We study backward uniqueness properties for equations of the form u' + Au = f. Under mild regularity assumptions on A and f, it is shown that u(0) = 0 implies u(t) = 0 for t < 0. The argument is based on α-log and log-log convexity. The results apply to mildly nonlinear parabolic equations and systems with rough coefficients and the 2D Navier-Stokes system.