## 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

### Keywords:

discontinuous nonlinearities; Galerkin method
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### References:

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