On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains (Q1411762)

The author of this interesting paper studies the initial-boundary value problem for the quasilinear wave equation \[ u_{tt}- \operatorname {div}\{\sigma (|\nabla u|^2)\nabla u\}+a(x)u_t=0 \] in \(\Omega \times [0,\infty)\) with initial data \(u(x,0)=u_0(x)\), \(u_t(x,0)=u_1(x)\) and boundary condition \(u|_{\partial\Omega }=0\), where \(\Omega \) is an exterior domain in \(\mathbb{R}^N\) with a smooth boundary \(\partial\Omega \), \(\sigma (v)=1/\sqrt{1+v}\) and \(a(x)\) is a nonnegative function. The dissipation term \(a(x)u_t\) is characterized with two assumptions: (A) there exist \(x_0\in \mathbb{R}^N\) and an open set \(\omega \) in \(\bar\Omega \) such that the closure of \(\Gamma (x_0)\) is a subset of \( \omega \) (\(\Gamma (x_0)\) is a part of the boundary \(\partial\Omega \)) and \(a(x)\geq \varepsilon_0>0\) for \(x\in\omega \) (\(\varepsilon_0\) is a constant). (B) there exist \(L>>1\) and \(\varepsilon_0>0\) such that \(a(x)\geq \varepsilon_0>0\) for \(|x|\geq L\). The summand \(a(x)u_t\) is required to be effective only in localized area and no geometrical condition is imposed on the boundary \(\partial\Omega \). The main result is the global existence of small amplitude solutions under the hypotheses A and B. The author pays attention to both cases \(N\geq 4\) and \(N=3\) and taking into account that \(\sigma \) is differentiable function on \(R^+\) and bounded (\(\sigma (v)\geq k_0>0\), \(\sigma (v)-2|\sigma^{\prime }(v)|v\geq k_0>0\), \(0\leq v\leq R\), \(R>0\), \(k_0\) is a positive constant) proves two existence and uniqueness theorems concerning these cases.