论文标题
非线性双曲偏微分方程的逆初始边界值问题
Inverse Initial Boundary Value Problem for a Non-linear Hyperbolic Partial Differential Equation
论文作者
论文摘要
在本文中，我们关注的是空间维度$ n \ geq 2 $中非线性波方程的反向初始边界值问题。特别是我们考虑所谓的内部确定问题。这个非线性波方程具有微不足道的解，即零解决方案。通过在微不足道的解决方案处将该方程线性化，我们具有具有独立电位的通常线性波方程。对于任何小解决方案$ u = u（t，x）$的这个非线性方程式，它是线性波方程的扰动，与时间无关的潜在潜在的扰动相对于$（t，x）$，其成分的$（t，x）$是四边形，而不是$ \ nabla__ {通过少量$ o（| \ nabla_ {t，x} u（t，x）|^3）$忽略术语，我们将通过许多边界测量在空间域的边界上的许多边界测量在有限时间间隔和最终超出$ t $ t $ t $ t $ t $ t = t $ t $ t = t $ t = t $ t = t $ timeplate的边界处，通过许多边界测量在空间域的边界上进行许多边界测量的潜力和系数。换句话说，测量由所谓的输入输出映射给出（请参见（1.5））。
In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension $n\geq 2$. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution $u=u(t,x)$ of this non-linear equation, it is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to $(t,x)$ of a vector whose components are quadratics with respect to $\nabla_{t,x} u(t,x)$. By ignoring the terms with smallness $O(|\nabla_{t,x} u(t,x)|^3)$, we will show that we can uniquely determine the potential and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval and the final overdetermination at $t=T$. In other words, the measurement is given by the so-called the input-output map (see (1.5)).
