一类非线性伪抛物型方程Cauchy问题的适定性

通过引入算子I-Δ的Bessel势将伪抛物型方程化成抽象的抛物型方程,然后利用算子半群理论讨论了一类非线性伪抛物型方程Cauchy问题的适定性问题.     

具有间断非线性项的拟线性抛物型方程(英)

本文研究具有间断非线性项的拟线性抛物型方程，利用Clarke广义梯度和伪单调算子理论证明了解的存在性．     

具有间断非线性项的拟线性抛物型方程

本文研究具有间断非线性项的拟线性抛物型方程，利用Ｃｌａｒｋｅ广义梯度和伪单调算子理论证明了解的存在性。     

二阶退化拟线性抛物型方程解的存在性

本文研究二阶退化拟线性抛物型方程的初－边值问题．在适当带权的Ｓｏｂｏｌｅｖ空间,我们利用伪单调算子理论证明了解的存在性．     

一类p-Laplace方程非局部边值问题解的性态研究

本文研究一类具非局部边值条件的p-Laplace抛物型方程解的性态.利用抛物型方程的上下解方法和一些基本理论,得到该问题解的整体存在性,有限时刻爆破以及爆破速率的估计等结论.     

具有伪单调算子的非线性微分包含约束的最优控制

本文研究一类受主算子为伪单调算子的非线性微分包含约束的最优控制问题．首先,探讨抛物型发展方程的柯西问题其解的性质及微分包含问题的容许轨线的存在性;然后,利用一个新的可测选择定理解决了受非线性微分包含约束的最优控制的存在性,最后,给一例子加以说明所获结果的应用性．     

非局部非线性发展方程解的存在性与唯一性

本文利用Ｂａｎａｃｈ不动点定理证明一类非线性发展方程的非局部Ｃａｕｃｈｙ问题解的存在性与唯一性，及其对非局部拟线性抛物方程的应用。     

一类三阶非线性伪抛物方程的初边值问题

非线性伪抛物方程由于其来源于一些重要的物理过程而成为研究热点.对于一类三阶非线性伪抛物方程的初边值问题,给出了Hilbert空间中相应的强制不等式,利用同胚理论及推广的反函数定理,得到了非线性方程初边值问题解的大范围存在定理.对于相应的半线性方程给出了初边值问题解的大范围存在性、唯一性定理.     

一类时滞非线性伪抛物型方程的有界解

本文讨论了一类时滞非线性伪抛物型方程Cauchy问题有界解的存在唯一性问题 ,给出了有界解存在唯一的充分条件 .     

抛物型Monge-Ampère型方程的Cauchy-Neumann问题

本文研究了一类抛物型Monge-Ampère型方程的Cauchy-Neumann问题.通过构造辅助函数,利用函数在极大值点的性质及柯西不等式等方法对方程的解进行估计,得到了方程解的全局二阶梯度估计.接着利用抛物方程的一般理论,进一步得到在光滑条件下,解的长时间存在性,推广了抛物型Monge-Ampère方程的结果.     

一非线性发展方程的反问题

研究一非线性发展方程的未知源项的反演问题.首先,把所考虑的初边值问题化成一等价非线性发展方程的Cauchy问题;然后,利用半群理论,论证反问题解的存在性和唯一性;最后,利用压缩映射不动点方法,得到反问题的可解性.     

一类非线性色散型发展方程反问题的整体解

将一类非线性色散型发展方程反问题转化为抽象空间非线性发展方程Cauchy问题。利用半群方法和赋等价范数技巧,建立了该类抽象发展方程整体解的存在唯一性定理,并应用于所论反问题,得到了该类非线性色散型发展方程反问题整体解的存在唯一性定理,本质地改进了袁忠信得出的解的局部存在唯一性结果。     

具非线性边界条件的非线性发展方程解的Blow up

讨论一类非线性发展方程具有非线性边界条件的初边值问题.在某些假设条件下,利用抛物型方程的最大值原理和凸性方法证明了其解在有限时间内具有爆破性质.     

非线性边界条件的非线性发展方程解的爆破

讨论一类非线性发展方程具有非线性边界条件的初边值问题.在某些假设条件下,利用抛物型方程的最大值原理和凸性方法证明了其解在有限时间内具有爆破性质.     

An Inverse Problem of Identifying the Radiative Coefficient in a Degenerate Parabolic Equation

The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.     

微分特征列法在拟微分算子和Lax表示中的应用

微分特征列法用于拟微分算子和非线性发展方程Lax表示的计算.首先,利用微分特征列法和微分带余除法计算拟微分算子的逆和方根,由于不必求解常微分方程组,并将解代入,因此,使得计算得以简化.其次,利用微分特征列法,约化从广义Lax方程和Zakharov-Shabat推出的非线性偏微分方程,并得到相应的非线性发展方程.在Mathematica计算机代数系统上,编写了相关程序,从而可以利用计算机辅助完成一些非线性发展方程Lax表示的计算.     

Degenerate first-order inverse problems in Banach spaces

We are concerned with an inverse problem for a degenerate linear evolution equation of first-order. Both hyperbolic and parabolic cases will be considered. We indicate sufficient conditions for the existence and uniqueness of a solution. All the results can be applied to inverse problems for equations from mathematical physics. As a possible application of the abstract theorems, some examples of partial differential equations are given.     

A posteriori error estimates with the finite element method of lines for a Sobolev equation

A posteriori error estimates for semidiscrete finite element methods for a nonlinear Sobolev equation are considered. The error estimates are obtained by solving local nonlinear or linear pseudo‐parabolic equations for corrections to the solution on each element. The ratios of these estimates and the true errors are proved to converge to 1, implying that the estimates can be used as indicators in adaptive schemes for the problem. Numerical results underline our theoretical results. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A class of degenerate parabolic equations with a concentrated nonlinear source

In this paper, we consider the Cauchy problem for a class of degenerate parabolic equations with a concentrated nonlinear source. We obtain the existence of the generalized solutions for the problem based on some a priori estimates on solutions.     

Existence of Global and Periodic Solutions for Delay Equation with Quasilinear Perturbation

Delay parabolic problems have been studied by many authors. Some authors investigated more general delay problem (refer to [1], [2]), some investigated concrete delay partial differential equations. Recently, we have done some work on delay parabolic problem. We discussed semilinear parabolic delay problem and obtained some results on the existence of solutions. In particular the results on existence of periodic solutions are characteristic (see [3], [4], [5], [6]). The purpose of this paper is to study delay equation with quasilinear perturbation. We present the existence of global and periodic solutions of abstract evolution equations in Section 2. The abstract results are used to obtain the existence of global and periodic solutions of delay parabolic problem with quasilinear perturbation in Section 3. We make preparation for our investigation and give a generalization of Gronwall inequality (Lemma 1.3) which is used in next section.