Parabolic variational inequalities with singular inputs (Q1374501)

The authors study the problem \[ dy(t) + Ay(t) dt +\partial \varphi (y(t))dt \ni f(t) dt +dM(t), \quad t\in [0,T],\qquad y(0)=y_0, \tag{1} \] where \(A: V\to V^*\) is a linear self-adjoint operator, \(\partial \varphi\) is a subdifferential of a lower semi-continuous convex function \(\varphi : H\to \overline {\mathbb R}\), \(f\in L^2(0,T;H)\), \(M\in C(0,T;H)\) with \(V,H,X\) real Hilbert separable spaces such that \(H=H^*\), \(V\subset H\subset V^*\), \(X\subset H\subset X^*\), \(V\cap X\) is dense both in \(V\) and \(X\). The weak form to (1) is introduced as \[ y(t) + \int_0^t Ay(s)ds +\eta (t) = y_0 + \int_0^t f(s)ds +M(t), \quad t\in [0,T],\qquad d\eta (t) \in \partial \varphi (y(t))dt.\tag{2} \] The authors prove that under some additional assumptions on \(A\), \(\varphi\) and \(M\) there exists a unique solution \((y,\eta)\) of (2) satisfying \(y\in C([0,T];V)\), \(Ay\in L^2(0,T;H)\), \(\eta \in C([0,T];H)\cap BV([0,T];X^*)\).