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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
##### Keywords:
nonlinear evolution system; maximal monotone operator
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##### References:
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