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Studies of solvability for hyperbolic Volterra equations

双曲 Volterra 方程的可解性研究

基本信息

批准号：
17540143
负责人：
OKA Hirokazu
金额：
$ 2.39万
依托单位：
Ibaraki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540143/
关键词：
finite difference approximations stability condition semilinear evolution equations semierouos of locally Lipschitz operators Cauchy problems mild solutions dissiiativity condition nonlinear hyperbolic systems 準線形双曲型方程式半閉作用素 Hyers-Ula

项目摘要

1. When we consider the initial-boundary problem for a partial differential equation in the space of continuous functions, the operator defined naturally from the equation has possibly non dense domain because of its boundary condition. This fact motivates us to study the initial-value problem for the equation of evolution governed by a family of closed linear operators whose common domain is not necessarily dense in the underlying Banach space. It is shown that an evolution operator is generated by such a family of operators, under the stability condition introduced from the viewpoint of finite difference approximations.2. We discuss the global solvability fir a class of semilinear evolution equations which is the abstract version of the quasilinear wave equation with strong damping. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives.3. We introduce the notion of semigroups of locally Lipschitz operators which provide us with mild solutions to the Cauchy problem for semilinear evolution equations, and characterize such semigroups of locally Lipschitz operators. This notion of the semigroups is derived from the well-posedness concept of the initial-boundary value problem for differential equations whose solution operators are not quasi-contractive even in a local sense but locally Lipschitz continuous with their initial data.4. A new dissipativity condition is proposed in terms of a family of metric-like functionals, and a necessary and sufficient condition is given of the existence of semigroups of locally Lipschitz operators which provide us with mild solution of Cauchy problem for nonlinear evolution equations. The advantage of using a family of metric-like functionals instead of the metric induced by the original norm lies in the fact that the obtained result may be applied to some nonlinear hyperbolic systems.

1.在连续函数空间中考虑偏微分方程的初边值问题时，由于其边界条件的限制，由方程自然定义的算子可能具有非稠密域。这一事实促使我们研究由闭线性算子族控制的发展方程的初值问题，该闭线性算子族的公共区域不一定在基本的Banach空间中稠密。在从有限差分近似的观点引入的稳定性条件下，证明了发展算子是由这样一族算子生成的。讨论了一类半线性发展方程的整体可解性，半线性发展方程是强阻尼拟线性波动方程的抽象形式。我们的公式的优点在于，可以通过检验一些只涉及低阶导数的能量不等式来获得全局解。我们引入了局部Lipschitz算子半群的概念，给出了半线性发展方程柯西问题的温和解，并刻划了这种局部Lipschitz算子半群。这个半群的概念源于微分方程组初边值问题的适定性概念，其解算子不是局部意义上的拟压缩的，而是与其初始数据局部Lipschitz连续的。利用一族度规泛函提出了一个新的耗散性条件，并给出了局部Lipschitz算子半群存在的一个充要条件，从而为我们提供了非线性发展方程柯西问题的温和解。用一族类度规泛函代替由原范数诱导的度规的优点在于所得到的结果可以适用于某些非线性双曲组。

项目成果

期刊论文数量（0）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

An analysis on the internal structure of the celebrated Furuta inequality via operator mean

用算子均值分析著名的Furuta不等式的内部结构

DOI：
发表时间：
2005
期刊：
Sci.Math.Jpn 62・3
影响因子：
0
作者：
H.Heyer;T.Jimbo;S.Kawakami;K.Kawasaki;M.Fujii
通讯作者：
M.Fujii

局所リプシッツ作用素半群の生成定理

局部 Lipschitz 算子半群的生成定理

DOI：
发表时间：
2006
期刊：
影响因子：
0
作者：
Hirokazu;Oka;Hirokazu Oka;Yoshikazu Kobayashi;Toshitaka Matsumoto;Naoki Tanaka;Torben Maack Bisgaard;Toshitaka Matsumoto;田中直樹;松本明美;Naoki Tanaka;Akemi Matsumoto;榊原暢久;Nobuhisa Sakakibara;榊原暢久;Nobuhisa Sakakibara;田中直樹
通讯作者：
田中直樹

Approximation of abstract quasilinear evolution equations in the sense of Hadamard

Hadamard意义上的抽象拟线性演化方程的逼近

DOI：
发表时间：
期刊：
Taiwanese J. Math. in press
影响因子：
0
作者：
Hirokazu;Oka;Hirokazu Oka;Yoshikazu Kobayashi;Toshitaka Matsumoto;Naoki Tanaka
通讯作者：
Naoki Tanaka

On the continuity of positive definite functions on conelike semigroups

锥状半群上正定函数的连续性

DOI：
发表时间：
2005
期刊：
影响因子：
0
作者：
Hirokazu;Oka;Hirokazu Oka;Yoshikazu Kobayashi;Toshitaka Matsumoto;Naoki Tanaka;Torben Maack Bisgaard;Toshitaka Matsumoto;田中直樹;松本明美;Naoki Tanaka;Akemi Matsumoto;榊原暢久;Nobuhisa Sakakibara;榊原暢久;Nobuhisa Sakakibara;田中直樹;Naoki Tanaka;Nobuhisa Sakakibara
通讯作者：
Nobuhisa Sakakibara

Abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators

摘要具有非稠密定义算子的拟线性演化方程的柯西问题