# Local boundedness for Cauchy problem

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

• I finally find a result in Theorem 11.17 in Chapter XI.6 in parabolic PDE book by Gary Lieberman. But it restricts on the case of one dimension space. Mar 1, 2022 at 10:22