本征值问题的统一推导方法及讨论

针对一般情形的本征方程X″(x)-2bX′(x)+λX(x)=0结合第一、二、三类齐次边界条件的统一形式,给出有关本征值问题的统一结果,从而可直接利用分离变量法求解2U/t2=a22U/x2+a1u/x+a2t/u+a3u型等含有ux项的泛定方程的定解问题.     

EXISTENCE AND UNIQUENESS OFWEAK SOLUTIONS OF UNIFORMLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS

In this paper we deal with the quasilinear parabolic equation u/t=/x_i[a_(ij)(x, t, u))u/x_j]+b_i(x, t, u)u/x_i+c(x, t, u) which is uniformly degenerate at u=O. Under some assumptions we prove existence anduniqueness of nonnegative weak solutions to the Cauchy problem and the first boundary valueproblem for this equation. Furthermore, the weak solutions are globally Holder continuous.     

图上的Li-Yau不等式的一些注记

对于图G上热方程(△-/t-q)u=0的正解u=u(x,t),得到图上改进的Li-Yau梯度估计不等式,这里q满足Γ(q)≤η~2,η是一个常数,进而得到改进的Harnack型不等式,推广了以前的结果.     

非周期Dirac 方程的稳态解

本文研究Dirac方程-iΣαkku＋aβu＋M（x）u=g（x,｜u｜）u的解,其中M（x）是位势函数,g（x,｜u｜）u在无穷远处关于u是超线性的.本文用变分法来研究这一问题.借助于与此方程的＂极限方程＂相关的某个辅助系统,构造了变分泛函ΦM的环绕水平,使得建立在ΦM环绕结构上的极小极大值CM满足0〈CM〈C,这里C是＂极限方程＂的最小能量.从而可以证明（C）c条件对所有c〈C成立,因此得到了方程的最小能量解.     

数形结合解复合函数的零点个数的常见解法

本文通过利用函数图像的方法研究复合函数y=g（f（x））的零点问题,即复合函数方程g（f（x））=0的根,令u=f（x）（内层方程）,这样g（f（x））=0就转化成g（u）=0.当外层方程g（u）=0容易求解时,可以先解方程g（u）=0,再解内层方程u=f（x）,这样方程的总个数即为复合函数y=g（f（x））的零点个数.     

一类强退化抛物方程弱解的惟一性

研究了如下形式的强退化抛物方程(C)(u)/(t)=(2A(u,x,t))/(x2)+(B(u,x,t))/(x),基于Holm gren方法,证明了弱解的惟一性.     

时间周期折现Hamilton-Jacobi方程里1-周期解的存在性

主要运用PDE方法,在时间1-周期的哈密尔顿函数H(x,t,p)关于(x,t,p)连续、关于p强制且关于t,x周期、关于t线性的条件下,证明了比较定理,从而得到了时间周期折现Hamilton-Jacobi方程λu(x,t)+u_t(x,t)+H(x,t,D_xu(x,t))=0里唯一1-周期解的存在性.     

一类变指数基尔霍夫型方程的无穷多解

本文研究带有各向异性p(x)-Laplace算子的基尔霍夫型方程Dirichlet边值问题-N∑i=1M_i(∫_Ω|_x_iu|~(pi(x)pi(x)dx)_x_i(|_x_iu|~(pi(x)-2_x_iu=H(∫_ΩF(x,u)dx)f(x,u),x∈Ω,u=0,x∈Ω其中Ω是R~N(N≥3)中具有光滑边界的有界区域,f(x,u)∈C(×R,R),,i=1,2,…,N,且M_i(t):R~+→R~+,H(t):R→R和p_i(x):→R为连续函数.当非线性项在零点附近次线性增长时,运用临界点理论中的Clark定理获得了新的多重解存在性结果.     

变时间分数阶扩散方程的非一致时间网格有限差分方法

本文在非一致时间网格上,使用有限差分方法求解变时间分数阶扩散方程?α(x,t)u(x,t)/tα(x,t)-2u(x,t)/x2=f(x,t),0α(x,t)q≤1,证明了该方法在最大范数下的稳定性与收敛性,收敛阶为C(Δt2-q+h2).数值实例验证了理论分析的结果.     

THE HOLDER CONTINUITY OF A CLASS OF 3-DIMENSION ULTRAPARABOLIC EQUATIONS

We obtained the Cα continuity for weak solutions of a class of ultraparabolic equations with measurable coeffcients of the form
δt u = δx（a（x, y, t）δx u） ＋ b0（x, y, t）δxu ＋ b（x, y, t）δyu, which generalized our recent results on KFP equations.     

Initial Boundary Value Problem of an Equation from Mathematical Finance

Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.     

Hölder regularity for a Kolmogorov equation

We study the interior regularity properties of the solutions to the degenerate parabolic equation,

which arises in mathematical finance and in the theory of diffusion processes.

     

Homoclinic orbits for an unbounded superquadratic

We consider the following nonperiodic diffusion systems

Remarks on the regularity criteria of the solutions of the 3D micropolar fluid equations

We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2< \alpha \leq\infty, 2\leq\beta< \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r< \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space.     

Continuous dependence on the boundary conditions for the Wentzell Laplacian

The solution u of the well-posed problem

Gradient Estimates for a Nonlinear Heat Equation Under Finsler-geometric Flow

This paper considers a compact Finsler manifold $(M^n, F(t), m)$ evolving under a Finsler-geometric flow and establishes global gradient estimates for positive solutions of the following nonlinear heat equation $$\partial_{t}u(x,t)=\Delta_{m} u(x,t),~~~~~~~~~~(x,t)\in M\times[0,T],$$where $\Delta_{m}$ is the Finsler-Laplacian. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Our results generalize and correct the work of S. Lakzian, who established similar results for the Finsler-Ricci flow. Our results are also natural extension of similar results on Riemannian-geometric flow, previously studied by J. Sun. Finally, we give an application to the Finsler-Yamabe flow.     

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: c(x, y) = inf x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation:V_t(t, x) + sup u∈UV_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t) = 0,V(0, x) = Φ0(x).     

Attractors for a class of semi-linear degenerate parabolic equations

We consider degenerate parabolic equations of the form $$\left. \begin{array}{ll}\,\,\, \partial_t u = \Delta_\lambda u + f(u) \\u|_{\partial\Omega} = 0, u|_{t=0} = u_0\end{array}\right.$$ in a bounded domain ${\Omega\subset\mathbb{R}^N}$ , where Δ_λ is a subelliptic operator of the type $$\quad \Delta_\lambda:= \sum_{i=1}^{N} \partial_{x_i}(\lambda_{i}^{2} \partial_{x_i}),\qquad \lambda = (\lambda_1,\ldots, \lambda_N).$$ We prove global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity.     

Global solutions of the heat equation with a nonlinear boundary condition

We consider the heat equation with a nonlinear boundary condition, $$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$ where ${\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}$ , p > 1 + 1/N, and (N ? 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P).     

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.