Abstract
In this article we consider the asymptotic behavior of solutions to second-order evolution inclusions with the boundary multivalued term u ″ ( t ) + A ( t , u ′ ( t ) ) + B u ( t ) + γ ̄ ∗ ∂ J ( t , γ ̄ u ′ ( t ) ) ∋ 0 and u ″ ( t ) + A ( t , u ′ ( t ) ) + B u ( t ) + γ ̄ ∗ ∂ J ( t , γ ̄ u ( t ) ) ∋ 0 , where A is a (possibly) nonlinear coercive and pseudomonotone operator, B is linear, continuous, symmetric and coercive, γ ̄ is the trace operator and J is a locally Lipschitz integral functional with ∂ denoting the Clarke generalized gradient taken with respect to the second variable. For both cases we provide conditions under which the appropriately defined energy decays exponentially to zero as time tends to infinity. We discuss assumptions and provide examples of multivalued laws that satisfy them.
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