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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.

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up vote 3 down vote accepted

The change $u=e^{\gamma t}v$ transforms the given equation into the diffusion equation $v_t=\Delta v$. Its solution is of the form $$ v=\sum_\lambda e^{-\lambda t}\,\Phi_\lambda $$ where $\lambda$ are the eigenvalues of the Laplacian on $[0,a]^3$, that is, they are the solutions of $$ \Delta\Phi=\lambda\,\Phi,\quad \Phi=0\text{ on the boundary of the cube.} $$ The eigenvalues can be found by separation of variables. Let $\lambda_0$ be the smallest one; it is known that $\lambda_0>0$, that the space of eigenfunctions corresponding to $\lambda_0$ is one-dimensional, and that the corresponding eigenfunction $\Phi_{\lambda_0}$ can be taken strictly positive in the interior of the cube. In order for $u$ to remain bounded, we must have $\gamma\le\lambda_0$. All is left is to find the value of $\lambda_0$ for the cube.

All of the above is valid for any (reasonable) domain $\Omega\subset\mathbb{R}^3$, but the computation of $\lambda_0$ might be more difficult than for a cube.

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Some pointers, but I'll leave the formalization to you:

  • You should prove that there is a unique solution
  • You should find a particular solution (HINT: try the separation of variables), which is then the unique solution
  • Then deduce a condition on $\gamma$. It will most likely depend on the initial conditions (i.e., the distribution of the particules at time $t=0$).
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