Monotonicity formula for a problem with hysteresis (Q1643772)

This short note states without proof a local monotonicity formula for the initial boundary value problem \[ \begin{aligned} \partial_t u - \Delta u + h[u] & = 0 \;\text{ in }\; (0,T)\times \mathcal{U}\;, \\ u(0) & = \varphi \;\text{ in }\; \mathcal{U}\;,\end{aligned} \] supplemented with either Dirichlet or Neumann boundary conditions. Here, \(h[u]\) is a discontinuous nonlinearity built upon a function \(f:\mathbb{R}\to \{-1,1\}\) and is defined as follows: \(h[u](t,x)=f(u(t,x))=-1\) if \(u(t,x)\leq \alpha\), while \(h[u](t,x)=f(u(t,x))=1\) if \(u(t,x)\geq \beta\). When \(u(t,x)\in (\alpha,\beta)\), we set \(h[u](t,x)=f(u(\hat{t}(x),x))\), where \(\hat{t}(x)\leq t\) is the largest time in \([0,t]\) such that \(u(\hat{t}(x),x)\in\mathbb{R}\setminus (\alpha,\beta)\).