A new class of doubly nonlinear evolution equations
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 \textit{P. Colli} and \textit{A. Visintin} [Commun. Partial Differ. Equations 15, No. 5, 737--756 (1990; Zbl 0707.34053)], \textit{N. Kenmochi} and \textit{I. Pawlow} [Nonlinear Anal. 10, 1181--1202 (1986; Zbl 0635.35043)] and \textit{E. Maitre} and \textit{P. Witomski} [Nonlinear Anal. 50, No. 2(A), 223--250 (2002; Zbl 1001.35091)].
scientific article; zbMATH DE number 2104156