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Doubly nonlinear equations with unbounded operators. (English) Zbl 1073.34076
The authors investigate the following nonlinear evolution system \[ \begin{align*}{& v'(t)+w(t)=\tilde{f}(t)\quad\text{for a.a. } t\in ]0,T[,\cr & v(t)\in {\cal A}(t)u(t),\quad w(t)\in {\cal B}(t)u(t)\quad\text{for a.a. } t\in ]0,T[,\cr & v(0)=v_0,}\end{align*} \] where the operators \({\mathcal A}(t)\) and \({\mathcal B}(t)\) are subdifferentials of proper convex lower semicontinuous functions on a real Hilbert space \(V\) such that \({\mathcal A}(t)+{\mathcal B}(t)\) is \(V\)-coercive. They consider the case, when \({\mathcal A}(t)\) are bounded and possibly degenerate subdifferentials and \({\mathcal B}(t)\) are unbounded subdifferentials. The authors formulate conditions which guarantee the existence of solutions of the above system if the initial value \(v_0\) belongs to the range of \({\mathcal A}(0)\) and \(\tilde{f}\) is differentiable. As an example a variational inequality is considered.

MSC:
34G25 Evolution inclusions
47J35 Nonlinear evolution equations
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