Consider the following "evolution problem"
$f(t) - u_t(t) \in \partial \psi(u(t))$
$u(0) = u_0$
Where $f:[0,T] \rightarrow H$
$ u:[0,T] \rightarrow H$
$ \psi:H \rightarrow (-\infty,\infty]$ is convex and $\psi(u_0) < \infty$ and $\partial$ denotes the subgradient of $\psi$.
My question is what is this equation supposed to represent and how is it related to the heat equation.
Can you please interpret what the subgradient of $\psi(u(t))$ is supposed to mean here. I am not looking for mathematical definitions. I am looking for what it means analogous to the following:
The heat equation: $u_t = \Delta u + f$ makes sense because if $\Delta u$ is positive then the average of u at points around x is greater than u at x so u should increase over time (just like in the physical heat sense). Also, the $f$ in the equation represents a forcing term. If $f$ is positive we can interpret $u$ is increasing even more over time and that $f$ is some extra "heat" being added to the system.
Please give an explanation similar to the above but involving the subgradient. Assume the reader knows the definition of the subgradient and its intuition on $\mathbb{R}^n$ but not in general.
Thank you