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 $\epsilon\to 0$.


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