THE FIRST BOUNDARY VALUE PROBLEM FOR STRONGLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS IN ANISOTROPIC SOBOLEV SPACES

This article concerns the existence of weak solutions of the first boundary value problem for a kind of strongly degenerate quasilinear parabolic equation in the anisotropic Sobolev Space. With the theory of anisotropic Sobolev spaces an existence result is proved.     

THE FIRST BOUNDARY VALUE PROBLEM FOR SOLUTIONS OF DEGENERATE QUASUJNEAR PARABOLIC EQUATIONS

In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation $$\[\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}} {}&{(x,t) \in [0,T]} \end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T} \end{array}} \right.\]$$ $$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}} {and}&{v(u) \to 0\begin{array}{*{20}{c}} {as}&{u \to 0} \end{array}} \end{array}} \right)\]$$ under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$     

THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, the first boundary value problem for quasilinear equation of the form △A（u,x）＋∑i=1^m δb^i（u,x）/δxi＋c（u,x）=0,Au（u,x） ≥0is studied. By using the compensated compactness theory, some results on the existence of weak solution are established. In addition, under certain condition the uniqueness of solution is proved.     

多维拟线性退化抛物方程Cauchy问题解的唯一性和稳定性

本文证明了拟线性退化抛物方程■的Cauchy问题BV解的唯一性和稳定性.     

多维拟线性退化抛物方程Cauchy问题解的唯一性和稳定性

本文证明了拟线性退化抛物方程 (e)u/(e)t=n∑i=1 (e)/(e)xi(aij(u)(e)u/(e)xi)+n∑i=1 (e)bi(u)/(e)xi -c(u), u(x,0)=u0(x),aij(u)ξiξj≥0,(A)ξ∈Rn 的Cauchy问题BV解的唯一性和稳定性.     

带有可测系数的二阶非线性抛物型方程组的初—混合边值问题

在多连通区域上研究带有可测系数的二阶非线性抛物型方程组的初－混合边值问题，首先我们将其化为复形式的方程组，并给出在一定条件下的上述初－这值问题解的先验估计，然后利用解的估计和列紧性原理，证明了这种初－边值问题解的存在性。在论证过程中，我们始终用复分析方法讨论文中所提出的问题，没有看到国外有人使用这种方法处理此类问题。     

THE INITIAL DIRICHLET BOUNDARY VALUE PROBLEM FOR GENERAL SECOND ORDER PARABOLIC SYSTEMS IN NONSMOOTH MANIFOLDS

Abstract

We consider some unilateral boundary value problems in polygonal and polyhedral domains with unilateral transmission conditions. Regularity results in terms of weighted Sobolev spaces are obtained using a penalization technique, similar regularity results for the penalized problems and by showing uniform estimates with respect to the penalization parameter.     

二阶拟线性混合（椭圆—双曲）型方程的Frankl问题

本文证明了一类二阶阶拟线性混合型方程Frankl边值问题解的唯一性与存在性,文中先给出解后一种表示式,据此证明边值问题解的唯一性,进而求得解的估计式,据此再使用复分析理论与“参数开拓法”证明了边值问题解的存在性,这里使用的方法有别于其他作者对混合型方程通常使用的积分方程方法。     

二阶拟线性混合(椭圆一双曲)型方程的Frankl问题

本文证明了一类二阶拟线性混合型方程Frankl边值问题解的唯一性与存在性.文中先给出解的一种表示式,据此证明边值问题解的唯一性.进而求得解的估计式,据此再使用复分析理论与"参数开拓法”证明了边值问题解的存在性.这里使用的方法有别于其他作者对混合型方程通常使用的积分方程方法.     

高阶拟线性椭圆型方程奇摄动问题

本文利用微分不等式和多重尺度，研究了一个高阶椭圆型偏微分方程摄动边值问题，并得到了一致有效的渐近展开式。     

INTEGRAL BOUNDARY VALUE PROBLEM FOR FIRST ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS

By developing a comparison result and using the monotone iterative technique, we obtain the existence of the minimal and the maximal solutions to an integral boundary value problem for first order impulsive integro-differential equations.     

PERIODIC BOUNDARY VALUE PROBLEM FOR SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper investigates the periodic boundary value problems for a class of second order functional differential equations. The monotone iterative technique and the maximum principle are applied to obtain the existence of maximal and minimal solutions.     

ANTI-PERIODIC BOUNDARY VALUE PROBLEM FOR FIRST ORDER DELAY DIFFERENTIAL EQUATIONS

The authors employ the method of upper and lower solutions coupled with the monotone iterative technique to obtain some results on the existence and uniqueness of the solution for anti-periodic boundary value problem of delay differential equations.     

ATTRACTIVITY PROPERTIES OF SOLUTIONS FOR DEGENERATE QUASILINEAR PARABOLIC EQUATIONS OF HIGHER ORDER

1IntroductionIn[1]theexistence0fweaksoluti0ns0fdegeneratequasilinearparabolicinitialboundaryvaluepr0blemonthespaceXT=Lp(O,T;V)hasbeenpr0ved,whereQT=(O,T)xfl,V=W:"(v,fl),whichisaweightedS0b0levspace.Thedegenerati0nisdeterminedbyavectorfunctionv(x)=(v.(x)),IaI=m,withp0sitivec0mponentsv.(x)inflsatisfyingcertainintegrabilityassumptions.Theaim0fthispaperistosh0wsomeattractivitypr0perties0ftheso1ution(ast-oo).Similarresultsf0rparab0licequati0nscanbefounde-g.in[2],[3].Letflbeab0unded0pensubset…     

二阶微分方程边值问题的多重正解

基于Leray-Schauder度理论和上下解方法讨论非线性边值问题(t)+g(t,y)=0,(0)=0,y(1)=b≥0的正解存在性,其中g局部Lipschitz连续,g(t,0)≥0,但是可以是变号函数。主要结论是:如果g(t,y)在y=+∞满足一个超线性增长条件,并且存在使得β(1)>0的非负上解β,则存在正数B使得当0B时,不存在正解。     

二阶非线性积分微分方程组边值问题解的渐近式

研究n-维二阶非线性向量积分微分方程组边值问题的奇摄动,在适当的条件下,利用改进了的对角化方法、逐步逼近法和不动点定理,求得并证明非线性向量积分微分方程组边值问题解的存在性及其渐近表达式,并给出渐近估计.     

