Studies of solvability for hyperbolic Volterra equations

1.在连续函数空间中考虑偏微分方程的初边值问题时，由方程自然定义的算子由于其边界条件的限制，可能具有非稠密区域。这一事实促使我们研究的初始值问题的发展方程由一个家庭的封闭的线性算子，其公共域不一定是密集的基本Banach空间。证明了在有限差分近似的稳定性条件下，演化算子可以由这类算子族生成.讨论了一类半线性发展方程的整体可解性，该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，它是可能的，以获得一个整体的解决方案，通过检查一些能量不等式只涉及低阶导数.我们引入了局部Lipschitz算子半群的概念，它为我们提供了半线性发展方程Cauchy问题的温和解，并刻画了局部Lipschitz算子半群的特征。这个半群的概念是从微分方程的初边值问题的适定性概念导出的，该微分方程的解算子即使在局部意义下也不是拟压缩的，而是局部Lipschitz连续的.利用度量泛函族给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，该算子半群为非线性发展方程Cauchy问题提供了温和解.用度量泛函族代替原范数诱导的度量的优点在于所得结果可应用于某些非线性双曲型方程组。

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.