# Maximum principle for heat equation with Neumann boundary conditions

Consider the initial-boundary value problem $$\frac{\partial u}{\partial t} = a\Delta u \;\mbox{ in } \;\Omega$$ $$a\frac{\partial u}{\partial n} = g \;\mbox{ on }\;\Gamma = \partial \Omega$$ $$u|_{t=0} = u_0$$ Is it possible to establish the maximum principle for this problem of the type $u(x,t) \leq M$, $M = \max\{u_0, g\}$ or something similar to this?

The function $g$ represents the rate of heat flow through the boundary; in physics terms, its units are different from the units of $u$. Thus, $M = \max\{u_0, g\}$ is never going to be useful.
If the net flow into the domain is positive, the temperature inside will grow without bound as $t\to\infty$ at a linear rate. Indeed, $$\frac{d}{dt}\int_\Omega u=\int_\Omega u_t= \int_\Omega a\Delta u = \int_{\partial \Omega} a\frac{\partial u}{\partial n} = \int_{\partial \Omega}g$$
So, if $\int_{\partial \Omega}g>0$, the solution is unbounded above.
If $\int_{\partial \Omega}g\le 0$, the solution is bounded above. But the bound cannot be expressed in terms of $u_0$ and $g$ alone: it depends in an intricate way on the geometry of the domain. Imagine $\Omega$ being the union of two disks joined by a narrow passage (width $\epsilon$). If $g=1$ on the boundary of one disk and $g=-1$ on the boundary of the other, the first disk will have very high temperature in the long run, with the maximum $M=M(\epsilon)$ such that $M(\epsilon)\to\infty$ as $\epsilon\to 0$.