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Existence and regularity results for doubly nonlinear inclusions with nonmonotone perturbation. (English) Zbl 1320.34092

The paper deals with a class of nonlinear evolution inclusions arising from enthalpy formulation of heat conduction problems with phase change and nonmonotone source. In particular, let \(V\) and \(Z\) be separable and reflexive Banach spaces with the dual spaces \(V^*\) and \(Z^*\) respectively. Let \(H\) be a Hilbert space which is identified with his dual \(H^*\) such that \[ V\subset Z \subset H \subset Z^* \subset V^*, \] where all embeddings are dense and continuous and \(V\) embeds to \(Z\) compactly.
In this context the author studies the existence and regularity for nonlinear evolution inclusion in the abstract form: \[ \frac{d}{dt} B(u(t))+A(t,u(t))+G(t,u(t)) \ni f(t) , \text{ for a.e. } t \in (0,T), \] where \(B\) is the subdifferential of a proper, convex and lower semicontinuous functional defined on \(Z\), \(A(t,.):V \to V^*\) is a pseudomonotone operator and \(G(t,.):Z \to 2^{Z^*}\) is a nonmonotone perturbation.
An application to a physical model described by hemivariational inequality is given.

MSC:

34G25 Evolution inclusions
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