On the quasilinear wave equation with a mixed nonhomogeneous condition (Q1893937)

The authors consider the quasilinear wave equations \[ u_{tt}- \Delta u+ f(u_t)= 0,\;x\in (0, 1),\;t\in (0, T) \] with initial and nonhomogeneous boundary values, where \(f(u_t)= |u_t|^\alpha\text{sign}(u_t)\), \(0< \alpha< 1\). They use the Galerkin method associated with monotonic arguments to consider the global existence and the uniqueness of the solution of the aforementioned initial-boundary value problem. When the boundary condition at \(x= 0\) is linear and nonhomogeneous: \[ u_x(0, t)= h\cdot u(0, t)+ g(t),\quad h> 0, \] the behavior of the solution of the problem is considered as \(h\) tends to \(0_+\).