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.     

Decay of Solutions to Damped Korteweg–de Vries Type Equation

In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when a initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H ^k -level, k∈ℕ. In addition, the decay of the energy regarding a initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect.     

Stability criterion for a family of nonlocal difference schemes

We consider finite-difference schemes for the heat equation with nonlocal boundary conditions that contain a real parameter γ. A stability criterion for finite-difference schemes with respect to the initial data was earlier obtained for |γ| ≤ 1. In the present paper, we consider the case in which γ ∈ (−cosh π,−1) and the original differential problem is stable, while the stability conditions for the finite-difference schemes substantially depend on γ. We obtain estimates for the energy norm of the solution of the finite-difference problem via the same norm of the initial data and prove the equivalence of the energy norm and the grid L ₂-norm.     

A dynamic boundary-value problem without initial conditions for almost linear parabolic equations

We study a dynamic boundary-value problem without initial conditions for linear and almost linear parabolic equations. First, we establish conditions for the existence of a unique solution of a problem without initial conditions for a certain abstract implicit evolution equation in the class of functions with exponential behavior as t → −∞. Then, using these results, we prove the existence of a unique solution of the original problem in the class of functions with exponential behavior at infinity.     

Enumerating (Multiplex) Juggling Sequences

We consider the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or landing schedule) of the balls. We first show that this problem is equivalent to choosing 1’s in a specified matrix to guarantee certain column and row sums, and then using this matrix, derive a recursion. This work is a generalization of earlier work of Chung and Graham.     

Initial value problem and relaxation limits of the hamer model for radiating gases in several space variables

We study the initial value problem for a hyperbolic-elliptic coupled system with L ^∞ initial data. We prove global-in-time existence and uniqueness for that model by means of contraction and comparison properties. Moreover, after suitable scalings, we analyze both the hyperbolic–hyperbolic and the hyperbolic–parabolic relaxation limits for the model itself.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

On the Riemann matrices of the hyperbolic system dual to the system of equations of one-dimensional gas dynamics

We consider the Cauchy problem for the quasilinear hyperbolic system describing a one-dimensional flow of a gas with the equation of state p = p(ϱ), p′(ϱ) > 0, and with initial data satisfying a monotonicity condition. We suggest an approach to solving it by reduction to the Cauchy problem for the linear hyperbolic system obtained from the original system by the hodograph transformation. These constructions are extended to a system of elasticity equations describing nonlinear vibrations of a one-dimensional medium. The main result is illustrated by two examples.     

The cauchy problem for a nonlinear integro-differential transport equation

In the present paper, we study the Cauchy problem for a nonlinear time-dependent kinetic neutrino transport equation. We prove the existence and uniqueness theorem for the solution of the Cauchy problem, establish uniform bounds int for the solution of this problem, and prove the existence and uniqueness of a stationary trajectory and the stabilization ast→∞ of the solution of the time-dependent problem for arbitrary initial data. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 677–686, May, 1997. Translated by A. M. Chebotarev     

Slow convergence to zero for a parabolic equation with a supercritical nonlinearity

We study the behaviour of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to zero from above as t → ∞. We show that any algebraic decay rate slower than the self-similar one occurs for some initial data.     

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|^−μ, μ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202.     

On the Cauchy Problem of Evolution P-Laplacian Equations with Strongly Non-linear Sources When 1 < p < 2

We study the solvability of the Cauchy problem (1.1)–(1.2) for the largest possible class of initial values, for which (1.1)–(1.2) has a local solution. Moreover, we also study the critical case related to the initial value u ₀, for 1 < p < ∞. Project supported by the National Natural Science Foundation of China (19971070)     

On the two-scale method for the problem of perturbed one-frequency oscillations

We consider the asymptotic behavior with respect to time of the solution to the initial problem for an ordinary differential equation with a small parameter ∈. We construct an asymptotic approximation that is valid for time valuest≫∈ up to any order in ∈. Translated from Teoreticheskaya i Matematicheskay Fizika, Vol. 118, No. 3, pp. 383–389, March, 1999.     

Existence of the best <Emphasis Type="Italic">n</Emphasis>-term approximants for structured dictionaries

We extend the Pizzetti formulas, i.e., expansions of the solid and spherical means of a function in terms of the radius of the ball or sphere, to the case of real analytic functions and to functions of Laplacian growth. We also give characterizations of these functions. As an application we give a characterization of solutions analytic in time of the initial value problem for the heat equation ∂ _t u = Δu in terms of holomorphic properties of the solid and/or spherical means of the initial data.     

Stabilization of a solution to the Cauchy problem for a nondivergence parabolic equation with growing lower order coefficients

We obtain sharp sufficient conditions on the growth of lower order coefficients of a second-order parabolic equation under which a solution to the Cauchy problem stabilizes to zero uniformly in x on every compact set K ∈ ℝ ^N in some classes of growing initial functions.     

Differential equations related to the Williams-Bjerknes tumour model

We investigate an initial value problem which is closely related to the Williams-Bjerknes tumour model for a cancer which spreads through an epithelial basal layer modeled onI ⊂ Z ². The solution of this problem is a familyp = (p _i(t)), where eachp _i(t)could be considered as an approximation to the probability that the cell situated ati is cancerous at timet. We prove that this problem has a unique solution, it is defined on [0, +∞[, and, for some relevant situations, lim_t→∞ P _i(t) = 1 for alli ∈ I. Moreover, we study the expected number of cancerous cells at timet.     

Admissibility spectra throughω 1

Jensen showed that any countable sequenceA ofA-admissibles is the initial part of the admissibility spectrum of a real. We considerω ₁-long sequences, to be realized byB ⊆ω ₁. The problem is similar to finding a club subset of a stationary set. We investigate when such aB can be forced and when one is already inV.     

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.     

Neumann boundary condition for a non-autonomous Hamilton-Jacobi equation in a quarter plane

We consider Hamilton-Jacobi equation u _t +H(u, u _x ) = 0 in the quarter plane and study initial boundary value problems with Neumann boundary condition on the line x = 0. We assume that p → H(u, p) is convex, positively homogeneous of degree one. In general, this problem need not admit a continuous viscosity solution when s → H(s, p) is non increasing. In this paper, explicit formula for a viscosity solution of the initial boundary value problem is given for the cases s → H(s, p) is non decreasing as well as s → H(s, p) is non increasing.     

A priori estimates for solutions of the first initial boundary-value problem for systems of fully nonlinear partial differential equations

We prove a priori estimates for a solution of the first initial boundary-value problem for a system of fully nonlinear partial differential equations (PDE) in a bounded domain. In the proof, we reduce the initial boundary-value problem to a problem on a manifold without boundary and then reduce the resulting system on the manifold to a scalar equation on the total space of the corresponding bundle over the manifold. St. Petersburg Architecture Building University, St. Petersburg. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 3, pp. 338–363, March, 1997.