L p Boundedness for the Multilinear Singular Integral Operator

L^p (\mathbbRⁿ )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by

Bounds for Hardy type differences

Let A _k be an integral operator defined by

Dynamics of neuronal waves

First of all, by studying the existence and stability of traveling wave fronts of the following nonlinear nonlocal model equation

Fujita exponents for evolution problems with nonlocal diffusion

We prove the existence of a critical exponent of Fujita type for the nonlocal diffusion problem

Solutions of a system of diffusion equations

We study existence and multiplicity of homoclinic type solutions to the following system of diffusion equations on \mathbbR ×W{\mathbb{R}} \times \Omega :

Quasi-linear parabolic problems in L 1 with non homogeneous conditions on the boundary

We are interested in parabolic problems with L¹ data of the type

Oscillatory integrals on unit square along surfaces

Let Q ² = [0, 1]² be the unit square in two-dimensional Euclidean space ℝ². We study the L ^p boundedness of the oscillatory integral operator T _α,β defined on the set ℒ(ℝ²⁺ⁿ ) of Schwartz test functions by

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

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.     

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      12. Some presentations of the real number field      A. S. Morozov 《Algebra and Logic》2012,51(1):66-88 It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .      13. Classification of positive solutions to an integral system with the poly-harmonic extension operator      Sufang Tang  Jingbo Dou 《中国科学 数学(英文版)》2018,61(9):1603-1616 In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R~+_n:{u(x)=c_n,a∫_?R_+~n(x_n~(1-a_v)(y)/|x-y|~(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n~(1-a_uθ)(x)/|x-y|~(n-a))dx,y∈ ?R_+~n,where n 2, 2-n a 1, κ, θ 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.      14. The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations      Yoshitaka Uda 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1995,46(3):366-383 We study nonnegative solutions of the initial value problem for a weakly coupled system       15. Nonlinear wavelet methods for high-dimensional backward heat equation      Rui Li  Jin Ru Wang 《数学学报(英文版)》2013,29(5):913-922 The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation ut (x, y, t) = u xx (x, y, t) + uyy (x, y, t), x ∈ R, y ∈ R, 0 ≤ t 1, u(x, y, 1) = (x, y), x ∈ R, y ∈ R. Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.      16. Ground states for Schrödinger–Poisson type systems      Giusi?VairaEmail author 《Ricerche di matematica》2011,60(2):263-297 In this paper we consider the following elliptic system in \mathbbR3{\mathbb{R}^3} $\qquad\left\{{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\right.$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.      17. On the convergence of double Fourier series of functions from Lp,p > 1      I. L. Bloshanskii 《Mathematical Notes》1977,21(6):438-444 It is proved that if a function from Lp, p > 1, satisfies the condition $$\frac{1}{{t \cdot \tau }}\int_0^t {\int_0^\tau {\left| {f(x + u,y + v) - f(x,y)} \right|} dudv = O\left( {\left[ {1n\frac{1}{{(t^2 + \tau ^2 )}}} \right]^{ - 2} } \right),}$$ then the double Fourier series of function f, under summation over a rectangle, converges almost everywhere.      18. On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II      A. K. Prykarpatsky  D. L. Blackmore  Ya. V. Mykytyuk 《Journal of Mathematical Sciences》2001,104(5):1411-1421 This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)]) for a solution to the Hamilton–Jacobi nonlinear partial differential equation where the Cauchy data are now a function semicontinuous from below, is the usual norm in , , and is a positive evolution parameter. We proved that the Lax formula solves the Cauchy problem (2) at all points , fixed save for an exceptional set of points R of the F type, having zero Lebesgue measure. In addition, we formulate a similar Lax-type formula without proof for a solution to a new nonlinear equation of the Hamilton–Jacobi-type: where is a diagonal positive-definite matrix, mentioned in Part I and having interesting applications in modern mathematical physics.      19. Matrix near-rings and 0-primitivity      W.-F. Ke  J. H. Meyer 《Monatshefte für Mathematik》2012,11(5):353-363 In this article, we study the implication of the primitivity of a matrix near-ring ${\mathbb{M}_n(R) (n >1 )}${\mathbb{M}_n(R) (n >1 )} and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and \mathbbMn(R){\mathbb{M}_n(R)} has the descending chain condition on \mathbbMn(R){\mathbb{M}_n(R)}-subgroups, then the 0-primitivity of \mathbbMn(R){\mathbb{M}_n(R)} implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on \mathbbMn(R){\mathbb{M}_n(R)} is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.      20. Explicit Strong Solutions of SPDE’s with Applications to Non-Linear Filtering      F. E. Benth  Th. Deck  J. Potthoff  G. VÅge 《Acta Appl Math》1998,51(2):215-242 A differential calculus for random fields is developed and combined with the S-transform to obtain an explicit strong solution of the Cauchy problem Here L is a linear second order elliptic operator, hi and c are real functions, and , where W t is a Brownian motion. An application of the solution to nonlinear filtering and mathematical finance is also considered.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Some presentations of the real number field

