the first one was constructed the theory of
potential. Here, in this paper, we aim at constructing theory of
double-layer potentials corresponding to the third fundamental
solution. By using some properties of one of Appell's
hypergeometric functions in two variables, we prove limiting
theorems and derive integral equations concerning a denseness of
double-layer potentials.
Date of submission: 31 July 2017 г.
- Bandura A.I., Skaskiv O.B. Exhaustion by balls and entire functions of bounded $\mathbf{L}$-index in joint variables
Status: reviewing
Abstract. We prove criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives
on sphere in $\mathbb{C}^n.$
Some obtained results are new even for entire functions
of bounded index in joint variables, i. e. $\mathbf{L}(z)\equiv 1,$
because we used an exhaustion of $\mathbb{C}^n$ by balls
instead an exhaustion of $\mathbb{C}^n$ by polydiscs.
Date of submission: 08 Avgust 2017 г.
- Baskakov A.G., Dikarev E.E. Spectral Theory of Functions in Research of Partial Differential Operators
Status: accepted в т.0 №0
Abstract. Spectral properties of differential operators with
constant coefficients defined on subspaces of space of bounded continuous functions are studied. Necessary and sufficient conditions of invertibility are
obtained under condition of regularity at the infinity (ellipticity type conditions) of polynomial which describes such operators. Spectrum, images and kernels are described. Conditions of compactness of resolvent of differential operators are obtained. Main results are obtained by methods of harmonic analysis and spectral theory of Banach modules.
Date of submission: 10 Avgust 2017 г.
- Rubinshtein A.I. On the Bary-Stechkin Theorem
Status: reviewing
Abstract. We concider the problem on the modulus of continuity for the analogue
conjugate functions in the case of functions in the case of functions defined
on the diadic group. It is shown that for this case no analogue a
Bary--Stechkin theorem.
Date of submission: 18 Avgust 2017 г.
- Das S. ON THE ZEROS OF A POLYNOMIALS
Status: reviewing
Abstract. In this paper we extend a classical result due to Cauchy [6] for
moduli of all zeros of a polynomial of degree $n$. our result is best possible and sharpen some well-known results. In many cases the new bounds are much better than some other known bounds.
Date of submission: 30 Avgust 2017 г.
- Salakhudinov R.G. Some properties of domain functionals on level sets
Status: reviewing
Abstract. For a plane domain $G$ we consider special functionals that are constructed with the help of domain functions, such as the distance function from a point to the boundary $\partial G$, and the warping function of $G$.
Functionals that depend on the distance function are considered in the case of simply-connected domains. Functionals that depend on the distance function are considered in the case of simply-connected domains. Functionals depending on the warping function of a finitely connected domain are also studied.
We prove that isoperimetric monotonicity properties with respect to a free parameter of the functionals generate another monotonicity of the functionals. Namely, we consider the functionals as functions defined on subdomains of $G$. Some special cases of inequalities were obtained earlier by Payne. We note that the inequalities have been successfully applied to justify new estimates of the torsional rigidity of simply connected and multiply connected domains. In particular, new functionals of the domain with monotonic property in both their arguments are constructed.In addition, exact estimates of the rate of change of the functionals are found, that is, exact estimates of the derivatives are obtained.
Date of submission: 26 September 2017 г.
- Rathod A. CHARACTERISTIC FUNCTION AND DEFICIENCY OF ALGEBROID FUNCTIONS ON ANNULI
Status: reviewing
Abstract. In this paper, the value distribution theory for meromorphic
functions with maximal deficiency sum will be considered for algebroid
functions on annuli and also the relationship between the deficiency of
algebroid function on annuli and that of their derivatives is studied.
Date of submission: 26 October 2017 г.
- Asylgareev A.S. On the application of comparison theorems to the study of stability with probability 1 of stochastic differential equations
Status: reviewing
Abstract. Comparison theorems for solutions of stochastic differential equations were proven. Based on the results obtained conditions of the stability with probability 1 of the perturbed solution of a stochastic differential equation were shown. The approach stated in the article is based on the fact that the solution of a stochastic differential equation can be represented as a deterministic function of a random argument. Due to the fact that this technique is based on properties of the individual trajectory, the results obtained in this work can be reformulated for deterministic analogs of stochastic differential equations.
Date of submission: 03 November 2017 г.
