Blow-up and lifespan estimates for solutions to the weakly coupled system of nonlinear damped wave equations outside a ball

In this paper, we consider the initial-boundary value problems with several fundamental boundary conditions (the Dirichlet/Neumann/Robin boundary condition) for the multi-component system of semi linear classical damped wave equations outside a ball. By applying a test function approach with a judicious choice of test functions, which approximates the harmonic functions being subject to these boundary conditions on partial derivative Omega, simultaneously we have succeeded in proving the blow-up result in a finite time as well as in catching the upper bound of lifespan estimates for small solutions in all spatial dimensions. Moreover, such kind of these results, which become sharp in the subcritical cases for one-dimensional case, will be discussed at the end of this paper.