It is proved that every two Σ-presentations of an ordered field \mathbbR \mathbb{R} of reals over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) , whose universes are subsets of \mathbbR \mathbb{R} , are mutually Σ-isomorphic. As a consequence, for a series of functions f:\mathbbR ? \mathbbR f:\mathbb{R} \to \mathbb{R} (e.g., exp, sin, cos, ln), it is stated that the structure \mathbbR \mathbb{R} = 〈R, +, ×, <, 0, 1, f〉 lacks such Σ-presentations over \mathbbH\mathbbF ( \mathbbR ) \mathbb{H}\mathbb{F}\,\left( \mathbb{R} \right) .     

Classification of positive solutions to an integral system with the poly-harmonic extension operator

In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R~+_n:{u(x)=c_n,a∫_?R_+~n(x_n~(1-a_v)(y)/|x-y|~(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n~(1-a_uθ)(x)/|x-y|~(n-a))dx,y∈ ?R_+~n,where n 2, 2-n a 1, κ, θ 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.     

The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations

We study nonnegative solutions of the initial value problem for a weakly coupled system

Nonlinear wavelet methods for high-dimensional backward heat equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation ut (x, y, t) = u xx (x, y, t) + uyy (x, y, t), x ∈ R, y ∈ R, 0 ≤ t 1, u(x, y, 1) = (x, y), x ∈ R, y ∈ R. Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.     

Ground states for Schrödinger–Poisson type systems

In this paper we consider the following elliptic system in \mathbbR³{\mathbb{R}^3}

On the convergence of double Fourier series of functions from Lp,p > 1

It is proved that if a function from L_p, p > 1, satisfies the condition $$\frac{1}{{t \cdot \tau }}\int_0^t {\int_0^\tau {\left| {f(x + u,y + v) - f(x,y)} \right|} dudv = O\left( {\left[ {1n\frac{1}{{(t^2 + \tau ^2 )}}} \right]^{ - 2} } \right),}$$ then the double Fourier series of function f, under summation over a rectangle, converges almost everywhere.     

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Matrix near-rings and 0-primitivity

In this article, we study the implication of the primitivity of a matrix near-ring ${\mathbb{M}_n(R) (n >1 )}${\mathbb{M}_n(R) (n >1 )} and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and \mathbbM_n(R){\mathbb{M}_n(R)} has the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)}-subgroups, then the 0-primitivity of \mathbbM_n(R){\mathbb{M}_n(R)} implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on \mathbbM_n(R){\mathbb{M}_n(R)} is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.     

Explicit Strong Solutions of SPDE’s with Applications to Non-Linear Filtering

A differential calculus for random fields is developed and combined with the S-transform to obtain an explicit strong solution of the Cauchy problem