Time integrability of an entropy-dissipation product (${\int }_0^{\infty }|H{H}^{\prime }|=\frac{H(0{)}^2-H_{\infty }^2}{2}$)

If finite nonnegative entropy $H(t)$ decreases, its limit $H_{\infty }$ exists. The fundamental theorem of calculus applied to $H^2$ proves the displayed absolute-integrability identity. This product conclusion does not require a zero limiting entropy.