Inverse coefficient problem for one-dimensional evolution equation vanishing initial condition

TL;DR

This paper proves the uniqueness of determining a spatially varying coefficient in a one-dimensional evolution PDE from boundary and initial data, under conditions on the initial zero set.

Contribution

It establishes the first uniqueness result for an inverse coefficient problem with vanishing initial conditions in a one-dimensional evolution equation.

Findings

Uniqueness of the coefficient p(x) under given data

Conditions on initial zero set are sufficient for uniqueness

The result applies to complex-valued coefficients and data

Abstract

We consider an inverse problem of determining a coefficient $p(x)$ of an evolution equation $\sigma\ppp_tu = a(x)\ppp_x^2u - p(x)u$ for $0<x<\ell$ and $0<t<T$, where $\sigma \in \C \setminus \{0\}$, $\ell>0$ and $T>0$ are arbitrarily given. Our main result is the uniqueness: by assuming that the zeros of initial value $b(x):= u(0,x)$ on $[0, \ell]$ is a finite set and each zero is of order one at most, if two solutions have the same Cauchy data at $x=0$ over $(0,T)$ and the same initial value $b(x)$, then the coefficient $p(x)$ is uniquely determined on $[0,\ell]$.