  We provide a detailed proof of Proposition 3.1 in the paper titled
``Backstepping control of a class of space-time-varying linear parabolic PDEs
via time invariant kernel functions''.
  In the paper titled ``Backstepping control of a class of space-time-varying
linear parabolic PDEs via time invariant kernel functions'', we addressed the
problem of exponential stabilization and continuous dependence of solutions on
initial data in different norms for a class of $1$-D linear parabolic PDEs with
space-time-varying coefficients under backstepping boundary control. In order
to stabilize the system without involving a Gevrey-like condition or the
event-triggered scheme, a boundary feedback controller was designed via a time
invariant kernel function. By using the approximative Lyapunov method, the
exponential stability of the closed-loop system was established in the spatial
$L^{p}$-norm and $W^{1,p}$-norm, respectively, whenever $p\in [1, +\infty]$. It
was also shown that the solution to the considered system depends continuously
on the spatial $L^{p}$-norm and $W^{1,p}$-norm, respectively, of the initial
data.
