Consider the Cauchy problem $$\left\{\hspace{5pt}\begin{aligned} &-\dfrac{\partial u }{\partial t} +a\dfrac{\partial^2 u}{\partial x^2} +b \dfrac{\partial u }{\partial x} +c u = f(u) \leq 0& \hspace {10pt} &\text{for $(x,t) \in \mathbb{R} \times (0,T]$} ;\\ &u(x,T) = e^x\geq 0 & \hspace{10pt} &\text{for $x \in \mathbb{R}$.} \end{aligned}\right.$$ Here we assume that $u$, $a>0$, $b$, $c<0$, $f \leq 0$ and $g\geq 0$ are smooth enough. Moreover, I know that the solution $u$ grows slower than the fundamental solution.
I am going to ask if the gradient $\partial_x u$ also grows slower than the fundamental solution. Due to the nonlinear $f(u)$, we can not differentiate the Green representation of $u$ to estimate the gradient bound.