Problem without initial conditions for linear and almost linear degenerate operator differential equations

We study the problem without initial conditions for linear and almost linear degenerate operator differential equations in Banach spaces. The uniqueness of a solution of this problem is proved in the classes of bounded functions and functions with exponential behavior as t → –∞. We also establish sufficient conditions for initial data under which there exists a solution of the considered problem in the class of functions with exponential behavior at infinity.     

Pseudoparabolic variational inequalities without initial conditions

We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a variable t. Under certain conditions imposed on the coefficients of the inequality, we prove theorems on the unique existence of a solution for a class of functions with exponential growth as t → ∞. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 919–929, July, 1998.     

The Fourier problem with nonlocal boundary conditions for a class of nonlinear parabolic equations

The paper contains theorems on the uniqueness and existence of a generalized solution to the Fourier problem (the problem without initial conditions) for a class of strongly nonlinear parabolic equations, without any restrictions on the behavior of the data and of the solution ast → ∞. A priori estimates of the solution are obtained as well. Bibliography: 8 titles. Dedicated to Olga Arsenievna Oleinik Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19, pp. 000-000, 0000.     

Systems of parabolic variational inequalities without initial conditions

We consider a parabolic variational inequality without initial conditions. We construct a class of existence and uniqueness for a solution of this inequality. This class is defined by the exponential decrease or increase of solutions as t→−∞, depending on the coefficients of the inequality. L’viv University, L’viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 540–547, April, 1997.     

A boundary-value problem in an unbounded strip

Using the generalized scheme of the method of separation of variables we propose a new approach to the study of a boundary-value problem in an unbounded strip for hyperbolic partial differential equations of even order with respect to time. In the space of entire functions of exponential type (with respect to the spatial variable) we single out a unique solution class for the problem and exhibit an algorithm for constructing a solution of it. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 16–21.     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

Problem of the motion of two compressible fluids separated by a closed free interface

We consider the problem of the simultaneous evolution for two barotropic capillary viscous compressible fluids occupying the space ℝ³ and separated by a closed free interface. Under some restrictions on the viscosities of the liquids, the local (in time) unique solvability of this problem is obtained in the Sobolev-Slobodetskii spaces. After the passage to Lagrangian coordinates it is possible to exclude the fluid density from the system of equations. The proof of the existence theorem for a nonlinear, noncoercive initial boundary-value problem is based on the method of successive approximations and on an explicit solution of a model linear problem with a plane interface between the liquids. The restrictions on the viscosities mentioned above appear in the intermediate estimation of this explicit solution in the Sobolev spaces with an exponential weight. Bibliography: 8 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 000, 1997, pp. 61–86. Translated by I. V. Denisova.     

Cauchy problem for \left\{ {\vec p;\vec h} \right\}-parabolic equations with time-dependent coefficients

We establish the existence of a unique solution continuously depending on the initial data to the Cauchy problem for -parabolic equations with time-dependent coefficients for which the initial data are generalized functions (distributions) of slow growth. For a particular class of equations, we state necessary and sufficient conditions for the existence of a unique solution of the Cauchy problem with properties of its spatial variable which are characteristic of its fundamental solution.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 395–411.Original Russian Text Copyright © 2005 by V. A. Litovchenko.This revised version was published online in April 2005 with a corrected issue number.     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L _p spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge in weighted L _p spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying mask, we find a characterization for the corresponding refinable function to lie in weighted L ₂ spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay, which can be viewed as an extension of the Butterworth filters.     

Problem with nonlocal conditions for weakly nonlinear hyperbolic equations

For weakly nonlinear hyperbolic equations of order n, n≥3, with constant coefficients in the linear part of the operator, we study a problem with nonlocal two-point conditions in time and periodic conditions in the space variable. Generally speaking, the solvability of this problem is connected with the problem of small denominators whose estimation from below is based on the application of the metric approach. For almost all (with respect to the Lebesgue measure) coefficients of the equation and almost all parameters of the domain, we establish conditions for the existence of a unique classical solution of the problem. Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L’viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 186–195, February, 1997.     

On the validity of the Ginzburg-Landau equation

Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when a control parameterR lies in the unstable regionO(ε ²) away from the critical valueR _c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic) or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense. For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem for G-L's equation on a finite interval of theO(1/ε²)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution equation with corresponding initial conditions, has a unique solutionO(ε²) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact solution with a relatively small error.     

Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables

We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to the space variables. The case of arbitrarily many space variables is considered. We establish conditions for the existence and uniqueness of a solution of this problem independent of the behavior of the solution as |x| → + ∞. The indicated classes of existence and uniqueness are the spaces of locally integrable functions, and, furthermore, the dimension of the domain does not limit the order of nonlinearity. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1523–1531, November, 2007.     

Singular Cauchy problem for the equation of flow of thin viscous films with nonlinear convection

For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the L ²-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact supports. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 250–271, February, 2006.     

Global solutions to the second initial boundary value problem for fully nonlinear parabolic equations

This paper is concerned with the global existence of the second initial boundary value problem for fully nonlinear parabolic equations. It is proved that when the initial data is sufficiently small, the problem admits a unique global solution. Moreover, ast goes to +∞, the solution exponentially decays to zero.     

Properties of two DC algorithms in quadratic programming

Some new properties of the Projection DC decomposition algorithm (we call it Algorithm A) and the Proximal DC decomposition algorithm (we call it Algorithm B) Pham Dinh et al. in Optim Methods Softw, 23(4): 609–629 (2008) for solving the indefinite quadratic programming problem under linear constraints are proved in this paper. Among other things, we show that DCA sequences generated by Algorithm A converge to a locally unique solution if the initial points are taken from a neighborhood of it, and DCA sequences generated by either Algorithm A or Algorithm B are all bounded if a condition guaranteeing the solution existence of the given problem is satisfied.     

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.     

On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations

We consider a family of two-point boundary-value problems for systems of ordinary differential equations with functional parameters. This family is the result of the reduction of a boundary-value problem with nonlocal condition for a system of second-order, quasilinear hyperbolic equations by the introduction of additional functions. Using the parametrization method, we establish necessary and sufficient conditions of the unique solvability of the family of two-point boundary-value problems for a linear system in terms of the initial data. We also prove sufficient conditions of the unique solvability of the problem considered and propose an algorithm for its solution. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 21–39, 2006.     

Higher-order parabolic variational inequality in unbounded domains

We prove the existence and uniqueness of a solution of a nonlinear parabolic variational inequality in an unbounded domain without conditions at infinity. In particular, the initial data may infinitely increase at infinity, and a solution of the inequality is unique without any restrictions on its behavior at infinity. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 949–968, July, 2008.     

The problem of recovering the temperature and moisture fields in a porous body from incomplete initial data

We propose a statement and computational scheme for the inverse problem of recovering the temperature field and the moisture distribution in a body with incompletely known initial conditions. We give additional relations on the integral values of the unknown functions and introduce a test for the choice of a unique solution of the problem from the set of admissible temperature and moisture functions. We state conditions for independence of the additional data and obtain systems of equations and conditions that close the initial indeterminate problem. We study in detail the example of heat-moisture conduction in a layer. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 66–73.     

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.