Stabilization of the semilinear wave equation with viscous damping (Q804811)

The author considers the problem of local exponential stability of solutions to the following nonlinear wave equation: u\({}_{tt}(x,t)=(\Delta -1)u(x,t)-F(u_ t(\cdot,t))(x)\), \((x,t)\in \Omega \times (0,\infty),\) u(x,0)\(=u_ 0(x)\), \(u_ t(x,0)=u_ 1(t)\), \(x\in \Omega\), \(u_ 0\in H^ 1(\Omega)\), \(u_ 2\in L_ 2(\Omega),\) \(\partial u/\partial n(x,t)=g(u(x,t))\), (x,t)\(\in \partial \Omega \times (0,\infty).\) It is assumed that \(\Omega\) is an open bounded domain in \({\mathbb{R}}^ 2\) and \(F\in {\mathcal L}(L_ 2(\Omega))\) is such that the solution to the problem with \(g\equiv 0\) decays exponentially to zero in \(H^ 1(\Omega)\times L^ 2(\Omega)\). It is then proved that local exponential stability holds in the same topology for a class of g which is small at the origin. The proof uses sharp regularity results for the solutions combined with semigroup methods. The methods are restricted to the case that \(\Omega \subset {\mathbb{R}}^ 2\). The case that \(\Omega \subseteq {\mathbb{R}}^ 3\) remains an open problem.