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On a class of implicit evolution variational inequalities. (English) Zbl 0844.35058

Let \(V\subset H\subset V'\) be an evolution triple, where \(V\) and \(H\) are Hilbert spaces. The Cauchy problem for the following nonlinear evolution equation \[ (I+ \partial\phi)(u'(t))+ Lu(t) \ni F(t, u(t)),\quad 0<t< T \] is studied. Here \(\partial\phi\) denotes the subdifferential of a proper, convex, and lower-semicontinuous function \(\phi: V\to (- \infty, +\infty]\), \(L: V\to V'\) is a linear, bounded, and selfadjoint operator, and \(F: (0, T)\times V\to V'\) is a nonlinear function. Qualitative results (via finite differences in time) and applications to PDEs are discussed.

MSC:

35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
34G20 Nonlinear differential equations in abstract spaces
35K55 Nonlinear parabolic equations
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