Decay of energy for second-order boundary hemivariational inequalities with coercive damping (Q622386)

The author considers two strongly damped partial differential inclusions \[ 0\in u''(t) + A(t,u'(t)) + Bu(t)+\gamma^*\partial J (\gamma u(t)) \] and \[ 0\in u''(t) + A(t,u'(t)) + Bu(t)+\gamma^*\partial J (\gamma u'(t)). \] In both cases the differential operator \(A\) is nonlinear pseudomonotone and coercive while the differential operator \(B\) is bounded, linear, monotone and symmetric. The multivalued term has the form of Clarke subdifferential of the functional defined on the space of boundary functions and \(\gamma\) is the trace operator. For both cases the conditions are given under which appropriately defined energy tends to zero exponentially as time tends to infinity. The methodology based on Nakao's lemma is used. For the inclusion with multivalued term dependent on \(u'\) it is shown that exponential energy decay occurs for every of possibly many solutions while for the (more difficult) case of inclusion with multivalued term dependent on \(u\) it is shown, basing on an approximation argument, that there exists a solution with exponential energy decay.