Blow-up of solutions to systems of nonlinear wave equations with supercritical sources

摘要

In this article, we focus on the life span of solutions to the following system of nonlinear wave equations: U_(tt) - Δu + g_1(u_t) = f_1(u, v); v_(tt) - Δv + g_2(v_t) = f_2(u, v) in a bounded domain Ω{is contained in}R~n with Robin and Dirichlet boundary conditions on u and v, respectively. The nonlinearities f_1(u, v) and f_2(u, v) represent strong sources of supercritical order, while g_1(u_t) and g_2(v_t) represent interior damping. The nonlinear boundary condition on u, namely {partial deriv}_vu+u+g(u_t)=h(u) on Γ, also features h(u), a boundary source, and g(u_t), a boundary damping. Under some restrictions on the parameters, we prove that every weak solution to system above blows up in finite time, provided the initial energy is negative.