Global existence and asymptotic behavior for a semilinear Bresse beam model with boundary constraints

Abstract

We study the semi-linear Signorini problem for the Bresse beam (curved beams) with frictional dissipation. We show that there exists at least one weak solution that decays exponentially to zero when the system is totally dissipative, otherwise, if the dissipative mechanisms are effective in only one or two equations of the system, we still show the exponential decay provided the propagation speeds of the model are equal to each other. If the speeds of propagations are different and the dissipative mechanisms effective only in one or two equations then the solution decays polynomially in general as 1/t. The exception is when the frictional dissipative mechanism is effective only over the axial force in which case the rate of decay is 1/t4. Our main tool is the Semigroup Theory applied to the hybrid model that approximates the Signorini problem. A numerical approach is presented to highlight our theoretical results.