Hyperbolic-parabolic problem with degenerate second-order boundary conditions. (Q2713037)

The authors study the existence and uniqueness of solutions to the initial boundary value problem in \(\Omega\times (0,T)\) with \(\Omega\in \mathbb R^n\): \(P(x,t)v_{tt}-\Delta v +\alpha v_t=f(x,t)\) together with initial conditions and boundary condition \(\left( {\partial v\over\partial\nu} +K(v)v_{tt} +g(v_t) \right)| _{\partial\Omega\times (0,T)}=0\). Here \(P(x,t)\) and \(K(u)\) are continuously differentiable non-negative functions, \(\alpha\) is a positive constant, \(\nu\) is the outward unity normal vector on \(\partial\Omega\). When \(K(v)>0\), the existence of solutions has been proved by the authors in [Electron. J. Differ. Equ. 1998, Paper No. 28, 10 p. (1998; Zbl 0915.35063)]. In this paper, the authors extend the result in the aforementioned paper and consider the degenerate case when \(K(v)\) is only non-negative. By exploiting the Faedo-Galerkin method, a priori energy estimates and compactness arguments, the authors prove the existence of global generalized solutions. And the uniqueness is obtained in the one-dimensional case.