We denote by $\mathcal{D}(\mathbb{R}^3)$ the collection smooth function in $\mathbb{R}^3$ which are compactly supported.
Fix $t>0$ and consider the linear operator $L:\mathcal{D}(\mathbb{R}^3) \to \mathcal{D}(\mathbb{R}^3)$ which maps $f\mapsto \partial_tu(\cdot, t)$ where $u$ is a solution to $$ \begin{cases} u_{tt} - \Delta u = 0\\ u(x, 0) = 0\\ u_t(x,0) = f. \end{cases} $$ For $p>1$, $p\neq 2$ (for $p=2$ the result is well known), can $L$ be extended to a map from $L^p(\mathbb{R}^3)\to L^p(\mathbb{R}^3)$?
I am not quite sure how to approach the problem, but I figured I should look for a sequence of funtions $(f_n)\subseteq \mathcal{D}(\mathbb{R}^3)$ such that $\lVert f_n \rVert_{L^p(\mathbb{R}^3)}$ is uniformly bounded but $$\lVert \partial_t u_n(\cdot, t)\rVert_{L^p(\mathbb{R}^3)}\to\infty.$$ I'm not quite sure how to find such a sequence or if this is even the correct approach. Any input is appreciated!