The Minimal Process Generated by a Differential Operator of the Second Order in R^n

For a differential operator $\Omega u=\sum\limits_i,j=1^n \frac{\partial}{\partial x_i}(a_ij(x)\frac{\partial u}{\partial x_j})+\sum\limits_{i=1}^n b_i(x)\frac{\partial u}{\partial x_i}+c(x)u$ with unbounded coefficients in R^n, a standard continuous paths process with infinitesimal operator \Omega has been constructed in this paper, and the invariance of such process under a transformation group of phase space has been discussed.     

The Generalized Riemann-Hilbert problem For a Multi-connected Region of Second Order Non-linear Elliptic Equations

In this paper, we consider the generalized Riemann-Hilbert problem for second order non-linear elliptic complex equation $\frac{\partial ^2 w}{\partial \bar z ^2}=F(z,w,\frac{\partial w}{\partial \bar z},\frac{\partial w}{\partial z},\frac{\partial ^2 w}{\partial z \partial \bar z}),z\in G$(1) with the boundary condition $Re[z^-n_1e^-\pii\alpha_1(z)w]=r_1(z),Re[z^-n_2e^\pi i \alpha_2(z) \frac{\partial w}{\partial \bar z}]=r_2(z),z\in \Gamma$ where $\Gamma=\Gamma_0+\Gamma_1+\cdots+\Gamma_m$ is the smooth boundary of a multi-connected region G,$n_i(i=1,2)$ are called the indices of the boundary value problem. we also obtain the following existence theorem of generalized solution. Theorem, suppose that the indices $n_i>m-1$, the coefficients of the complex equation (1) and the boundary condition (2) satisftes the condition (c),and q^0 is sufficiently small, then the seneralized Riemann-Hilbert problem.(1), (2)is solvable and the solution has theexpression (7).     

Global integrability for minimizers of anisotropic functionals

We consider integral functionals in which the density has growth p _i with respect to ${\frac{\partial u}{\partial x_i}}$ , like in $$\int\limits_{\Omega}\left( \left| \frac{\partial u}{\partial x_1}(x) \right|^{p_1} + \left|\frac{\partial u}{\partial x_2}(x)\right|^{p_2} + \cdots + \left|\frac{\partial u}{\partial x_n}(x) \right|^{p_n} \right) dx.$$ We show that higher integrability of the boundary datum forces minimizer to be more integrable.     

广义Greiner算子的几类Hardy型不等式

对构成广义Greiner算子的向量场$X_j = \frac{\partial }{\partial x_j} + 2ky_j \vert z\vert ^{2k - 2}\frac{\partial }{\partialt}$, $Y_j = \frac{\partial }{\partial y_j } - 2kx_j \vert z\vert^{2k - 2}\frac{\partial }{\partial t}$, j = 1,... ,n, x,y∈ Rⁿ, $z = x + \sqrt { - 1} \,y$, t ∈ R, k ≥1, 得到了拟球域内和拟球域外的Hardy型不等式;建立了广义Picone型恒等式,并由此导出比文献[3]更一般的全空间上的Hardy型不等式;并在$p = 2$时建立了具最佳常数的Hardy型不等式.     

Γ-Convergence of some super quadratic functionals with singular weights

We study the Γ-convergence of the following functional (p > 2)

Existence of Nonnegative Solutions for a Fractional Parabolic Equation in the Entire Space

Ukrainian Mathematical Journal - We study the existence of nonnegative solutions of a parabolic problem $$ \frac{\partial u}{\partial t}=-{\left(-\Delta \right)}^{\frac{\alpha...     

On the Growth Properties of Solutions for a Generalized Bi-Axially Symmetric Schr\"{o}dinger Equation

In this paper, we have considered the generalized bi-axially symmetric Schr\"{o}dinger equation $$\frac{\partial^2\varphi}{\partial x^2}+\frac{\partial^2\varphi}{\partial y^2} + \frac{2\nu} {x}\frac{\partial \varphi} {\partial x} + \frac{2\mu} {y}\frac{\partial \varphi} {\partial y} + \{K^2-V(r)\} \varphi=0,$$ where $\mu,\nu\ge 0$, and $rV(r)$ is an entire function of $r=+(x^2+y^2)^{1/2}$ corresponding to a scattering potential $V(r)$. Growth parameters of entire function solutions in terms of their expansion coefficients, which are analogous to the formulas for order and type occurring in classical function theory, have been obtained. Our results are applicable for the scattering of particles in quantum mechanics.     

Evans functions and bifurcations of standing wave fronts of a nonlinear system of reaction diffusion equations

Consider the following nonlinear system of reaction diffusion equations arising from mathematical neuroscience $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u]-w,~ \frac{\partial w}{\partial t}=\varepsilon(u-\gamma w).$ Also consider the nonlinear scalar reaction diffusion equation $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u].$ In these model equations, $\alpha>0$, $\beta>0$, $\gamma>0$, $\varepsilon>0$ and $\theta>0$ are positive constants, such that $0<2\theta<\beta$. In the model equations, $u=u(x,t)$ represents the membrane potential of a neuron at position $x$ and time $t$, $w=w(x,t)$ represents the leaking current, a slow process that controls the excitation.\\indent The main purpose of this paper is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing wave fronts) and Evans functions (complex analytic functions) to establish the existence, stability, instability and bifurcations of standing wave fronts of the nonlinear system of reaction diffusion equations and to establish the existence and stability of the standing wave fronts of the nonlinear scalar reaction diffusion equation.     

