Given the following heat equation and boundary conditions:
$u_t= \alpha(x) u_{xx}, \quad x \geqslant 0 \,\,\text{and}\,\, \exists b \geqslant a > 0, a \leqslant \alpha(x) \leqslant b\\ u_x(0,t) = \begin{cases} 1 , \quad 0 \leqslant t \leqslant t_h \\ 0 ,\quad t_h < t \leqslant t_f \end{cases} \\ u(x,0)=0$
It seems that distinct $\alpha(x)$ could give rise to distinct $u(x,t)$. However, if only two identical surface heat responses $u_1(0,t) = u_2(0,t)$ are observed over the same time interval $[0, t_f]$, is it possible to derive a bound around the difference between $\alpha_1(x)$ and $\alpha_2(x)$(or $\int_{x} |\alpha_1(x) - \alpha_2(x)|dx$) in terms of $a$, $b$, $t_h$ and $t_f$?
Thanks in advance!