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

Guo, Yanqiu; Rammaha, Mohammad

doi:10.1090/S0002-9947-2014-05772-3

Systems of nonlinear wave equations with damping and supercritical boundary and interior sources
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Trans. Amer. Math. Soc. 366 (2014), 2265-2325 Request permission

Abstract:

We consider the local and global well-posedness of the coupled nonlinear wave equations \begin{align*} u_{tt}-\Delta u+g_1(u_t)=f_1(u,v),\\ v_{tt}-\Delta v+g_2(v_t)=f_2(u,v) \end{align*} in a bounded domain $\Omega \subset \mathbb {R}^n$ 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.

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