## heat equation equilibrium temperature [closed]

Consider a uniform bar of length L which has been insulated both ends. If the inital temperature is given by u(x,0)=f(x) (some general distribution), and there is a constant, nonzero source of thermal energy.

a)Demonstrate mathematically that the equilibrium temperature cannot be found.

I know that the energy from the internal source cannot escape through the boundaries and so the temperature will continue to increase at all locations for all time never reaching equilibrium. However, I cannot demonstrate it. Please help.

b)Suggest what kind of boundary condition change would allow for an equilibrium to be reached.

Solution is $\frac{\partial u}{\partial x}(0,t) = A$ and $\frac{\partial u}{\partial x}(0,t)=B$ but I couldn't understand why. Could you please explain. Thank you.

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You could try asking this question at one of the sites listed in the FAQ, where it would be more appropriate. However, if you do choose to ask at one of those sites, it would help if you said precisely why you "cannot demonstrate it", and what you have tried. Right now, it looks like you are solving a homework problem, and haven't thought about it in a very structured way. – S. Carnahan Jun 15 at 17:27