Consider the Cauchy problem $$\left\{\hspace{5pt}\begin{aligned} &-\dfrac{\partial u }{\partial t} +a\dfrac{\partial^2 u}{\partial x^2} +b \dfrac{\partial u }{\partial x} +c u = f(u) \leq 0& \hspace {10pt} &\text{for $(x,t) \in \mathbb{R} \times (0,T]$} ;\\ &u(x,T) = g(x)\geq 0 & \hspace{10pt} &\text{for $x \in \mathbb{R}$.} \end{aligned}\right.$$ Here we assume that $u$, $a>0$, $b$, $c<0$, $f \leq 0$ and $g\geq 0$ are smooth enough. Moreover, I have the local bound $\max_{t}\|u\|_{L^2(-R,R)} \lesssim 1$ and $\|\partial_x u\|_{L^2((-R,R) \times [0,T])} \lesssim 1$ for any fixed $R >0$. Also, I know that $u\geq0$.

I am going to prove that $\sup_{[a,b] \times [0,T]} u \lesssim 1$ for any fixed $a<b$. But in the parabolic PDE book by Gary Lieberman (Theorem 6.17 in Chapter VI.6), I only have $\sup_{[a,b] \times [\delta,T-\delta]} u \lesssim 1$. Is there some theorem or method to extend it globally in time?