= Solution
We use the homogeneous version of the <Sobolev embedding theorem> in three dimensions:
$$
\|h\|_{L^6(\mathbb R^3)}\leq C_S\|\nabla h\|_{L^2(\mathbb R^3)},\qquad h\in\dot H^1(\mathbb R^3).
$$
Here <homogeneous Sobolev space> $\dot H^1$ is the completion of compactly supported <smooth functions> in the <L2 norm> of the <gradient>, identified with its $L^6$ representative. Apply this <Sobolev inequality> both to $\phi$ and to its <spatial derivatives>.
Let
$$
B(t)=\left(\sum_{i=1}^3\|\partial_i\phi_t(t)\|_2^2+
\sum_{i,j=1}^3\|\partial_i\partial_j\phi(t)\|_2^2\right)^{1/2}.
$$
The <Plancherel theorem> identifies the <L2 norm> of the <Hessian matrix> with $\|\phi\|_{\dot H^2}$, because $\sum_{i,j}\xi_i^2\xi_j^2=|\xi|^4$. In particular,
$$
\|\phi\|_{\dot H^2}+\|\phi_t\|_{\dot H^1}\leq\sqrt2 B(t),\qquad B(0)\leq D.
$$
Differentiating the <defocusing semilinear wave equation> gives $\Box(\partial_i\phi)=3\phi^2\partial_i\phi$. By the <Holder inequality> with exponents $3$ and $6$, the preceding <Sobolev inequality>, and conservation of the positive <wave energy>,
$$
\begin{aligned}
\|\phi^2\nabla\phi\|_2
&\leq\|\phi\|_6^2\|\nabla\phi\|_6\\
&\leq C\|\nabla\phi\|_2^2\|D_x^2\phi\|_2
\leq 2CE B(t).
\end{aligned}
$$
Use the inhomogeneous <wave energy estimate> simultaneously for the three <spatial derivatives>. It gives
$$
B(t)\leq B(0)+C E\int_0^t B(s)\,ds.
$$
The <Gronwall inequality> therefore yields $B(t)\leq D\exp(C E t)$. We may take the continuous, locally bounded function
$$
\boxed{f(T)=\sqrt2 D\exp(C E T).}
$$
The constant is universal; the dependence on the initial data is only through $D$ and $E$. The <H2 bound for the defocusing cubic wave equation> holds on every existing smooth interval, without assuming the global conclusion. If $D=0$ or $E=0$, the zero solution satisfies the same estimate.
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