Temperature on rods with Robin boundary conditions

We consider solutions uf to the one-dimensional Robin problem with heat source f∈L1[−π,π] and parameter α>0. For given m, M, s, 0≤m<s<M, we identify sources f0, such that uf0 maximizes temperature gap max[−π,π]⁡uf−min[−π,π]⁡uf over all f m≤f≤M ‖f‖L1=2πs. In particular, this answers a question raised by J. Langford P. McDonald in [5]. also sources, which maximize/minimize at point x0∈[−π,π] same class of as above discuss few related questions.