= Solution
The sign convention makes this a <defocusing semilinear wave equation>:
$$
\phi_{tt}-\Delta\phi+\phi^3=0.
$$
Its <wave energy> includes a nonnegative potential term:
$$
\boxed{E(t)=\int_{\mathbb R^3}\left(\frac12\phi_t^2+\frac12|\nabla\phi|^2+\frac14\phi^4\right)\,dx=E(0).}
$$
For a <smooth function> solution, <finite propagation speed> preserves <compact support> on finite time intervals. Differentiate under the integral and use <integration by parts>:
$$
\frac{dE}{dt}=\int_{\mathbb R^3}\phi_t\bigl(\phi_{tt}-\Delta\phi+\phi^3\bigr)\,dx=0.
$$
Equivalently the local <conservation law> has density $e=\phi_t^2/2+|\nabla\phi|^2/2+\phi^4/4$ and flux $-\phi_t\nabla\phi$. All three terms are nonnegative, and $E=0$ forces both <Cauchy data> to vanish. Hence this is a positive conserved <energy>, rather than the indefinite energy associated with the opposite sign.
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