Inverse Initial Boundary Value Problem for a Non-linear Hyperbolic Partial Differential Equation

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)).