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''

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 $p\in [1,+\mathrm{\infty }]$. 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.