Global existence and blow up of positive initial energy solution of a quasilinear wave equation with nonlinear damping and source terms (Q2019894)

Summary: In this paper, we consider a quasilinear wave equation having nonlinear damping and source terms \[ u_{tt} - \Delta u_{t} - \sum^n_{i=1} \frac{\partial}{\partial x_i} \Big[ \sigma_i(x, u_{x_i})+ \beta_i(x, u_{tx_i})\Big] + f(x, u_t) = g(x, u) \] and obtained global existence and blow up results under certain polynomial growth conditions on the nonlinear functions \(\sigma_i\), \(\beta_i\), (\(i = 1, 2,\dots, n\)), \(f\) and \(g\). We obtain global existence result for positive initial energy solution using Galerkin approximation procedure and nonexistence (blow up) result using the technique introduced by \textit{V. Georgiev} and \textit{G. Todorova} [J. Differ. Equations 109, No. 2, 295--308 (1994; Zbl 0803.35092)] with little modification for our problem.