I am trying to show uniqueness of solutions for the homogeneous heat equation with unhomogeneous Robin Boundary Conditions, i.e.
$$ u_t(x,t) = u_{xx}(x,t), \quad x>0, t>0 $$ $$ u(x,0) = 0, \quad x>0, $$ $$ u_x(0,t) + \alpha u(0,t) = f(t), \quad t>0, $$
with $\alpha < 0$.
I am aware that the Energy method and Maximum principle are usually used to prove uniqueness of this kind of problems but I was not able to find any proper reference for the maximum principle used in this specific case and I was not able to understand it the Energy method applied to this problem (see below).
I found this post that tackles this question using the Energy method, but I do not understand the concept of logarithmic convexity of the Energy.
I would also be very happy if someone could provide some reference of the Maximum principle applied to this specific problem