Global existence for a class of quasilinear hyperbolic-parabolic equations (Q1916089)

The author proves that, given \(u_0\) and \(u_1\), there is an \(\varepsilon_0> 0\) such that for any \(\varepsilon\leq \varepsilon_0\), the initial value problem \[ \varepsilon u_{tt}+ u_t- u_{xx}- (f(u_x))_x= 0,\;u(x, 0)= u_0(x),\;u_t(x, 0)= u_1(x), \] where \(f\) is a smooth increasing function on \(\mathbb{R}\), has a unique classical solution defined for all \(t\geq 0\); moreover, its derivatives decay to 0 as \(t\to \infty\).