Systems of nonlinear wave equations with damping and supercritical boundary and interior sources

We consider the local and global well-posedness of the coupled nonlinear wave equations u t t − Δ u + g 1 ( u t ) = f 1 ( u , v ) , v t t − Δ v + g 2 ( v t ) = f 2 ( u , v ) {align*} uₜₜ-Δ u+g_1(u_t)=f_1(u,v),\\ vₜₜ-Δ v+g_2(v_t)=f_2(u,v) {align*} in a bounded domain Ω ⊂ R n Ω ⊂ R^n with Robin and Dirichlét boundary conditions on u u and v v respectively. The nonlinearities f 1 ( u , v ) f_1(u,v) and f 2 ( u , v ) f_2(u,v) have supercritical exponents representing strong sources, while