Wellposedness of the Cauchy problems for hyperbolic systems with large data

1. 局部Lipschitz算子的半群用类度量泛函族的次切条件和半线性稳定条件来表示。将所得结果应用于复Ginzburg-Landau方程的混合问题。用数值方法计算偏微分方程的柯西问题的解并讨论在这种情况下出现的收敛问题是很有趣的。用半群算子理论来处理这类问题。给出了Lipschitz算子半群的收敛定理和近似定理。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法。给出了Hadamard意义下抽象拟线性演化方程的一个近似定理。将所得结果应用于简并Kirchhoff方程的近似问题。在研究连续函数空间中某些偏微分方程的混合问题时，会出现微分算子的域在底层空间中不密集的情况。这导致了具有非密集定义算子的拟线性演化方程的抽象柯西问题的研究。将所得结果应用于具有声学边界条件的Kirchhoff型拟线性波动方程的全局适定性和具有Wentzell边界条件的拟线性波动方程在连续函数空间中的局部可解性。

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.