Oscillation of solutions of nonlinear partial differential equations of neutral type

In this paper, we deal with the oscillatory behavior of solutions of the neutral partial differential equation of the form $$\begin{gathered} \frac{\partial }{{\partial t}}\left[ {p\left( t \right)\frac{\partial }{{\partial t}}(u\left( {x,t} \right) + \sum\limits_{i = 1}^t {p_i \left( t \right)u\left( {x,t - \tau _i } \right)} )} \right] + q\left( {x,t} \right)f_j (u(x,\sigma _j (t))) \hfill \\ = a\left( t \right)\Delta u\left( {x,t} \right) + \sum\limits_{k = 1}^n {a_k \left( t \right)} \Delta u\left( {x,\rho _k \left( t \right)} \right), \left( {x,t} \right) \in \Omega \times R_ + \equiv G \hfill \\ \end{gathered} $$ where Δ is the Laplacian in EuclideanN-spaceR ^N, R₊=(0, ∞) and Ω is a bounded domain inR ^N with a piecewise smooth boundary δΩ.     

Construction of a solution of a mixed problem for the generalized heat conduction equation by the fourier method

By the Fourier method a solution of the equation

On boundary values of solutions of the dirichlet problem for second order elliptic equation

The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for the general elliptic equation of the second order

Numerical Approximation of a Reaction-Diffusion System with Fast Reversible Reaction

The authors consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction.It is deduced from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the kinetic rate.It follows that the solution converges to the solution of a nonlinear diffusion problem,as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.     

An optimization algorithm for the pile driver problem

In this paper we present the analysis of an algorithm of Uzawa type to compute solutions of the quasi variational inequality $$\begin{gathered} (QVI)\left( {\frac{{\partial ^2 u}}{{\partial t^2 }},\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + \left( {\frac{{\partial u}}{{\partial x}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \left( {\frac{{\partial ^2 u}}{{\partial x\partial t}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \hfill \\ + \left[ {u(1,t) + \frac{{\partial u}}{{\partial t}}(1,t)} \right]\left[ {\upsilon (1) - \frac{{\partial u}}{{\partial t}}(1,t)} \right] + J(u;\upsilon ) - J\left( {u;\frac{{\partial u}}{{\partial t}}} \right) \geqslant \hfill \\ \geqslant \left( {f,\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + F(t)\left[ {\upsilon (0) - \frac{{\partial u}}{{\partial t}}(0,t)} \right],t > 0,\forall \upsilon \in H^1 (0,1), \hfill \\ \end{gathered} $$ which is a model for the dynamics of a pile driven into the ground under the action of a pile hammer. In (QVI) (...) is the scalar product inL ²(0, 1) andJ(u;.) is a convex functional onH ¹(0, 1), for eachu, describing the soil-pile friction effect.     

Mean value theorem and a maximum principle for Kolmogorov's equation

For an equation of the form $$\begin{gathered} \frac{{\partial u}}{{\partial t}} - \sum\nolimits_{ij = 1}^n {{\text{ }}\alpha ^{ij} } \frac{{\partial ^2 u}}{{\partial x^i \partial x^j }} + \sum\nolimits_{ij = 1}^n {\beta _j^i x^i } \frac{{\partial u}}{{\partial x^i }} = 0, \hfill \\ {\text{ }}x \in R^n ,{\text{ }}t \in R^1 , \hfill \\ \end{gathered}$$ where α=(α^ij) is a constant nonnegative matrix andΒ=(Β _i ⁱ ) is a constant matrix, subject to certain conditions, we construct a fundamental solution, similar in its structure to the fundamental solution of the heat conduction equation; we prove a mean value theorem and show that u(x₀, t₀) can be represented in the form of the mean value of u(x, t) with a nonnegative density over a level surface of the fundamental solution of the adjoint equation passing through the point (x₀, t₀); finally, we prove a parabolic maximum principle.     

