Interpreting a PDE for Heat/Diffusion with Vanishing Neumann Boundary Conditions

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The discussion centers on interpreting the partial differential equation (PDE) \(\triangle u = f(x,y,z)\) under vanishing Neumann boundary conditions, specifically in the context of heat and diffusion. The user correctly identifies that \(u\) represents the concentration of heat or substance, and the PDE indicates that there is no escape of heat or substance from the body, implying a steady-state condition. The absence of a time term suggests that the distribution of heat or substance is time-invariant, governed by the Laplacian operator equating to the function \(f\).

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Given \triangle u = f(x,y,z) on a given body with vanishing neumann boundary conditions. I'm asked to interpret it in terms of heat and diffusion.

Since heat/diffusion take the form u_t = k \triangle u, I am a little confused as I there is no time term here. I think the answer is that u denotes the concentration of heat/substance and the PDE is saying that no heat/substance will escape the body. The process is time invariant, so the PDE is just defining the distribution of heat/substance inside the body to follow this strange rule that its laplacian is f?

Is this reasoning correct? Can I assign some physical intuition to the distribution of heat/substance inside the body following the rule that its laplacian be f?
 
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