Maxwell-Klein-Gordon energy estimates in Klainerman and Machedon's 1994 paper

Question

In the 1994 paper On the Maxwell-Klein-Gordon Equation with Finite Energy of Klainerman and Machedon, the proof of Proposition 1.1 contains the following statement. For $\phi$ (the scalar field of) a classical solution of the Maxwell-Klein-Gordon system, if $$ \| \phi \|_{L^3(\mathbb{R}^3)} \leqslant C(1+t) \left(1 +\| \phi \|_{L^3(\mathbb{R}^3)}\right)^{1/2}, $$ then $$ \| \phi \|_{L^3(\mathbb{R}^3)} \leqslant C(1+t). $$ I don't understand how they deduce this. For a start, it certainly doesn't follow if one ignores the fact that $\phi$ solves the Klein-Gordon equation: take, for example $\phi(t,x) = t^2 \psi(x)$, where $\psi(x)$ is any $L^3(\mathbb{R}^3)$ function with unit norm. The $(1+t)$ looks like it should come from some sort of linear wave equation estimate, but there's no indication they are using anything like that in that proof! What's going on?

Answer 1

There is a typo in the paper. Look at the bottom two lines of page 22 which I transcribe here

\begin{align} \|\phi\|_{L^3} & \leq \ldots \\ & \lesssim \mathscr{I}_0^{1/2} (1+t)^{1/2} ( \mathscr{I}_0 + \|A\|_{L^6} \|\phi\|_{L^3} )^{1/2}\\ & \lesssim \mathscr{I}_0 \color{red}{(1+t)} (1 + \|\phi\|_{L^3})^{1/2} \end{align}

If we use Sobolev to control $\|A\|_{L^6} \lesssim \|\nabla A\|_{L^2} \lesssim \mathscr{I}_0$, to get from the second to last line to the last line, the final line should in fact have the term that I marked in red replaced by $\color{green}{(1+t)^{1/2}}$. With this (1.5.c) follows.