论文标题
（1+1） - 维波方程的良好性具有准线性边界条件
Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition
论文作者
论文摘要
我们考虑线性波方程$ v（x）u_ {tt}（x，x，t） - u_ {xx}（x，x，x，t）= 0 $ on $ [0，\ infty）\ times [0，\ infty）$具有初始条件和非线性noumant neumann neumann neumann neumann noumann noumann noumann noumann边界条件$ _x（0，t）（0，t）=（0，f（f（f（f（f（u_t）），这个问题是在电动力学中的非线性麦克斯韦问题的准确降低。在$ f \ colon \ mathbb {r} \ to \ mathbb {r} $的情况下，我们通过特征和固定点方法来研究初始价值问题的全球存在，独特性和最初价值问题的良好性。我们还证明了能量和动力的保护，并讨论了为什么在$ f $是同态同态下降的情况下，没有良好的情况。最后，我们表明，波浪方程的先前已知的时间周期性的，空间局部的解决方案（呼吸），非线性Neumann边界条件为$ x = 0 $具有足够的规律性，可以用自己的初始数据解决初始值问题。
We consider the linear wave equation $V(x) u_{tt}(x, t) - u_{xx}(x, t) = 0$ on $[0, \infty)\times[0, \infty)$ with initial conditions and a nonlinear Neumann boundary condition $u_x(0, t) = (f(u_t(0,t)))_t$ at $x=0$. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where $f\colon\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where $f$ is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at $x=0$ have enough regularity to solve the initial value problem with their own initial data.
