The question is
Suppose $U=\Omega \times (0,T)$ where $\Omega \subseteq \Bbb{R}^n$ is a bounded domain. Let $u\in C_1^2(U)\bigcap C(\bar U)$ satisfy $u_t \le\Delta u + cu$ in $U$ where $c \le 0$ is a constant. If $u \ge 0$, show that $u$ contains maximum on the parabolic boundary of $U$. Give a counter example if the condition $u \ge 0$ is not satisfied.
I once solved this question but today when I looked at my solution again, I got a problem. The following is a scan of my solution. Zoom it in you can see it clearly.
The inequality underlined blue "$\frac{\partial }{{\partial t}}u - \Delta u - 2n\varepsilon < c(u + \varepsilon |x{|^2})$" is my problem, why this inequity holds?
It is only given that $\frac{\partial }{{\partial t}}u - \Delta u \le cu$. In order for the inequality to hold, $ - 2n\varepsilon < c\varepsilon |x{|^2}$ must hold, thus $ - 2n < c|x{|^2}$. But we don't have any information about $x$ except for it is bounded.
Thank you!
You can adapt your proof by using
$$V_{\epsilon}(x,t) = u(x,t) + \epsilon \left( |x|^2 -\sup_{x\in\Omega} |x|^2 \right) $$
Then you have
$$\Delta \left( |x|^2 -\sup_{x\in\Omega} |x|^2 \right) = -2n < 0 \leq c\left( |x|^2 -\sup_{x\in\Omega} |x|^2 \right) $$