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. 当我们考虑连续函数空间中的偏微分方程的初始边界问题时，由于其边界条件，从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一系列闭合线性算子控制的演化方程的初值问题，这些算子的公共域在基础 Banach 空间中不一定是稠密的。证明了在从有限差分近似的角度引入的稳定性条件下，由该算子族生成演化算子。 2．我们讨论了一类半线性演化方程的全局可解性，半线性演化方程是强阻尼拟线性波动方程的抽象版本。我们的公式的优点在于，可以通过检查一些仅涉及低阶导数的能量不等式来获得全局解。3．我们引入了局部 Lipschitz 算子半群的概念，它为半线性演化方程的柯西问题提供了温和的解，并刻画了这种局部 Lipschitz 算子半群的特征。半群的概念源自微分方程的初始边值问题的适定性概念，其解算子即使在局部意义上也不是拟收缩的，而是与其初始数据局部Lipschitz连续的。 4．根据类度量泛函族提出了新的耗散条件，并给出了局部Lipschitz算子半群存在的充要条件，为非线性演化方程的柯西问题提供了温和的解。使用一系列类度量泛函代替由原始范数导出的度量的优点在于，所获得的结果可以应用于某些非线性双曲系统。