THE GENERALIZED RIEMANN-HILBERT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF SECOND ORDER QUASI-LINEAR ELLIPTIC EQUATIONS

In this paper, we consider the generalized Riemann-Hilberij problem for second order quasi-linear elliptic complex equation \[\begin{array}{l} \frac{{{\partial ^2}w}}{{\partial {{\bar z}^2}}} + {q_1}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial {z^2}}} + {q_2}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {q_3}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + {q_4}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \gamma (z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}}),z \in G \end{array}\] satifying the boundary condition \[{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_1}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _1}(z),{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_2}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _2}(z),z \in \gamma {\kern 1pt} {\kern 1pt} {\kern 1pt} (2)\] Many authors (see that papers 1, 4-6) have studied the Diriohlet problem and Riemann-Hilbert problem for linear elliptic complex equation. In our papers 2, 3 we also considered the generalized Riemann-Hilbert problem of the general second order linear elliptic complex equation. We obtained the existence theorem, the explicit form of generalized solution and the sufficient and necessary conditions for the solvability of the above mentioned boundary value problem. Based on these results and applying the property of the introduced integral operators and Schauder's fixed-point principle, it can be proved that the analogous deductions in 3 also hold for the generalized Riemann-Hilber problem (1), (2) of the quasi-linear complex equation, i, e., we have the following theorem: Theorem, If the coefficients of second order quasi-linear elliptic complex equation (1) satifies some conditions then i) When index \({n_1} \ge 0,{n_2} \ge 0\), the boundary value problem (1), (2) is always solvable and the solution depends on 2 \(2({n_1} + {n_2} + 1)\) arbitrary real constants. ii) When index \({n_1} \ge 0,{n_2} < 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (or{\kern 1pt} {\kern 1pt} {\kern 1pt} {n_1} < 0,{n_2} \ge 0{\kern 1pt} )\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1),(2) consists of \( - 2{n_2} - 1{\kern 1pt} {\kern 1pt} {\kern 1pt} ( - 2n, - 1)\) real equalities, if and only if the equalities are satisfied, the boundary value problem is solvable and the solution depends on \(2{n_1} + 1{\kern 1pt} {\kern 1pt} (2{n_2} + 1)\) arbitrary real constants. iii)When index \({n_1} < 0,{n_2} < 0\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1) , (2) consists of \( - 2({n_1} + {n_2} + 1)\) real equalities, if and only if the equalitieis are satisfied, the boundary-value problem is solvable. Finally, in the similar way, we may farther extend the result to the case of the nonlinear uniform elliptic complex equation.     

ON DIRICHLET PROBLEMS FOR SECOND ORDER QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

The purpose of this paper is to study the existence of the classical solutions of some Dirichlet problems for quasilinear elliptic equations $$\[{a_{11}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial {x^2}}} + 2{a_{12}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial x\partial y}} + {a_{22}}(x,y,u)\frac{{{\partial ^2}u}}{{\partial {y^2}}} + f(x,y,u,\frac{{\partial u}}{{\partial x}},\frac{{\partial u}}{{\partial y}}) = 0\]$$ Where $\[{a_{ij}}(x,y,u)(i,j = 1,2)\]$ satisfy $$\[\lambda (x,y,u){\left| \xi \right|^2} \le \sum\limits_{i,j = 1}^2 {{a_{ij}}(x,y,u)} {\xi _i}{\xi _j} \le \Lambda (x,y,u){\left| \xi \right|^2}\]$$ for all $\[\xi \in {R^2}\]$ and $\[(x,y,u) \in \bar \Omega \times [0, + \infty ),i.e.\lambda (x,y,u),\Lambda (x,y,u)\]$ denote the minimum and maximum eigenvalues of the matrix $\[[{a_{ij}}(x,y,u)]\]$ respectively, moreover $$\[\lambda (x,y,0) = 0,\Lambda (x,u,0) = 0;\Lambda (x,y,u) \ge \lambda (x,y,u) > 0,(u > 0).\]$$ Some existence theorems under tire “ natural conditions imposed on $\[f(x,y,u,p,q)\]$ are obtained.     

二阶完全非线性蜕化抛物方程

本文研究了一个空间变量的二阶完全非线性蜕化抛物方程ｕｔ＝Ｆ（ｕｘｘ，ｕｘ，ｘ，ｔ）的第一边值问题。在仅要求Ｆ及其一阶导数满足结构条件的情形，给出了蜕化问题连续解的存在唯一性。这个工作将渗流方程的结果推广到非常一般的情形。     

二阶混合型泛函微分方程边值问题解的存在性

本文利用迭合度方法讨论二阶混合型泛函微分方程 x=f(t,x^t,x^t) 0≤t≤T的边值问题（BVP），得到这个边值问题解的存在性的一个充分条件。