A lower semicontinuity result for some integral functionals in the space SBD

In this paper, we obtain a lower semicontinuity result with respect to the strong L1-convergence of the integral functionals F（u,Ω）=Ωf（x,u（x）,εu（x））dx defined in the space SBD of special functions with bounded deformation. Here Su represents the absolutely continuous part of the symmetrized distributional derivative Eu. The integrand f satisfies the standard growth assumptions of order p 〉 1 and some other conditions. Finally, by using this result,we discuss the existence of an constrained variational problem.     

Quenching Phenomenon for a Parabolic MEMS Equation

This paper deals with the electrostatic MEMS-device parabolic equation u_t-?u =λf(x)/(1-u)~p in a bounded domain ? of R~N,with Dirichlet boundary condition,an initial condition u0(x) ∈ [0,1) and a nonnegative profile f,where λ 0,p 1.The study is motivated by a simplified micro-electromechanical system(MEMS for short) device model.In this paper,the author first gives an asymptotic behavior of the quenching time T*for the solution u to the parabolic problem with zero initial data.Secondly,the author investigates when the solution u will quench,with general λ,u0(x).Finally,a global existence in the MEMS modeling is shown.     

一类双重退化抛物方程局部解的存在性

This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate p- Laplace equation(p＞2) u_t-div(b(x,t,u)|▽u|~(p-2)▽u)=f(x,u,t). The initial-boundary value problem in a bounded domain of R~N is considered under mixed boundary conditions.The existence of local-in-time weak solutions is obtained.     

Homogenization of a nonlinear degenerate parabolic differential equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problem ɑtu∈-div(D(x/∈, u∈,▽u∈)+ K(x/∈, u∈))= f(x) with Dirichlet boundary condition. Here the operator D(y, s,s) is periodic in y and degenerated in ▽s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈→ 0 and also prove the corrector results of ▽u∈ to strong convergence.     

EXISTENCE,MULTIPLICITY AND STABILITY RESULTS FOR POSITIVE SOLUTIONS OF NONLINEAR p-LAPLACIAN EQUATIONS

This paper studies the existence of positive solutions of the Dirichlet problem for the nonlinear equation involving p-Laplacian operator:-△pu=λf(u) on a bounded smooth domain Ω in Rn. The authors extend part of the Crandall-Rabinowitz bifurcation theory to this problem. Typical examples are checked in detail and multiplicity of the solutions are illustrated. Then the stability for the associated parabolic equation is considered and a Fujita-type result is presented.     

一类新四阶边值问题的唯一性结果

In this paper, we investigate the existence and uniqueness of solutions for a new fourth-order differential equation boundary value problem:{u(4)(t) = f(t, u(t))-b, 0 t 1,u(0) = u′(0) = u′(1) = u(3)(1) = 0,where f ∈ C([0,1] ×(-∞,+∞),(-∞, +∞)),b ≥ 0 is a constant. The novelty of this paper is that the boundary value problem is a new type and the method is a new fixed point theorem ofφ-(h,e)-concave operators.     

THE BOUNDARY VALUE PROBLEMS FOR THE HIGHER ORDER QUASILINEAR PARABOLIC AND ELLIPTIC SYSTEMS SATISFYING GENERAL BOUNDARY CONDITIONS

In this paper, the existence and uniqueness of the boundary value problems for the higher order quasilinear parabolic systems satisfying general boundary conditions and initial value condition are considered. We have concluded the problems to the following: if all the solutions of a family of problems of the same type which are derived from a substitution of τf(s, t, u, ...,Dx~(2b-1)u), τgj(y, t, u) and τ(x) (0≤τ≤1) for f, g, and φ respectively are uniformly bounded, then the original problems have a unique solution in H~(2b a,1 a/2b)((?)_T). Under assumption that the linear problems have a unique solution, we have proved the existence and uniqueness of the solution of the boundary value problems for the quasillnear elliptic systems.     

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations(b(x,u))/(t)-div(a(x,t,u,▽u))+div(φ(x,t,u))=f in Q_T,where b(x,u) is bounded function of u,the term-div(a(x,t,u,▽u)) is a Leray-Lions type operator and the function φ is a nonlinear lower order and satisfy only the growth condition.The second term f belongs to L~1(Q_T).The proof of an existence solution is based on the penalization methods.     

INITIAL VALUE PROBLEM FOR A CLASS OF NONLINEAR PARABOLIC SYSTEM OF FOURTH-ORDER

In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u_1=-A(x, t)u_(x4)+B(x, t)u_(x2)+(g(u))_(x2)+(grad h(u))_x+f(u)are studied, where u(x, t)=(u_1(x, t).…, u_J(x, t) is a J-dimensional unknown vector valued function, f(u) and g(u) are the J-dimensional vector valued function of u(x, t), h(u) is a scalar function of u, A(x, t) and B(x, t) are J×J matrices of functions. The existent, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J=1, h(u)=0, g(u)=au~3, A=a_1, B=a_2, where a_1, a_2 a are constants, the system is a generalized diffusion model equation in population problem.     

单圈图的无符号狄利克雷谱半径

Let G be a simple connected graph with pendant vertex set ?V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ≠ 0 on V(G) such that Q(G)f(u) = λf(u) for u ∈ V_0 and f(u) = 0 for u ∈ ?V. The signless Dirichlet spectral radiusλ(G) is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.     

