Suppose $u : [0,1] \times [0,T] \to \mathbb{R}$, let's consider a simplifed problem \begin{equation} \left\{\hspace{5pt}\begin{aligned} &\dfrac{\partial u}{\partial t} - \partial_x^2 u - \partial_x u - u= 0& \hspace{10pt} &\text{for $(x,t) \in \big(0,1\big) \times (0,T]$} ;\\ &u(x,0) = g(x) & \hspace{10pt} &\text{for $x \in \big(0,1\big)$.}\\ &u(0,t) = p(t) & \hspace{10pt} &\text{for $t \in [0,T]$.}\\ &\partial_x u(1,t) = \partial_x q(1,t) & \hspace{10pt} &\text{for $t \in [0,T]$.} \end{aligned}\right. \end{equation} Here $g,p,q$ are some suitable functions, but we do not know their signs.
I would like to have some maximum principle for this. Since we do not know the sign for the neumann condition at $x=1$, so I take $r(x,t) = p(t) + x \partial_x q(1,t)$. Therefore, the system for $u-r$ has zero boundary values at $x=0$ and $x=1$. Then I can use maximum principle to get the upper and lower bound of the solution. However, the equation has terms involving the time derivative of $r$. The bounds depends on $\partial_t r$. I think I do not have much information of $\partial_t r$. Is there any method to avoid this?
Also, may I have some reference for maximum principle for parabolic equation with mixed boundary condition?