I need help working out the following question, I don't know exactly what to do.
In the cube $(0,a)^3$, a substance is diffusing whose molecules multiply at a rate proportional to the concentration. It therefore satisfies the PDE $u_t=k\Delta u+\gamma u$, where $\gamma$ is a constant. Assume that $u=0$ on all six sides. What is the condition on $\gamma$ so that the concentration does not grow without bound?
Thank you for any and all help, it is greatly appreciated. The more I help I can get with the actual mathematical methods involved the better.