Strong solutions of a quasilinear wave equation with nonlinear damping

We study the following initial and boundary value problem \[ \begin{gathered} u_{tt} - \Delta u_t - \sum\limits_{i = 1}^N {\sigma '_i } \left( {u_{x_i } } \right)u_{x_i x_i } + \left| {u_t } \right|^\alpha \operatorname{sgn} u_t = 0, \hfill \\ 0 < \alpha < 1\quad (x,1)\quad {\text{in }}\Omega \times ] {0,T} [, \hfill \\ u = 0\quad {\text{on }}\partial \Omega , \hfill \\ u(x,0) = u_0 (x),\qquad u_t (x,0) = u_1 (x) \hfill \\ \end{gathered} \] where $\Omega $ is a bounded domain in $R^N $ with a sufficiently regular boundary $\partial \Omega $. In § 1, it is proved that for $u_0 $ in $H_0^1 (\Omega )$, $u_1 $ in $L_2 (\Omega )$, $\sigma _i $ in $C(\mathbb{R},\mathbb{R})$ nondecreasing and inducing mappings of $L_2 (\Omega )$ into itself, taking bounded sets into bounded sets, the problem admits a global weak solution. If, in addition, the $\sigma _i $’s are assumed locally Lipschitzian, then the solution is unique.In § 2, it is proved that for $N = 1$, $u_0 $ in $H_0^1 (\Omega ) \cap H^2 (\Omega )$, $u_1 $ i...