Behavior of solutions at the initial time in nonlinear parabolic differential equation (Q1324947)

The nonlinear parabolic differential equation \[ {d \over dt} u(t) + \partial \varphi \bigl( u(t) \bigr) \ni 0, \quad t > 0, \tag{1} \] is considered where \(\varphi\) is a proper lower semicontinuous convex functional defined on a real Hilbert space \(H\) and \(\partial \varphi\) denotes the subdifferential of \(\varphi\). Existence results of (1) are proved, such that \(u(t)\) converges weakly, but not strongly, to a point of \(D(\partial \varphi)\) as \(t \downarrow 0\).