Quenching phenomena for a non-local diffusion equation with a singular absorption

In this paper we study the quenching problem for the non-local diffusion equation

Fractional Modified Dyadic Integral and Derivative on ℝ<Subscript>+</Subscript>

For functions in the Lebesgue space L(ℝ₊), a modified strong dyadic integral J _α and a modified strong dyadic derivative D ^(α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ₊), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J _α and D ^(α) is indicated. The formulas D ^(α)(J ^α(f)) = f and J _α(D ^(α)(f)) = f are proved for each α > 0 under the condition that . We prove that the linear operator is unbounded, where is the natural domain of J _α. A similar statement for the operator is proved. A modified dyadic derivative d ^(α)(f)(x) and a modified dyadic integral j _α(f)(x) are also defined for a function f ∈ L(ℝ₊) and a given point x ∈ ℝ₊. The formulas d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ₊ of f.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 64–70, 2005Original Russian Text Copyright © by B. I. GolubovSupported by the Russian Foundation for Basic Research (grant no. 05-01-00206).     

Time periodic solutions to a nonlinear wave equation with <Emphasis Type="Italic">x</Emphasis>-dependent coefficients

In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients on under the boundary conditions a ₁ y(0, t)+b ₁ y _x (0, t) = 0, ( for i = 1, 2) and the periodic conditions y(x, t + T) = y(x, t), y _t (x, t + T) = y _t (x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For , we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients. This work was supported by the 985 Project of Jilin University, the Specialized Research Fund for the Doctoral Program of Higher Education, and the Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University.     

Continuous time random walks and the Cauchy problem for the heat equation

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ?ⁿ. A particle moves in ?ⁿ in such a way that the probability density function u(·, t) of finding it in region Ω of ?ⁿ is given by ∫_Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation

$$u\left( {x,t} \right) = \left[ {\left( {J - \delta } \right)*u} \right]\left( {x,t} \right)$$

, where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.

     

On a multiplicative functional transformation arising in nonlinear filtering theory

Summary This paper concerns the nonlinear filtering problem of calculating estimates E[f(x_t)¦y _s, st] where {x _t} is a Markov process with infinitesimal generator A and {y _t} is an observation process given by dy _t=h(x_t)dt +dw_twhere {w _t} is a Brownian motion. If h(x_t) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup T _s,t ^y obtained by a certain y-dependent multiplicative functional transformation of the signal process {x _t}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is A _t ^y f=e^y(t)h(A– 1/2h²)(e^–y(t)h f).Work partially supported by the U.S. Department of Energy (Contract ET-76-C-01-2295) at the Massachusetts Institute of Technology     

Multiparameter acoustic imaging in the born approximation

Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?₀ = 1 and c(x) ? c₀ = 1 for |x| ? δ, and |γ_n(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γ_n(x) = c^?2(x) ? 1. The samples are insonified by plane pulses s(x · θ₀ – t) where x = |θ₀| = 1 and the scattered pulse is shown to have the form |x|^?1 e_s(|x| – t, θ, θ₀) in the far field, where x = |x| θ. The response e_s(τ, θ, θ₀) is measurable. The goal of the work is to construct the sample parameters γn and γ? from e_s(τ, θ, θ₀) for suitable choiches of s, θ and θ₀. In the limiting case of constant density: γ?(x)_? 0 it is shown that Where δ represents the Dirac δ and S² is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x).     

A new family of Padé-type approximants in

This paper gives the definition and some properties of a new family of Padé-type approximants (PTA) for k-variate formal power series (FPS). These PTA have the form P(t)/Q(t) where Q(t) = Π^r_{i = 0}(1 ? x(i)·t), {x(i), 0 ? i ? r} being a given set of points in

, and x·t is the scalar product of x and t in

. Some results about the approximation order, the unicity and some invariance properties of these PTA are proved together with a convergence result when the FPS is defined by a Stieltjes integral.     

The Evolution of the Weyl and Maxwell Fields in Curved Space—Times

The covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (u, φ) of a space-time (M, g) are considered. It is shown that the condition g⁰⁰(x) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δ_tφ = L_(s)φ. Here L_(s) is a linear first order differential operator on the pre—Hilbert space (C (U_t, ^2s+1). (…)), where U_t ? IR³ is the image of the coordinate map of the spacelike hyper-surface t = const, and (φ, C) = ?U_t ? *Q d⁽³⁾x with a suitable Hermitian n × n- matrix Q = Q(t,x). The total energy of the spinor field ? with respect to U_t is then simply given by E = 〈?,?〉. In this way inequalities for the energy change rate with respect to time, δ_t|?|² = 2Re (?, L_(s)?) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.     

