Consider the following problems.
Problem 1. Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary, L an elliptic operator defined by $$Lu=-D_j(a^{ij}D_iu)$$ where $a^{ij}=a^{ji}$ is smooth in $\bar\Omega$ and suppose that u is a smooth solution to
\begin{cases} u_t + Lu = 0 \ \text{ in } \Omega \times (0,\infty) \\ u=0 \text{ in } \partial\Omega\times(0,\infty)\\ u=g \text{ in } \Omega\times {0} \end{cases}
Prove that there exist $C,\gamma>0$ such that $|u(x,t)|\leq Ce^{-\gamma t} \text{ for } (x,t) \in \Omega\times(0,\infty)$
Problem 2. Let $\Omega \subset \mathbb{R}^n$ be open and bounded with smooth boundary. Suppose u is a smooth solution of
\begin{cases} u_t + \Delta u + cu = 0 \ \text{ in } \Omega \times (0,\infty) \\ u=0 \text{ in } \partial\Omega\times(0,\infty)\\ u=g \text{ in } \Omega\times {0} \end{cases}
and the function c satisfies $c>\gamma>0$. Prove the exponential decay estimate $|u(x,t)|\leq Ce^{-\gamma t} \text{ for } (x,t) \in \Omega\times(0,T]$
I could solve the problem 2 (Evans - chapter 7 exercise 7) with the maximum principle for parabolic operators. You can do that by looking at the PDE that $u(x,y)-Ce^{-\gamma t}$ satisfies. I tried a similar idea with the problem 1 but did not succeed. I also tried to study the function $e^{\gamma t}u(x,t)$.
If problem 1 is correct, you can consider the particular case $a^{ij} = \delta^{ij} $. It says that the hypothesis of $c\geq\gamma>0$ in problem 2 is not necessary. Whats going on? how one could solve problem 1?