Exponential decay estimate for $(x,t) \in U_T$
I have question from the above mentioned problem:
I take $v=e^{\gamma t}u(x,t)$, then
$v_t= u_t e^{\gamma t}+ \gamma u e^{\gamma t}$
$\Delta v= e^{\gamma t} \Delta u+ 2 \gamma u_t+\gamma^2e^{\gamma t} u$
So, $v_t -\Delta v-(\gamma -c)v=-\gamma^2u e^{\gamma t}- \gamma u_te^{\gamma t}$.
I stuck here, how do I claim $v_t -\Delta v-(\gamma -c)v \geq 0$. I want to show this to apply maximum principle.
Can anyone suggest some hint to handle the last step?
The Laplacian is only in terms of the spatial variables. In particular $\Delta v= e^{\gamma t}\Delta u$. This, together with your formula for $v_t$, gives $$ v_t-\Delta v= -cue^{\gamma t} + \gamma e^{\gamma t} u= (\gamma-c)v. $$ Therefore $v_t-\Delta v + (c-\gamma)v =0$. Since $c-\gamma\geq 0$, we can apply the weak maximum and minimum principles.