In this paper, we first define the notion of viscosity solution for the
following system of partial differential equations involving a subdifferential
operator:\[\{[c]{l}\dfrac{\partial u}{\partial
t}(t,x)+\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\in\partial\phi (u(t,x)),\quad
t\in[0,T),x\in\mathbb{R}^d, u(T,x)=h(x),\quad x\in\mathbb{R}^d,\] where
$\partial\phi$ is the subdifferential operator of the proper convex lower
semicontinuous function $\phi:\mathbb{R}^k\to (-\infty,+\infty]$ and
$\mathcal{L}_t$ is a second differential operator given by
$\mathcal{L}_tv_i(x)={1/2}\operatorname
{Tr}[\sigma(t