Existence and concentration result for Kirchhoff equations with critical exponent and Hartree nonlinearity

This paper is concerned with the following Kirchhoff-type equations $$ \left\{ \begin{array}{ll} \displaystyle -\big(\varepsilon^{2}a+\varepsilon b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\mathrm{d}x\big)\Delta u + V(x)u+\mu\phi |u|^{p-2}u=f(x,u), &\quad \mbox{ in }\mathbb{R}^{3},\(-\Delta)^{\frac{\alpha}{2}} \phi=\mu|u|^{p},~u>0, &\quad \mbox{ in }\mathbb{R}^{3},\\end{array} \right. $$ where $f(x,u)=\lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~\mu\geq0$ are constants, $\alpha\in(0,3)$, $p\in[2,3),~q\in[2p,6)$ and $\varepsilon,~\lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{\varepsilon}$ with exponential decay at infinity for $\lambda>0$ and $\varepsilon$ small enough. Furthermore, $u_{\varepsilon}$ concentrates around a global minimum point of $V(x)$ as $\varepsilon\rightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature.     

LOCALLY EKELAND'S VARIATIONAL PRINCIPLE AND SOME SURJECTIVE MAPPING THEOREMS

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔ（Ｍ，ｄ）ｂｅａｃｏｍｐｌｅｔｅｍｅｔｒｉｃｓｐａｃｅ，ｌｅｔｆ：Ｍ→Ｒ∪｛＋∞｝ｂｅａｌｏｗｅｒｓｅｍｉｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎ，ｎｏｔｉｄｅｎｔｉｃａｌｙ＋∞ａｎｄｂｏｕｎｄｅｄｆｒｏｍｂｅｌｏｗ...     

On the numerical computation of blowing‐up solutions for semilinear parabolic equations

Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end , with a finite unknown ‘blow‐up’ time T_b have been studied in a previous work. Specifically, for ε a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ‘mass control’ property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve‐\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon‐u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end , \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow‐up time T_b and the blow‐up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end , with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end . Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end , we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end , defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end . In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end , the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end , verifies, $\|(u‐u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1‐\alpha}$\nopagenumbers\end , \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end . The two special cases of a single‐point blow‐up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p‐1})$\nopagenumbers\end . The estimate $|T_\eta‐T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd.     

Heat Kernel Upper Estimates for Symmetric Jump Processes with Small Jumps of High Intensity

We consider the following non-local operator

Integral inequalities for algebraic and trigonometric polynomials

Doklady Mathematics - In this paper, we consider sharp estimates of integral functionals $\int_0^{2\pi } {\phi (L|Lf_n (t)|)dt} $ for functions φ defined on the semiaxis (0, ∞) and...     

Besov空间中双参数Volterra型重分数过程的弱逼近

In this paper, we prove that two-parameter Volterra multifractional process can be approximated in law in the topology of the anisotropic Besov spaces by the family of processes{B_n(s,t)},n∈N defined by B_n(s,t)=∫_0~s ∫_0~tk_(a(s))(s,u)K_(β(t))(t,u)θ_(n(u,v))dudv,here {θ_n(u, v)}n∈N is a family of processes, converging in law to a Brownian sheet as n→∞,based on the well known Donsker's theorem.     

Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \ u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.     

Uniqueness of solutions for an integral boundary value problem with fractional $q$-differences

This paper deals with uniqueness of solutions for integral boundary value problem$\left\{\begin{array}{l}(D_q^{\alpha}u)(t)+f(t, u(t))=0,\ \ \ t\in(0,1),\ u(0)=D_qu(0)=0,\ \ u(1)=\lambda\int_0^1u(s){\mbox d}_qs, \end{array}\right.$ where $\alpha\in(2,3]$, $\lambda\in (0,[\alpha]_q)$, $D_q^{\alpha}$ denotes the $q$-fractional differential operator of order $\alpha$. By using the iterative method and one new fixed point theorem, we obtain that there exist a unique nontrivial solution and a unique positive solution.     

ε_0-Regularity for Mean Curvature Flow from Surface to Flat Riemannian Manifold

In this paper we prove an ε0-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫Σ0 |A|2 ≤ε0 and the initial area μ0(Σ0) is not large, then along the mean curvature flow, we have ∫Σt|A|2 ≤ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.     

对带混合非线性项的基尔霍夫方程规范解问题的补充

在本文中,我们研究了带有质量约束条件的基尔霍夫方程规范解的存在性问题, $$-\Big(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} \text{d} x\Big)\Delta u=\lambda u+\mu|u|^{q-2} u+|u|^{p-2} u,~~x\in \mathbb{R}^{3},$$ 其中质量约束条件为$$S_{c}:=\Big\{u \in H^{1}(\mathbb{R}^{3}):\int_{\mathbb{R}^{3}}|u|^{2} \text{d} x=c\Big\},$$ 这里$a$, $b$, $c>0$, $\mu\in \mathbb{R}$, $20$, 当$(p, q)$ 属于$\mathbb{R}^{2}$中的某个域时. 我们通过使用约束最小化、集中紧性原理和Minimax方法证明了规范解的存在性.我们部分地扩展了已经被研究的结果.