非线性微分方程的可解性研究
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摘要
本论文的结果主要概括为以下几个方面：
1 第一、二章讨论形式为Δu+f(x，u，▽u)=0的椭圆方程存在整体解的条件。第一章考察一维情况下此方程存在无穷多个正整体解的条件，给出了两个存在性定理(定理1．1—1．2)。第二章将此结果推广到高维情形，给出了一个线性增长解的存在性定理(定理2．1)；另外，本章还给出了方程存在对数增长解的两个定理(定理2．2—2．3)。所有这些结果都是考虑了此方程存在无界解的条件，这与已有的关于此方程存在有界解的结果是完全不同的，我们的结果进一步完善了相关的工作。
2 第三章与第四章研究了形式更加广泛的二阶非线性椭圆方程Au—m2u+f(x，u，Vu)=0的指数增长解与衰退解的存在性。得到了此类方程存在指数增长解的三个定理(定理3．1—3．3)与衰退解的两个定理(定理4．1—4．2)。这些结果不仅推广了已有的结果，而且近一步发展了上下解方法与不动点定理在研究微分方程的可解性方面的应用。
3 第五章讨论了一类源于刻划可扩充杆横截挠度的非线性双曲型方程utt+A2u+M(x，ttAl／211帕1)Au=0，Cauchy问题解的存在唯一性，给出了此方程有唯一局部解的存在定理(定理5．1)。与已有的关于此方程的结果相比较，我们对非线性项的要求要宽松得多。实际上，这里的结果是在打破了以前的所有限制得到的。
4 第六章利用寻求偏微分方程的相似约化的直接方法讨论了广义Burgers方程的相似约化以及相应的精确特解，并对所得到的约化提供了非经典对称群解释。就广义Burgers方程而言，本章的工作部分地回答了Clarkson所提出的一个公开问题：如何用直接方法寻求具有任意函数的非线性PDEs的对称约化问题。
5 最后的两章考虑了无约束总体极小化问题的微分方程方法，也称为神经网络方法。首先给出了非凸梯度神经网络平衡点集合的H—稳定性结果(定理7．1)，另外的两个定理(定理7．2—7．3)给出了不同平稳点的吸引域估计。这些结果的意义在于两个方面，其一是解决了利用梯度神经网络求解总体极值问题的网络稳定性问题；其二是为网络的设计提供了一个可行框架。第八章就是在此基础上给出了求解这一问题的两种神经网络设计，并给出了一些典型算例验证了网络的可行性与有效性。
The results of this research paper can be outlined mainly as the following aspects:
1 The existence of entire solutions for the nonlinear elliptic equations in the form of
Au + f(x,u,Vu) = 0
is discussed in chapter 1 and chapter 2. First of all, chapter 1 deals with the one-dimentinal case, and twe existing theorems are obtained (theoreml .1-1.2). This result is browden to n-dimentional case in chapter 2 first and an existence theorem (theorem 2.1) for linearly increasing solutions is given. Furthermore, two existing theorems for logarithmic increasing solutions are also obtained in this chapter (theorem 2.2-2.3). All these results are focused in the unbounded solution, which is different from the existing boundary ones, and so our work perfects the concerned results about the existence of solutions to this equation.
2 The existence theorems of exponently increasing solutions and decaying solutions for a class of more general nonlinear elliptic equations in the form of
Au - m2u + f(x, u,Vu) = 0
are studied in chapter 3 and chapter 4. Three theorems (theorem3.1-3.3) are obtained for exponentlv increasing solutions, and two (theorem4.1-4.2) for decaying ones. These results not only extend the existing ones, but also develop the applications of the sup-sub solution methods and fixed theorems in reseach on the solvability to differential equations.
3 Chaper 5 deals with the hyperbolic equation in the form of
2 2
utt+A2u-i-M(x, IA" u112 )Au0,
which comes from the mathematical descripition of the tension of a extensible beam, and the existence and uniqueness of Cauchy problems for this equation are presented here. An existence and uniqueness theorem (theorem 5.1) of local solvability is proven for this equation, compared with the existing results, our conditions restricted to the nonlinear term of this equation are much general. In fact, the results given here are obtained by breaking all the former restrictions on the nonlinear term.
4 The similarity reductions of generalized Burgers equation is treated in chapter 6 by the direct method for finding similarity reductions of patial differential equations, and the corresponding exact special solutions are given also. More over, a group interpretation
is provided to the similarity reductions in terms of nonclassical symmetry. Accomplishing with the generalized Burges equation, this work answers partially an open problem presented by Clarckson: that is how to find the similarity reductions of nonlinear PDEs with arbitrary functions by the direct method.
5 In the last two chapters, chapter7-8, the global minimized problems of the unconstrained optimization are studied by approching the solutions with the solution curves of a differential equation, which is also called the neural network method. Chapter 7 gives the H-stability of the eqilibrium point set first (theorem7.1), and the other two theorems (theorem7.2-7.3) provide an estimation result for the attractive region of different eqilibrium points. All these results own two essential meanings, the first is that this complete fully the stability of the neural network, the second is that provides a feasible frame for the neural network designs. Just based on these results, two models of neural networks for solving the problems are proposed in chapter 8, and moreober, the feasibility and the validity of our designs are shown by some typical computation examples.