- Petrosova M.A., Tikhonov I.V., Sherstyukov V.B. A rate of growth of the coefficients in the Bernstein polynomials
of the standard module function on a symmetric interval
Status: reviewing
Abstract. We study the Bernstein polynomials for the standard module function
on a symmetric interval.
The question is a~rate of growth of the coefficients in these polynomials
with an explicit algebraic representation.
Particular attention is paid to the behaviour of the maximum coefficient
for which exact exponential asymptotics
and corresponding two-sided estimates are established.
It is shown that the coefficients ``neighboring'' with the maximum
have the same rate of growth.
The asymptotics for the sum of absolute values of all coefficients is obtained.
Date of submission: 17 December 2017 г.
- Gorbatkov S.A., Polupanov D.V. ИССЛЕДОВАНИЕ УСТОЙЧИВОСТИ РЕШЕНИЯ НЕЛИНЕЙНОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ
Status: reviewing
Abstract. Получено аналитическое решение задачи анализа устойчивости решений нелинейной начально-краевой задачи теплопроводности в твердых телах, описываемой параболическим уравнением. Использован разработанный ранее авторами итеро-аппроксимативный метод (ИАМ) и метод функций Ляпунова. ИАМ позволяет выразить решение на каждом шаге итерации в виде рядов по собственным функциям линейной части параболического оператора задачи и создает все предпосылки для применения математического аппарата функций Ляпунова. Приведены результаты расчетов устойчивости теплофизического процесса в трехмерном металлическом теле с переменными по объему теплофизическими свойствами при возмущении начального состояния.
Date of submission: 28 December 2017 г.
- Singh G., Singh G., A New Subclass of Univalent Functions
Status: reviewing
Abstract. In this paper, a new subclass $\chi_t(A,B)$
of close-to-convex functions, defined by
means of subordination is investigated. Some results such as coefficient estimates, inclusion
relations, distortion theorems, radius of convexity and Fekete-Szego problem for this class
are derived. The results obtained here is extension of earlier known work.
Date of submission: 02 January 2018 г.
- Khakimova A.R. К задаче описания обобщенных инвариантных многообразий нелинейных уравнений
Status: reviewing
Abstract. В статье обсуждается задача построения обобщенных инвариантных многообразий для нелинейных уравнений в частных производных. Обобщенное инвариантное многообразие является аналогом понятия симметрии и имеет приложения в теории интегрируемости. Обобщенные инвариантные многообразия позволяют эффективно строить пары Лакса и операторы рекурсии для интегрируемых уравнений. В работе дано полное описание обобщенных инвариантных многообразий порядка $(2,2)$ для уравнения Кортевега-де Фриза. Показано как связано это многообразие с парой Лакса и с оператором рекурсии.
Date of submission: 15 January 2018 г.
- Alhouzani M., Chuprunov A.N. ПУАССОНОВСКИЕ ПРЕДЕЛЬНЫЕ ТЕОРЕМЫ В СХЕМАХ РАЗМЕЩЕНИЯ РАЗЛИЧИМЫХ ЧАСТИЦ
Status: reviewing
Abstract. Рассматривается случайная величина - число ячеек, содержащих $r$ частиц, среди первых $K$ ячеек
в равновероятной схеме размещения не более $n$ различимых частиц по $N$ различным ячейкам. Найдены условия, обеспечивающие сходимость этих случайных величин
к пуассоновской случайной величине. Получено описание предельного распределения. Показано, что эти результаты переносятся на схему размещения различимых частиц по различным ячейкам.
Date of submission: 18 January 2018 г.
- Galkina V.S., Polyntseva S.V. Two problems of identification of two lower coefficients in the many-dimensional parabolic equation of a special type
Status: reviewing
Abstract. We consider two problems of identification of two lower coefficients of the many-dimensional parabolic equa\-tions of a special type. In the first problem the overdetermination conditions are given on the same hyperplane, and in the second problem this conditions are given on two various hyperplanes. The inverse problems are reduced to Cauchy's direct auxiliary problems by means of the overdetermination conditions. The resolvability of direct auxiliary problems are proved. The theorems of existence and uniqueness of classical solutions of the inverse problems are proved in the classes of smooth bounded functions.
The solutions of the inverse problems are represented explicitly in terms of the solutions of the direct problems.
