Suppose that $u$ satisfies the Wave Equation in $R^{1+3}$ with initial conditions $u(0, x) = 0$, $u_t(0, x) = g(x)$ for $x \in R^3$ and $g \in L^p(R^3)$. Show that $u(t, ·) \in L^p(R^3)$ for all $t$ and $||u(t, ·)||_p ≤ || g ||_p|t|$.
I missed the last few lectures in this class and would like to know what the dot in $u(t, ·)$ means.
Any pointers on doing the problem would be great, thanks.
I used Kirchoff's formula to write $u$ as $u(t,x)=\frac{1}{4\pi t^2}\int_{\partial S}tg(y)dS$. I then took the norm of it but it's not really giving me something usable, just that expression within the norm expression. I know I should use Minkowski's inequality but I can't get it in a usable form. Any tips?