Principle of Maximum for $u_t - \Delta u \leq 0$.

Question

I would like to ask you a question. When working with the heat equation $u_t - \Delta u = 0$ in $\Omega_T$, with $u \in C^{2,1}(\Omega_T) \cap C^0(\overline{\Omega_T}) $, we have the weak maximum principle that guarantees $$\max_{\overline{\Omega_T}} u = \max_{\partial \Omega_T} u.$$ However, if $u_t - \Delta u \leq 0$, then is there a similar result that answers this problem? I ask this because, for harmonic and subharmonic functions, we have similar results involving maxima and minima. If possible, give me references. Thanks!