How do I apply this maximum principle? I have the maximum principle: $$\text{If } \psi\geq 0 \text{ on }\Gamma. \text{Then }L\psi\geq0\text{ implies }  \psi\geq 0 \text{ in } \bar{D},$$ where $D=(0,1)\times (0,T], \Gamma$ is parabolic boundary. The operator is defined as $$Lu:=-u_{xx}+bu+du_t=f,$$ where $b,d,f$ are functions and $b\geq\beta>0,d\geq\delta>0$. I would like to show that $$||u||\leq(1+\alpha T)\max\{||Lu||,||u||_\Gamma\},$$ where $\alpha=\max\{0,(1-b)/d\}\leq1/\delta$. It is supposed to be an immediate consequence of the Maximum principle but I am stuck.
I have tried to apply the Maximum principle on $$\phi^{\pm}=\max\{||Lu||,||u||_\Gamma\}\pm u$$ but I don't seem to go anywhere.
 A: The boundary $\Gamma$ is
$$
\Gamma=\{(t,x)\in \{0\}\times (0,1)\}\cup \{(t,x)\in (0,T)\times \{0\}\}\cup
\{(t,x)\in (0,T)\times \{1\}\}.
$$
You consider the problem 
$$
-u_{xx}+bu+du_t=f \mbox{ in }(0,1)\times(0,T), \quad u=\phi \mbox{ on } \Gamma.
$$
Consider $v=u+M$. It satisfies
$$
-v_{xx}+bv+dv_t=f+bM \mbox{ in }(0,1)\times(0,T), \quad u=\phi+M \mbox{ on } \Gamma.
$$
Choose $M$ so that $f+bM\geq0$ and $\phi+M\geq0$ and you obtain from the maximum principle you stated that $v\geq0$, in other words 
$$
u\geq-M=-\max(\inf(-\phi),\inf(-f/b))\geq -\max(\|\phi\|_\infty,\|\frac{f}{b}\|_\infty).
$$
Symmetrically, consider now the problem satisfied by $-u$, (which means just replacing $f$ and $\phi$ by $-f$ and $-\phi$). By exactly the same argument,
$$
-u\geq -N = - \max(\inf(\phi),\inf(f/b)) \geq -\max(\|\phi\|_\infty,\|\frac{f}{b}\|_\infty)..
$$ 
so altogether
$$
\|u\|_\infty\leq \max(\|\phi\|_\infty,\|\frac{f}{b}\|_\infty)\leq \max\left(1,\frac{1}{\beta}\right)\max(\|\phi\|_\infty,\|f\|_\infty).
$$
This is not the constant you wanted but your constant is puzzling me. You seem to say
$$
b\leq\beta,d\geq\delta\implies\max(0,\frac{b-1}{d})\leq\delta^{-1},
$$
this is not true in general, take for example $\delta<1$, and $b=\beta=1+\frac{1}{\delta^N}$, you obtain $$\max(0,\frac{b-1}{d})=\delta^{-N-1}>\delta^{-1}.$$
If you want to put some $\delta$ in the formula, you can, before you start, consider the problem satisfied by $\tilde{u}=u\exp(-\lambda t)$. Then, $f$ is changed to $\exp(-\lambda t)f$, $\phi$ is changed to $\exp(-\lambda t)\phi$, and $b$ is changed to $b+\lambda d$. The same argument then applies to $\tilde u$
Then you have instead, as long as $\beta+\lambda \delta >0$,
$$
\|u\|_\infty\leq\exp(\lambda T)\|\tilde{u}\|_\infty\\ \leq \max\left(1,\frac{1}{\beta+\lambda \delta}\right)\exp(\lambda T)\max(\|\exp(-\lambda t)\phi\|_\infty,\|\exp(-\lambda t)f\|_\infty),
$$
so if $f$ and $\phi$ do not depend on time, you obtain, for $\lambda>0$,
$$
\|u\|_\infty\leq\max\left(1,\frac{1}{\beta+\lambda \delta}\right)\exp(\lambda T)\max(\|\phi\|_\infty,\|f\|_\infty),
$$.
(and a worse bound than the direct one if $-\beta/\delta<\lambda<0$).
