This question came up while studying for a qualification exam:
Let $B_1(0)$ denote the unit ball in $\mathbb{R}^n$ ,centered at the origin, and let $u$ be a smooth solution of
\begin{cases} u_{tt} + a^2(x)u_t - \Delta u = 0 \text{ in } B_1(0) \times(0,\infty) \\ u(x,t) = 0 \text{ on } \partial B_1(0) \times(0,\infty) \\ u(x,0) = g(x), u_t(x,0) = h(x) \text{ for } x \in B_1(0) \end{cases} Here $h,g$ are smooth functions such that $h$ and $g$ vanish on $\partial B_1(0)$. Prove that $\int_{B_1(0)} u^2(x,t) \text{d}x \leq C\exp(-At)$ where $A = min\{a^2(x)\}$ and $C$ only depends on $h,g,n$.
$\textbf{My Attempt}$ It seems that we are to apply Poincare's Lemma since we have trace zero functions and $C$ can depend on $n$. I tried the standard energy method:
\begin{equation} E(t) := \frac{1}{2} \int_{B_1(0)} u_t^2 + |\nabla u|^2 \end{equation} and got \begin{equation} \partial_t E(t) = \int_{B_1(0)} -a^2(x) u_t^2 \leq \int_{B_1(0)} -A^2 (u_t^2 + |\nabla u|^2) + A^2 |\nabla u|^2 \end{equation} I want to apply Gronwall's Inequalityy to get exponential decay on $E(t)$ which implies exponential decay on $\int_{B_1(0)} |\nabla u|^2$ then use Poincare's Lemma to get bound on $\int_{B_1(0)} u^2$ by a constant multiple of $\int_{B_1(0)} |\nabla u|^2$. However, we do not exactly have the energy on the right hand side inequality and the extra $A^2 |\nabla u|^2$ term ruins the Gronwall Inequality by adding a term of the form $A^2\int_0^t \int_{B_1(0)} |\nabla u|^2$ onto the exponential decay. I am unsure on how to either modify the energy to get the argument to work or to bound $A^2\int_0^t \int_{B_1(0)} |\nabla u|^2$ in terms of $g$ and $h$. Any help would be greatly appreciated. We can actually obtain the desired damping off by a linear factor. Indeed,
\begin{equation} \partial_t E(t) \leq 0 \Rightarrow \frac{1}{2} \int_{B _1(0)} |\nabla u|^2 + u_t^2 \leq \frac{1}{2} \int_{B_1(0)} u_t(x,0)^2 + |\nabla u(x,0)|^2 := C \end{equation} so
\begin{equation} \int_0^t \int_{B_1(0)} |\nabla u(x,t)|^2 \leq Ct \end{equation} But Gronwall implies
\begin{equation} E(t) \leq \exp(-A^2t) [E(0) + \int_0^t \int_{B_1(0)} |\nabla u(x,t)|^2] \leq \exp(-A^2t) [E(0) + Ct] \end{equation} i.e. \begin{equation} \int_{B_1(0)} |\nabla^2 u| + u_t^2 \leq \exp(-A^2t) [E(0) + Ct] \end{equation} so \begin{equation} \int_{B_1(0)} u^2 \leq \alpha \int_{B_1(0)} |\nabla^2u| \leq \alpha(exp(-A^2t) [E(0) + Ct]) \end{equation} where $\alpha$ is the Poincare constant. This is very close to their formulation but off by a linear factor. Any help in resolving this would be greatly appreciated!