Let \(X\) be a Banach space, \(A:X\supset \text{dom}(A)\to X\) be the generator of an analytic semigroup, \(Y\) be an energy space associated with \(A\) and \(F:Y\to X\) be locally Lipschitz. The authors establish sufficient conditions for the boundedness of global solutions of \[ \dot u+Au=Fu \] in a setting which is motivated by the parabolic initial-boundary value problem \[ \partial_t u=\Delta u+ | u|^{p-1}u\quad \text{in }\Omega\times (0,\infty), \qquad u|_{\partial\Omega}\equiv 0,\quad u(\cdot,0)=u_0\geq 0\tag{1} \] (solutions with finite time blow-up may occur for certain initial conditions) and allows to carry over the reasoning employed in earlier work on (1) and related problems. Applications are given to higher order semilinear parabolic equations and to parabolic equations with a nonlocal term.