SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH DAMPING AND SUPERCRITICAL BOUNDARY AND INTERIOR SOURCES

摘要

We consider the local and global well-posedness of the coupled nonlinear wave equations utt - Δu + g1(ut) = f1(u, v), vtt - Δv + g2(vt) = f2(u, v) in a bounded domain Ω ? ?~n with Robin and Dirichlét boundary conditions on u and v respectively. The nonlinearities f_1(u, v) and f_2(u, v) have supercritical exponents representing strong sources, while g_1(u_t) and g_2(v_t) act as damping. In addition, the boundary condition also contains a nonlinear source and a damping term. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data.