Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary
A family of IBV hyperbolic problems of the form \(u^\epsilon_{tt}+\frac{1}{\epsilon}\chi_{\omega_\epsilon}u^\epsilon_t-\Delta u^\epsilon+\lambda u^\epsilon= f_\epsilon+\frac{1}{\epsilon}\chi_{\omega_\epsilon}g_\epsilon\) in \(\Omega \times(0,T)\), \(\frac{\partial u^\epsilon}{\partial n} =0\) on \(\Gamma \times(0,T)\), \(u^\epsilon(0,x)= u^\epsilon_0(x)\), \(u^\epsilon_t(0,x)= v^\epsilon_0(x)\) in \(\Omega\) is considered. Here \(\Gamma\) is the boundary of \(\Omega\), and \(\omega_\epsilon\) denotes a neighborhood of \(\Gamma\) that converges to the boundary when \(\epsilon\) goes to zero, and \(\chi_{\omega_\epsilon}\) denotes its characteristic function. The goal of this work is to prove that, when \(\epsilon\to 0\), the family of solutions \(u^\epsilon\) converges to a limit function \(u^0\) which is the solution of the IBV hyperbolic problem with boundary feedback damping term: \(u^0_{tt}-\Delta u^0+\lambda u^0 =f\) in \(\Omega\times(0,T)\), \(u^0_t+\frac{\partial u^0}{\partial n}=g\) on \(\Gamma\times(0,T)\), \(u^0(0,x)=u_0(x)\), \(u^0_t(0,x)=v_0(x)\) in \(\Omega\). Initial values are limits of the initial data in the considered system. Well-posedness results for both systems are proved. Mild and strict solutions for both problems are built. Energy estimates are considered. Convergence of mild solutions as well as of strict solutions are investigated.
scientific article; zbMATH DE number 7074655