1 第一、二章讨论形式为Δu+f(x，u，▽u)=0的椭圆方程存在整体解的条件。第一章考察一维情况下此方程存在无穷多个正整体解的条件，给出了两个存在性定理(定理1．1—1．2)。第二章将此结果推广到高维情形，给出了一个线性增长解的存在性定理(定理2．1)；另外，本章还给出了方程存在对数增长解的两个定理(定理2．2—2．3)。所有这些结果都是考虑了此方程存在无界解的条件，这与已有的关于此方程存在有界解的结果是完全不同的，我们的结果进一步完善了相关的工作。
2 第三章与第四章研究了形式更加广泛的二阶非线性椭圆方程Au—m2u+f(x，u，Vu)=0的指数增长解与衰退解的存在性。得到了此类方程存在指数增长解的三个定理(定理3．1—3．3)与衰退解的两个定理(定理4．1—4．2)。这些结果不仅推广了已有的结果，而且近一步发展了上下解方法与不动点定理在研究微分方程的可解性方面的应用。
3 第五章讨论了一类源于刻划可扩充杆横截挠度的非线性双曲型方程utt+A2u+M(x，ttAl／211帕1)Au=0，Cauchy问题解的存在唯一性，给出了此方程有唯一局部解的存在定理(定理5．1)。与已有的关于此方程的结果相比较，我们对非线性项的要求要宽松得多。实际上，这里的结果是在打破了以前的所有限制得到的。
4 第六章利用寻求偏微分方程的相似约化的直接方法讨论了广义Burgers方程的相似约化以及相应的精确特解，并对所得到的约化提供了非经典对称群解释。就广义Burgers方程而言，本章的工作部分地回答了Clarkson所提出的一个公开问题：如何用直接方法寻求具有任意函数的非线性PDEs的对称约化问题。
5 最后的两章考虑了无约束总体极小化问题的微分方程方法，也称为神经网络方法。首先给出了非凸梯度神经网络平衡点集合的H—稳定性结果(定理7．1)，另外的两个定理(定理7．2—7．3)给出了不同平稳点的吸引域估计。这些结果的意义在于两个方面，其一是解决了利用梯度神经网络求解总体极值问题的网络稳定性问题；其二是为网络的设计提供了一个可行框架。第八章就是在此基础上给出了求解这一问题的两种神经网络设计，并给出了一些典型算例验证了网络的可行性与有效性。
The results of this research paper can be outlined mainly as the following aspects:
1 The existence of entire solutions for the nonlinear elliptic equations in the form of
Au + f(x,u,Vu) = 0
is discussed in chapter 1 and chapter 2. First of all, chapter 1 deals with the one-dimentinal case, and twe existing theorems are obtained (theoreml .1-1.2). This result is browden to n-dimentional case in chapter 2 first and an existence theorem (theorem 2.1) for linearly increasing solutions is given. Furthermore, two existing theorems for logarithmic increasing solutions are also obtained in this chapter (theorem 2.2-2.3). All these results are focused in the unbounded solution, which is different from the existing boundary ones, and so our work perfects the concerned results about the existence of solutions to this equation.
2 The existence theorems of exponently increasing solutions and decaying solutions for a class of more general nonlinear elliptic equations in the form of
Au - m2u + f(x, u,Vu) = 0
are studied in chapter 3 and chapter 4. Three theorems (theorem3.1-3.3) are obtained for exponentlv increasing solutions, and two (theorem4.1-4.2) for decaying ones. These results not only extend the existing ones, but also develop the applications of the sup-sub solution methods and fixed theorems in reseach on the solvability to differential equations.
3 Chaper 5 deals with the hyperbolic equation in the form of
2 2
utt+A2u-i-M(x, IA" u112 )Au0,
which comes from the mathematical descripition of the tension of a extensible beam, and the existence and uniqueness of Cauchy problems for this equation are presented here. An existence and uniqueness theorem (theorem 5.1) of local solvability is proven for this equation, compared with the existing results, our conditions restricted to the nonlinear term of this equation are much general. In fact, the results given here are obtained by breaking all the former restrictions on the nonlinear term.
4 The similarity reductions of generalized Burgers equation is treated in chapter 6 by the direct method for finding similarity reductions of patial differential equations, and the corresponding exact special solutions are given also. More over, a group interpretation
is provided to the similarity reductions in terms of nonclassical symmetry. Accomplishing with the generalized Burges equation, this work answers partially an open problem presented by Clarckson: that is how to find the similarity reductions of nonlinear PDEs with arbitrary functions by the direct method.
5 In the last two chapters, chapter7-8, the global minimized problems of the unconstrained optimization are studied by approching the solutions with the solution curves of a differential equation, which is also called the neural network method. Chapter 7 gives the H-stability of the eqilibrium point set first (theorem7.1), and the other two theorems (theorem7.2-7.3) provide an estimation result for the attractive region of different eqilibrium points. All these results own two essential meanings, the first is that this complete fully the stability of the neural network, the second is that provides a feasible frame for the neural network designs. Just based on these results, two models of neural networks for solving the problems are proposed in chapter 8, and moreober, the feasibility and the validity of our designs are shown by some typical computation examples.
引文
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