Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term
The authors of this interesting paper investigate the nonlinear wave equation \( y_{tt}-\triangle y+f_0(\nabla y)=0 \) in \(\Omega \times (0,\infty)\), \(y=0\) on \(\Gamma_1\times (0,\infty)\) with nonlinear boundary condition \(\partial y/\partial\nu +g(y_t)=\int\limits_{0}^{t}h(t-\tau)f_1(y(\tau))d\tau \) on \(\Gamma_0\times (0,\infty)\), \(y(x,0)=y^0(x)\), \(y_t(x,0)=y^1(x)\) in \(\Omega \), where \(\Omega \) is a bounded star-shaped domain of \(\mathbb R\)\(^n\) (\(n\geq 1\)), with a smooth boundary \(\Gamma =\Gamma_0\cup \Gamma_1\), \(\Gamma_0\) and \(\Gamma_1\) are closed and disjoint, \(\nu \) is the unit outward normal to \(\Gamma \); \(f_0,g,f_1\) are nonlinear functions satisfying some general properties as smoothness, continuity, boundedness etc. The main result is that there exists at least a strong solution \(y:(0,\infty)\times \Omega \to \mathbb R\), that is, \(y, y'\in L^{\infty }(0,\infty; V)\) and \(y^{\prime\prime }\in L^{\infty }(0,\infty; L^{2}(\Omega))\). Moreover, the strong solution \(y\) decays exponentially, \(E(t)\leq Ce^{-\gamma t}\) (\(E(t)\) is the energy) for all \(t\geq t_0\) (\(\gamma , C>0\)) and in addition, if \(f_1\) is globally Lipschitz then the solution is unique. Another result is the existence of a weak solution in the space \(C([0,\infty);V)\cap C^1([0,\infty);L^2(\Omega))\).
scientific article; zbMATH DE number 1837318