On the Existence of Solutions for Some Nondegenerate Nonlinear Wave Equations of Kirchhoff Type

Let be a bounded domain in #x211D;ⁿ with a smooth boundary . In this work we study the existence of solutions for the following boundary value problem:

Analytic solution of vector model kinetic equations with constant kernel and their applications

Exact solutions are obtained for the first time for the half-space boundary-value problem for the vector model kinetic equations

Défaut d’estimation a priori pour un problème de dirichlet

Résumé  Dans un célèbre papier ([3]), B. GIDAS et J.SPRUCK ont utilisé-sous des hypothèses adéquates- la technique du “blow up” pour montrer que les solutionsu ∈C ⁰ ∩C ¹ (Ω) du problème admettent une estimation a priori dansC ⁰ . Dans ce travail, on montre que, si les solutionsu sont juste supposéesC ⁰ , une telle estimation a priori n’existe plus. In a famous paper ([3]), B. GIDAS and J. SPRUCK used a “blow-up” argument to show that, under appropriate assumptions, all the solutionsu ∈C ⁰ ∩C ¹ (Ω) of the problem admit an a priori estimate inC ⁰ . In this work, we show that, if one supposes the solutions are only inC ⁰ , such an a priori estimate does not hold.     

Existence,uniqueness and convergence as vanishing viscosity for a reaction-diffusion-convection system

A simple qualitative model of dynamic combustion

On the rate of convergence of some difference schemes for second order elliptic equations

The purpose of this paper is to estimate the rate of convergence for some natural difference analogues of Dirichlet's problem for uniformly elliptic differential equations, $$\begin{gathered} \sum\limits_{j,k = 1}^N {\frac{\partial }{{\partial x_j }}} \left( {a_{jk} \frac{{\partial u}}{{\partial x_k }}} \right) = F in R, \hfill \\ u = f on B, \hfill \\ \end{gathered}$$ in aN-dimensional domainR with boundaryB. These schemes will in general not be of positive type, and the analysis will therefore be carried out in discreteL ²-norms rather than in the maximum norm. Since our approximation of the boundary condition is rather crude, we will only arrive at a rate of convergence of first order for smoothF andf. Special emphasis will be put on appraising the dependence of the rate of convergence on the regularity ofF andf.     

Sums of Five, Seven and Nine Squares

Let r _k(n) denote the number of representations of an integer n as a sum of k squares. We prove that

Precise rate in the law of iterated logarithm for <Emphasis Type="Italic">ρ</Emphasis>-mixing sequence

Let {X, X _n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim _n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,

Another method for computing the densities of integrals of motion for the Korteweg-de Vries equation

In the first section of this article a new method for computing the densities of integrals of motion for the KdV equation is given. In the second section the variation with respect to q of the functional ∫ ₀ ^π w (x,t,x,;q)dx (t is fixed) is computed, where W(x, t, s; q) is the Riemann function of the problem $$\begin{gathered} \frac{{\partial ^z u}}{{\partial x^2 }} - q(x)u = \frac{{\partial ^2 u}}{{\partial t^2 }} ( - \infty< x< \infty ), \hfill \\ u|_{t = 0} = f(x), \left. {\frac{{\partial u}}{{\partial t}}} \right|_{t = 0} = 0. \hfill \\ \end{gathered} $$      

Brownian motion with negative drift and convex level sets in space-time

Suppose a, b, and are reals witha<b and consider the following diffusion equation

Some Tauberian conditions for Cesàro summability method

Let u = (u _n ) be a sequence of real numbers whose generator sequence is Cesàro summable to a finite number. We prove that (u _n ) is slowly oscillating if the sequence of Cesàro means of (ω _n ^(m−1)(u)) is increasing and the following two conditions are hold:

Low Regularity Solution of the Initial-Boundary-Value Problem for the ＂Good＂ Boussinesq Equation on the Half Line

we study an initial-boundary-value problem for the ＂good＂ Boussinesq equation on the half line
{δt^2u-δx^2u＋δx^4u＋δx^2u^2=0,t〉0,x〉0.
u（0,t）=h1（t）,δx^2u（0,t） =δth2（t）,
u（x,0）=f（x）,δtu（x,0）=δxh（x）.
The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data （f, h, h1, h2） belong to the product space
H^5（R^＋）×H^s-1（R^＋）×H^s/2＋1/4（R^＋）×H^s/2＋1/4（R^＋）
1 The analyticity of the solution mapping between the initial-boundary-data and the with 0 ≤ s 〈 1/2. solution space is also considered.     

Approximating periodic functions in the uniform metric by Jackson type polynomials

Let C be the space of continuous 2π-periodic functions f with the norm . Let , where , be the Jackson polynomials of a function f, E _n (f) be the best approximation of f in the space C by trigonometric polynomials of order n, and let , be the function trigonometrically conjugate to the primitive of f. The paper establishes results of the following types:

Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations

In this paper, we consider the problem of the existence of non-negative weak solution u of