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A new class of doubly nonlinear evolution equations. (English) Zbl 1073.35142
The authors consider an evolution equation of the form: \[ u'(t)+K(t,\theta(t))+G(t,u(t))=f(t) \text{ in }V^{*} \text{ for a.e. }t\in[0,T] u(0)=u_0, \] where for each \(t\in[0,T],\) \(K(t,.)\) is a weakly continuous operator from a reflexive Banach space \(V\) into its dual space \(V^{*},\) \(G(t,.)\) is a weakly continuous operator from a Hilbert space \(H\) into \(V^{*}\), where \(V\) is densely and compactly imbedded in \(H\), \(f\) is a given source function and \(u_0\) is an initial data. It is assumed that \(\theta(t)=\partial\psi^{t}(u(t))\) in \(V\) for a.e. \(t\in[0,T]\), where \(\{\psi^{t}\}\) is a family of proper lower semicontinuous and convex functions on the space \(V^{*}\) and \(\partial\psi^{t}\) is the subdifferential of \(\psi^{t}\) from \(V^{*}\) into \(V\). By using a time discretization scheme, the authors give an existence result for the above problem which complements previous results on other types of doubly nonlinear evolution equations by other authors such as P. Colli and A. Visintin [Commun. Partial Differ. Equations 15, No. 5, 737–756 (1990; Zbl 0707.34053)], N. Kenmochi and I. Pawlow [Nonlinear Anal. 10, 1181–1202 (1986; Zbl 0635.35043)] and E. Maitre and P. Witomski [Nonlinear Anal. 50, No. 2(A), 223–250 (2002; Zbl 1001.35091)].

35K90 Abstract parabolic equations
47J35 Nonlinear evolution equations
35K65 Degenerate parabolic equations
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