An inverse problem for the heat equation

Abstract

Let $u_t=u_{xx}-q(x)u,0\le x\le 1$, $t>0$, $u(0,t)=0,u(1,t)=a(t),u(x,0)=0$, where $a(t)$ is a given function vanishing for $t>T$, $a(t)\not\equiv 0$, ${\int }_0^{T}a(t)dt<\infty$. Suppose one measures the flux $u_x(0,t):=b_0(t)$ for all $t>0$. Does this information determine $q(x)$ uniquely? Do the measurements of the flux $u_x(1,t):=b(t)$ give more information about $q(x)$ than $b_0(t)$ does? The above questions are answered in this paper.