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A parabolic hemivariational inequality. (English) Zbl 0858.35072

A quasilinear nonmonotone parabolic initial boundary value problem of the form \[ u'(t)+Au(t)+\Sigma(t)=f(t), \quad u(0)=u_0, \quad \Sigma(x,t)\in\widehat{b} (u(x,t)) \quad\text{a.e. }(x,t)\in Q_T, \] is considered taking into account the existence of its solutions. This is a generalization of stationary hemivariational inequalities to the dynamic case in which \(A\) is assumed to be linear and continuous, and the multivalued mapping \(\widehat{b}\) is constructed by filling the gaps of a function \(b\in L^\infty_{\text{loc}}(R)\). In the nonpotential case, \(A\) being nonsymmetric, a linear growth condition on the behavior of \(b\) at infinity has been imposed whereas in the potential case, \(A\) being symmetric, a sign condition on \(b\) has been introduced. The Galerkin method in the setting of the evolution triple “\(V\subset H\subset V' \)” with \(V\hookrightarrow H^1(\Omega)\) and \(H=L^2(\Omega)\), combined with compactness technique are applied as the main mathematical tools.

MSC:

35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35R70 PDEs with multivalued right-hand sides
49J40 Variational inequalities
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