Let the solution of the $d$-dimensional $(d = 1, 2, 3)$ wave equation Cauchy problem be $u_d(\vec{x}, t)$ with initial conditions $\phi(\vec{x}), \psi(\vec{x})$ non-zero at $|\vec{x}| \leq 1$ and zero at $|\vec{x}| > 1$, i.e. \begin{equation} (P):\begin{cases} u_{tt} - a^2 \Delta x = 0,\\ u(\vec{x}, 0) = \begin{cases} \phi(\vec{x}), & |x| \leq 1,\\ 0, & \text{otherwise} \end{cases},\\ u_t(\vec{x}, 0) = \begin{cases} \psi(\vec{x}), & |x| \leq 1,\\ 0, & \text{otherwise} \end{cases},\\ \end{cases} \end{equation} $u_d(\vec{x}, t), (d = 1, 2, 3)$ is the solution of $(P)$. Prove that there exists a constant $C_d$ such that \begin{equation} |u_d(\vec{x}, t)| \leq C_d(t + 1)^{-\frac{d-1}{2}}, \vec{x} \in \mathbb{R}^d, t \in \mathbb{R}. \end{equation} This is my PDE teacher assigned as a possible final exam topic, is about the solution of the maximum mode estimation, 1-D case proof is very simple, but the 2-D and 3-D case is beyond my ability, 3-D case I can get $|u(\vec{x}, t)| \leq C_3 t ^{-1}$, but the conclusion is still a part of the difference, 2-D case is unable to start, could you give me some hints or answers?
By the way, I found a material that uses Sobolev space in the proof, but we didn't study this, so please use integrals and fundamental inequalities only for the proof if possible, thanks!