On the existence of solutions of strongly damped nonlinear wave equations (Q1574350)

The authors investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation: \[ u_{tt}(t, x)- \Biggl(\alpha+ \beta\Biggl( \int_\Omega|\nabla u(t, y)|^2 dy\Biggr)^\gamma\Biggr) \Delta u(t,x), \] \[ -\lambda\Delta u_t(t, x)+ \mu|u(t, x)|^{q- 1} u(t,x)= 0\quad\text{for }x\in \Omega,\;t\geq 0; \] \[ u(0, x)= u_0(x),\quad u_t(0,x)= u_1(x)\quad\text{for }x\in\Omega;\;u=0\quad\text{on }\partial\Omega, \] where \(q> 1\), \(\lambda> 0\), \(\mu\in\mathbb{R}\), \(\alpha, \beta\geq 0\), \(\alpha+ \beta> 0\). Previous results for some particular cases were obtained by \textit{K. Nishihara} and \textit{Y. Yamada} [Funkc. Ekvacioj, Ser. Int. 33, 151-159 (1990; Zbl 0715.35053)], \textit{K. Ono} and \textit{K. Nishihara} [Adv. Math. Sci. Appl. 5, No. 2, 457-476 (1995; Zbl 0842.45005)] and \textit{K. Ono} [J. Differ. Equations 137, No. 2, 273-301 (1997; Zbl 0879.35110)].