 Solution

The sign convention makes this a defocusing semilinear wave equation:Its wave energy includes a nonnegative potential term:For a smooth function solution, finite propagation speed preserves compact support on finite time intervals. Differentiate under the integral and use integration by parts:Equivalently the local conservation law has density $e={\varphi }_t^2/2+|\nabla \varphi {|}^2/2+{\varphi }^4/4$ and flux $-{\varphi }_t\nabla \varphi$. 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.