On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory

Theoretical and Mathematical Physics - We study two Cauchy problems for nonlinear equations of the Sobolev type, of the form $$ \frac{\partial}{\partial t}\frac{\partial^2u}{\partial x_3^2} +...     

On a semilinear double fractional heat equation driven by fractional Brownian sheet

In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=(\Delta_\alpha+\Delta_\beta)u+\frac{\partial f}{\partial x}(t,x,u)+\frac{\partial^2W}{\partial t\partial x},$$ where $1<\beta<\alpha< 2$, $W(t,x)$ is a fractional Brownian sheet, $\Delta_\theta:=-(-\Delta)^{\theta/2}$ denotes the fractional Lapalacian operator and $f:[0,T]\times \mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear measurable function. We introduce the existence, uniqueness and H\"older regularity of the solution. As a related question, we consider also a large deviation principle associated with the above equation with a small perturbation via an equivalence relationship between Laplace principle and large deviation principle.     

关于一个典型函数Fourier级数的Gibbs现象

In this paper,we point out that the Fourier series of a classical function∑∞k=1 sin kx/k has the Gibbs phenomenon in the neighborhood of zero.Furthermore,we estimate the upper bound of its partial sum and get:supn≥1‖n∑k=1sin kx/k‖∫x0sin x/xdx=1.85194,which is better than that in[1].     

A remark on global existence of solutions of shadow systems

Let Ω be an open, bounded domain in \mathbbRⁿ  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d ₁, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system:

Local Hölder continuity of weak solutions for an anisotropic elliptic equation

We prove, following DiBenedetto’s intrinsic scaling method, that a local bounded weak solution of the equation $$-\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\left[\left| \frac{\partial u}{\partial x_i} \right|^{p_{i}-2} \frac{\partial u}{\partial x_i} \right]=f $$ in Ω, is locally Hölder continuous, where f is a given bounded function and p _i  ≥ 2, for any i = 1, . . . , N.     

The Cauchy–Kowalewski Theorem in the Space of Pseudo Q-holomorphic Functions

In this work, we prove the Cauchy–Kowalewski theorem for the initial-value problem

$$\begin{aligned} \frac{\partial w}{\partial t}= & {} Lw \\ w(0,z)= & {} w_{0}(z) \end{aligned}$$

where

$$\begin{aligned} Lw:= & {} E_{0}(t,z)\frac{\partial }{\partial \overline{\phi }}\left( \frac{ d_{E}w}{dz}\right) +F_{0}(t,z)\overline{\left( \frac{\partial }{\partial \overline{\phi }}\left( \frac{d_{E}w}{dz}\right) \right) }+C_{0}(t,z)\frac{ d_{E}w}{dz} \\&+G_{0}(t,z)\overline{\left( \frac{d_{E}w}{dz}\right) } +A_{0}(t,z)w+B_{0}(t,z)\overline{w}+D_{0}(t,z) \end{aligned}$$

in the space \(P_{D}\left( E\right) \) of Pseudo Q-holomorphic functions.

     

On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory

The equation arising from Prandtl boundary layer theory $$\frac{\partial u}{\partial t} -\frac{\partial }{\partial x_i}\left( a(u,x,t)\frac{\partial u}{\partial x_i}\right)-f_i(x)D_iu+c(x,t)u=g(x,t)$$ is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since $a(\cdot,x,t)$ may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kružkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if $a_{x_i}(\cdot,x,t)\mid_{x\in \partial \Omega}=a(\cdot,x,t)\mid_{x\in \partial \Omega}=0$ and $f_i(x)\mid_{x\in \partial \Omega}=0$, the stability can be proved even without any boundary value condition.     

Evans functions and bifurcations of standing wave solutions in delayed synaptically coupled neuronal networks

Consider the following nonlinear singularly perturbed system of integral differential equations &amp;\frac{\partial u}{\partial t}+f(u)+w\\ =&amp;(\alpha-au)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &amp;+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau,\\ &amp;\frac{\partial w}{\partial t}=\varepsilon[g(u)-w], and the scalar integral differential equation &amp;\frac{\partial u}{\partial t}+f(u)\\ =&amp;(\alpha-a u)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &amp;+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau. There exist standing wave solutions to the nonlinear system. Similarly, there exist standing wave solutions to the scalar equation. The author constructs Evans functions to establish stability of the standing wave solutions of the scalar equation and to establish bifurcations of the standing wave solutions of the nonlinear system.     

A note on a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with existence, positivity property and long-time behavior of solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions      

Hyperbolicity of solution semigroups for linear neutral differential equations

Consider the linear neutral functional differential equation of the form

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation but much less on those of the Hermite heat equation due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). Supported partially by 973 project (2004CB318000)