Existence, blow-up and exponential decay for a nonlinear Love equation associated with Dirichlet conditions

In this paper we consider a nonlinear Love equation associated with Dirichlet conditions. First, under suitable conditions, the existence of a unique local weak solution is proved. Next, a blow up result for solutions with negative initial energy is also established. Finally, a sufficient condition guaranteeing the global existence and exponential decay of weak solutions is given. The proofs are based on the linearization method, the Galerkin method associated with a priori estimates, weak convergence, compactness techniques and the construction of a suitable Lyapunov functional. To our knowledge, there has been no decay or blow up result for equations of Love waves or Love type waves before.