Exponential decay estimate 
Assume $u$ is a smooth solution of 
  $$\begin{cases} u_t - \Delta u = 0 & \text{in }U \times (0,\infty) \\ \qquad \quad u=0 & \text{on }\partial U \times [0,\infty) \\ \qquad \quad u = g  &\text{on }U \times \{t=0\}.\end{cases}$$ Prove the expoentital decay estimate: $$\|u(\cdot,t)\|_{L^2(U)} \le e^{-\lambda_1 t} \|g\|_{L^2(U)} \quad (t \ge 0),$$ where $\lambda_1 > 0$ is the principal eigenvalue of $-\Delta$ (with zero boundary conditions) on $U$.

This is from PDE Evans, 2nd edition: Chapter 7, Exercise 2.
My first intuition is that $u \in H_0^1(U)$, since all the derivatives of $u$ belong in $L^2(U)$. I also invoked Poincare's inequality to obtain $$\int_U |u|^2 \, dx \le C\int_U |Du|^2 \, dx.$$ Is this a good first step? If so, how can I continue from here? Furthremore, what can I do with the fact that $\lambda_1 > 0$ is the "principal eigenvalue of $-\Delta$"? 
 A: Here is a solution more related to Evan's approach in the mentioned textbook. Let $u = u(x,t)$ be a solution of the PDE. Then using integration by parts and that $u_t = \Delta u$ we get
$$\begin{align}\frac{d}{dt} \left(\frac{1}{2} \|u\|_{L^2(U)}^2\right) &= \int_U u_tu\  dx = \int_U u \Delta u\  dx\\&= -\int_U |Du|^2 dx \overset{(\ast)}\leq -
 \lambda_1 \|u\|^2_{L^2(U)}\end{align}$$
where $(\ast)$ comes from Rayleigh's Formula
$$\lambda_1 = \underset{\substack {u \in H_0^1 (U)\\ u\neq 0}}\min \frac{B[u,u]}{\|u\|^2_{L^2(U)}} = \underset{\substack{u \in H_0^1 (U)\\ u\neq 0}} \min \frac{\int_U |Du|^2 dx}{\|u\|^2_{L^2(U)}} $$
Now let $\eta (s) = \|u(\cdot,s)\|^2_{L^2(U)}$. Then
$$\frac{d}{ds} \left(\eta(s) e^{2\lambda_1 s}\right) = e^{2\lambda_1 s} (\eta'(s) +2\lambda_1 \eta(s)) \leq 0$$
Integrating from $0$ to $t$ w.r.t. $s$ we obtain
$$\eta(t)e^{2\lambda_1 t} \leq \eta (0)$$
Since $\eta (0) = \|u (\cdot, 0)\|^2_{L^2(U)} = \|g\|^2_{L^2(U)}$ the result follows.
A: Write 
$$
u(x,t)=\sum_k c_k e^{- \lambda_k t} \varphi_k(x),
$$
where $\lambda_k\leq \lambda_{k+1}$ are the eigenvalues of the laplacian with zero Dirichlet boundary conditions, $\varphi_k$ are the corresponding eigenfunctions and 
$$
c_k=\int_U g(x) \varphi_k(x) dx
$$
are the Fourier coefficients of $g$ with respect to the orthonormal basis $(\varphi_k)_k$. Then we see, by Pythagoras' and Parseval's theorems,
$$
\| u(\cdot, t)\|_{L^2(U)}^2 = \sum_k c_k^2 e^{-2\lambda_k t} \leq e^{-2 \lambda_1 t}\sum_k c_k^2 = e^{-2\lambda_1 t} \| g\|_{L^2(U)}^2,
$$
where we also used the monotonicity of the eigenvalues in the second to last step. This is what you want.
