Abstract
utt −∆u + g1(ut) = f1(u, v) vtt −∆v + g2(vt) = f2(u, v), in a bounded domain Ω ⊂ R with Robin and Dirichlet boundary conditions on u and v respectively. The nonlinearities f1(u, v) and f2(u, v) are with supercritical exponents representing strong sources, while g1(ut) and g2(vt) 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.
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