A general oscillation theorem for self-adjoint differential systems with applications to Sturm-Liouville eigenvalue problems and quadratic functionals

For given 2n×2n matricesS ₁₃,S ₂₄ with rank(S ₁₃,S ₂₄)=2n we consider the eigenvalue problem:u′=A(x)u+B(x)v,v′=C ₁(x;λ)u-A ^T(x)v with

A central limit theorem for solutions of the porous medium equation

We study the large–time behavior of the second moment (energy) for the flow of a gas in a N-dimensional porous medium with initial density v₀(x) 0. The density v(x, t) satisfies the nonlinear degenerate parabolic equation v_t = v^m where m > 1 is a physical constant. Assuming that for some > 0, we prove that E(t) behaves asymptotically, as t , like the energy E_B(t) of the Barenblatt-Pattle solution B(|x|, t). This is shown by proving that E(t)/E_B(t) converges to 1 at the (optimal) rate t^{–2/(N(m-1)+2)}. A simple corollary of this result is a central limit theorem for the scaled solution E(t)^N/2v(E(t)^1/2x, t).     

A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u _t + uu _x ? u _xx ) _x + s(t)u _yy = 0 is transformed into the Riccati equation $H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0$ via the ansatz $u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , provided that s(t) = t ^?3/2. Further, a generalized Cole-Hopf transformations $u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}$ , $\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , r(t) = log t is derived to linearize (u _t + uu _x ? u _xx ) _x + t ^?3/2 u _yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ .     

Nonstationary Stokes system in weighted Sobolev spaces

We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??³ with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L_{2, ?µ}(Ω^T), initial velocity v₀∈H(Ω), µ∈?₊\? there exist velocity v∈H(Ω^T) and the pressure p, ?p∈L_{2, ?µ}(Ω^T) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:

• 1. the slip boundary condition and
• 2. the Helmholtz–Weyl decomposition.

Copyright © 2010 John Wiley & Sons, Ltd.     

Sojourns above a high level for a gaussian process with a point of maximum variance

Let X(t), 0≦t≦ 1, be a measurable Gaussian process with mean 0 and variance σ²(t) = EX²(t). Suppose that σ²(t) assumes a unique maximum value σ² at a point τ [0,1]. Define L_u = mes(t: 0≦t≦1, X(t)>)Under appropriate conditions, there exists a function fσ(u) such that fσ(u) → ∞ for u ∞, and The function fσ and the constant δ > 0 are determined by the behavior of the function R(x) = mes(t: 0 ≦ t ≦ 1, σ² - σ²(t) ≦ x) for x > 0.     

From Chaos to Global Convergence

The main purpose of this paper is to investigate dynamical systems F : \mathbbR² ? \mathbbR²{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR² ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x ₀, y ₀), such that the orbit

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

Embedding graphs with bounded degree in sparse pseudorandom graphs

In this paper, we show the equivalence of somequasi-random properties for sparse graphs, that is, graphsG with edge densityp=|E(G)|/( ₂ ⁿ )=o(1), whereo(1)→0 asn=|V(G)|→∞. Our main result (Theorem 16) is the following embedding result. For a graphJ, writeN _J(x) for the neighborhood of the vertexx inJ, and letδ(J) andΔ(J) be the minimum and the maximum degree inJ. LetH be atriangle-free graph and setd _H=max{δ(J):J⊆H}. Moreover, putD _H=min{2d _H,Δ(H)}. LetC>1 be a fixed constant and supposep=p(n)≫n ⁻¹ D _H. We show that ifG is such that

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Let C( \mathbbR^m ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbR^m ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

Orthogonality via Transforms

Let e(x, t) = ∑p_n(x)tⁿ be the generating function of a polynomial sequence, and the transform of multiplication by x relative to e(x, t). We show that the sequence p_n(x) is orthogonal precisely when is a t-variable, i.e., maps K[t] into itself and increases degree by 1. We also show how transform techniques can shed light on the recursion relations and differential equations for p_n(x).