Date of submission: 29 January 2018 г.
- Gekkieva S.Kh., Kerefov M.A. First boundary-value problem for Aller – Lykov moisture transfer equation with time fractional derivative
Status: reviewing
Abstract. In this paper we consider the first boundary value problem for the Aller – Lykov moisture transfer equation with the Riemann – Liouville fractional derivative with respect to time. The equation under study presents generalization for the Aller – Lykov equation employing the idea of the fractal speed change in humidity that explains the existence of moisture flows opposing the humidity potential.
The existence of a solution to the first boundary-value problem is proved by the Fourier method. With the method of energy inequalities, an a priori estimate is obtained for the solution to the problem in terms of the Riemann – Liouville fractional derivative that implies the uniqueness of the solution.
Date of submission: 20 February 2018 г.
- Biswas T. RELATIVE ORDER AND RELATIVE TYPE ORIENTED GROWTH
PROPERTIES OF GENERALIZED ITERATED ENTIRE FUNCTIONS
Status: reviewing
Abstract. The main aim of this paper is to study some growth properties of
generalized iterated entire functions in the light of their relative orders, relative
types and relative weak types.
Date of submission: 21 February 2018 г.
- Poptsova M.N., Habibullin I.T. Algebraic Properties of Quasilinear Two-Dimensional Lattices
Status: reviewing
Abstract. In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied within the frame of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we deal with the quasilinear lattices of the form $u_{n,xy}=\alpha(u_{n+1} ,u_n,u_{n-1} )u_{n,x}u_{n,y} + \beta(u_{n+1},u_n,u_{n-1})u_{n,x}+\gamma(u_{n+1} ,u_n,u_{n-1} )u_{n,y}+\delta(u_{n+1} ,u_n,u_{n-1})$. We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption on the structure of the characteristic Lie ring we described the class of the lattices integrable in the sense indicated above.
Date of submission: 27 February 2018 г.
- Абдуллаева З.~Ш., Фаязов К.С. Условная корректность внутренней краевой задачи для псевдо-дифференциального
уравнения с меняющимся направлением времени
Status: reviewing
Abstract. We consider a problem with data inside of the regularity domain for a pseudo-differential equation, spectral problems related to like thise equations. The uniqueness of the solution of the problem is proved, and the conditional stability of the solution of the problem on the set of correctness is obtained. Using the results of the generalized spectral problem, the form of the solution of the unknown problem is constructed and the incorrectness is proved, namely, the lack of stability of the solution from the data. The conditional stability of the solution on the set of correctness is proved by the methods of functional analysis. The obtained estimates characterizing the conditional stability of the solution of the required problem.
Date of submission: 28 February 2018 г.
- Khusnullin I.Kh. The perturbation of the quantum and the acoustic waveguide narrow potential
Status: reviewing
Abstract. We consider boundary value problems in an n-dimensional cylinder,
Modeling quantum and acoustic waveguides with a potential-
which depend on two parameters, small and large. Small
the parameter corresponds to the diameter of the carrier of the potential, and the large -
its maximum value. The ratios of the parameters are as follows: the product of a small parameter by a square root of a large parameter tends to zero. In this formulation, the problem is different
from the previously investigated topics, that the ratio of the parameters of the tax-
wives are weaker, and different
types of boundary conditions. The main content of the work is
constructing a special transformation that translates the
operator to an operator with a small localized perturbation.
Moreover, this transformation does not change the spectrum of the original operation.
Torah. The condition on the potential is obtained, under which from the edge of the non-
an eigenvalue arises, as well as condition
absence of such an intrinsic value. In case of occurrence,
the principal terms of its asymptotics are constructed. Results
are formulated as a theorem.
Date of submission: 01 Mart 2018 г.
- Lyakhov L.N., Polovinkina M.V., Roshchupkin S.A. On one Lizorkin's weighted class
Status: reviewing
Abstract. A new class of test functions ${\Phi^+_\gamma}$ is introduced.
This class is constructed by means of the mixed
Fourier--Bessel--Kipriyanov--Katrahov transform on the principle
of Lizorkin's space. Functional class ${\Phi^ + _\gamma}$ \,
($\gamma=(\gamma_1,\ldots,\gamma_n)$, $\gamma_i>0$) consists of
functions that are orthogonal to polynomials
of arbitrary order (in the $L_2^\gamma$ scalar product with weight
$\prod|x_i|^{\gamma_i}$) and only of them. We prove a theorem
that the class ${\Phi^+_\gamma}$ is dense in the Lebesgue space
$L_p^\gamma$ with this weight.
