Lifespan for solutions to 4-th order hyperbolic systems with time dependent coefficients

We study blow-up solutions of a nonlinear hyperbolic system of fourth order with time dependent coefficients under Dirichlet or Navier boundary conditions. We prove that under some restrictions on the data there exists a safe interval of existence of the solution and a lower bound of the lifespan is derived. The results are extended to a more general class of systems, where powers of the gradient of the solution are introduced. The proofs are based on some inequalities and coupled estimates techniques.