Wellposedness of the Cauchy problems for hyperbolic systems with large data

1.局部Lipschitz算子半群的特征是用度量泛函族表示的次切条件和半线性稳定性条件。将所得结果应用于复Ginzburg-Landau方程的混合问题.有趣的是计算解的柯西问题的偏微分方程数值和讨论的问题的收敛性出现在这种情况下。这类问题由算子半群理论来处理。给出了Lipschitz算子半群的一个收敛定理和一个逼近定理。将这些定理应用于一类带阻尼的拟线性波动方程的半离散逼近问题和一类Kirchhoff型拟线性方程的有限差分方法.给出了抽象拟线性发展方程在Hadamard意义下的一个逼近定理。并将所得结果应用于一类退化的Kirchhoff方程的逼近问题.在连续函数空间中研究某些偏微分方程的混合问题时，会出现微分算子域在基空间中不稠密的情况.这导致了拟线性发展方程的抽象柯西问题的研究与非稠密定义的运营商。应用所得结果得到了连续函数空间中具有声学边界条件的拟线性波动方程的整体适定性和具有Wentzell边界条件的拟线性波动方程的局部可解性.

1. Semigroups of locally Lipschitz operators are characterized by subtangential conditions and semilinear stability conditions in terms of a family of metric-like functionals. The result is applied to the mixed problem for the complex Ginzburg-Landau equation.2. It is interesting to compute solutions of Cauchy problems for partial differential equations numerically and to discuss the question of convergence which arises in that case. Such problems are treated by the theory of semigroups of operators. A convergence theorem and an approximation theorem for semigroups of Lipschitz operators are given. These theorems are applied to a semi-discrete approximation problem for a quasi-linear wave equation with damping and a finite difference method for a quasi-linear equation of Kirchhoff type.3. An approximation theorem is given for abstract quasi-linear evolution equations in the sense of Hadamard. The result is applied to an approximation problem for a degenerate Kirchhoff equation.4. The situation that the domains of differential operators are not dense in the underlying space arises when the mixed problems for certain partial differential equations in the space of continuous functions are studied. This leads to the study of the abstract Cauchy problems for quasi-linear evolution equations with non-densely defined operators. The result is applied to obtain the global wellposedness for quasi-linear wave equations of Kirchhoff type with acoustic boundary conditions and the local solvability of quasi-linear wave equations with Wentzell boundary conditions in the space of continuous functions.