Date of submission: 12 Mart 2018 г.
- Biswas D., Dutta S. Mobius action by $SL(2;{\mathbb{R}})$ on different homogeneous
spaces
Status: reviewing
Abstract. In this paper, we have considered all the possible subgroups of SL(2;R) (upto
conjugacy) from dimension zero to three. For each of the classification, we
have defined group action on the same line as Vladimir V. Kisil. M¨ obius
transformation have been taken as the corresponding action. This action is
defined on the homogeneous spaces of various dimensions generated by the
subgroups.
Date of submission: 15 Mart 2018 г.
- Aldweby H., Darus M., Elhaddad S. A Subclass of Harmonic Univalent Functions Defined by a
Generalized Differential Operator Involving $q$-Mittag-Leffler
function
Status: reviewing
Abstract. The starlike class of complex-valued harmonic univalent functions is
defined in this paper by using a rather generalized operator that involve
q-Mittag-Leffler function. In a more precise approach, a necessary and
sufficient coefficient for functions f is given to be included in this class.
Growth bounds and neighborhoods are also consider.
Date of submission: 18 Mart 2018 г.
- Chourdhary A., Raj K. ORLICZ DIFFERENCE TRIPLE LACUNARY IDEAL SEQUENCE
SPACES OVER N-NORMED SPACES
Status: reviewing
Abstract. In the present article, we introduce and study some Lacunary I−convergent
and Lacunary I−bounded triple difference sequence spaces defined by Orlicz function
over n−normed spaces. We shall investigate some algebraic and topological properties
of newly formed sequence spaces. We also make an effort to obtain some inclusion
results between these spaces.
Date of submission: 26 Mart 2018 г.
- Godase A.D. ON GENERALIZED $k$- LUCAS SEQUENCES
Status: reviewing
Abstract. The k- Lucas sequence is companion sequence of k- Fibonacci
sequence defined with the k- Lucas numbers which are defined with the
recurrence relation L k,n = kL k,n−1 + L k,n−2 with the initial conditions
L k,0 = 2 and L k,1 = k. In this paper, we introduce a new generalisation
M k,n of k-Lucas sequence. We present generating functions and Binet
formulas for generalized k-Lucas sequence, and establish binomial and
congruence sums of generalized k-Lucas sequence.
Date of submission: 27 Mart 2018 г.
- Vil'danova V.F. On the uniqueness of a weak solution for the integro-differential aggregation equation
Status: reviewing
Abstract. In a well-known paper A.~Bertozzi, D.~Slepcev (2010)
the existence and uniqueness of solution to a mixed problem for
the aggregation equation
$$u_t - \triangle A(x, u) + {\rm
div} (u\nabla K \ast u)=0$$ are established. The equation describe
the evolution of a colony of bacteria in a bounded convex domain
$\Omega$. In this paper we prove the existence and uniqueness of
the solution of a mixed problem for the more general equation
$$\beta(x,u)_t={\rm
div}(\nabla A(x,u)-\beta(x,u)G(u))+f (x,u).$$ Summand $f
(x,u)$ in the equation models the processes of
"birth-destruc\-tion" of bacteria. The class of integral operators
$G (v)$ is wide enough and contains, in particular, convolution
operators $\nabla K \ast u$. The vector kernel $g (x,y)$ of the
$G(u) $ operator can have singularities:
$$|\nabla g (x,y)| \le C(1+|x-y| ^{- \lambda}),\
\lambda\in (0,n),\ x,y\in\Omega.$$ Proof of the uniqueness of the
solution from the work of A.~Bertozzi, D.~Slepcev is based on the
fact of conservation of "masses" $\int_\Omega u(x,t)dx=const$ of
bacteria and uses the convexity of $\Omega$ and properties of the
convolution operator. Presence in equation the "inhomogeneity" $f
(x,u)$ violates the conservation of "mass". The proof of
uniqueness proposed in the paper is suitable for a nonuniform
equation that does not use the convexity of $\Omega$ and
properties of the convolution operator.
Date of submission: 19 Aprel 2018 г.