 Wave energy

For a real semilinear wave equation $u_{tt}-\Delta u+V^{\prime }(u)=0$, its energy density and energy flux areMultiply the equation by $u_t$ and use the product rule to obtain the local balance ${\partial }_{t}e+\mathrm{div}j=0$. On a fixed regular domain $\Omega$ this givesThus total wave energy is conserved when the boundary term vanishes: for example homogeneous Dirichlet boundary conditions, homogeneous Neumann boundary conditions, paired periodic boundary conditions, or suitable decay on all of space. A driven boundary can instead inject energy.^[ref]

For the linear wave equation, take $V=0$ and obtain $E_{\mathrm{lin}}=\frac{1}{2}\int (u_t^2+|\nabla u{|}^2)dx$. A defocusing potential such as $V(u)=|u{|}^{p+1}/(p+1)$ adds a nonnegative term. A focusing negative potential still obeys the balance law but need not control positive norms; conservation and positivity are separate properties.