Abstract quasilinear equations of second order with Wentzell boundary conditions (Q1762617)

Second-order differential operators with Wentzell-type boundary conditions received intense attention in the last decades. Recently, in the paper by K.-J- Engel and the reviewer [J. Differ. Equations 207, No. 1, 1-20 (2004; Zbl 1063.35104)], an abstract framework was presented to show that second-order differential operators with (generally, nonlocal) Wentzell boundary conditions generate cosine functions, and hence, the corresponding wave equations are well-posed. In the paper under review, the authors use this framework to consider quasilinear wave equations with Wentzell boundary conditions. The model problem is the equation \[ u_{tt}(x,t) = \phi(x,u_x(x,t))u_{xx}(x,t) + \psi(x,u(x,t),u_x(x,t)), \quad x\in [0,1],\, t\geq 0, \] with the boundary conditions \[ \phi(j,u_x(j,t))u_{xx}(j,t)+ \psi(j,u(j,t),u_x(j,t)) = \beta_j(u_x(j,t)) + \gamma_j(u(j,t)),\quad j=0,1,\, t\geq 0, \] in the space \(C[0,1]\). Using an abstract setting, analogous to the one used for linear problems, the authors transform the problem to a first-order one on a product space of four Banach spaces. The main technical difficulty is then to show the dissipativity of the corresponding nonlinear operator.