Stability of degenerate stationary waves for viscous gases (Q717195)

The Dirichlet-Cauchy problem for second order partial differential equations of the type, the parabolic \(u_t+f(u)_x=\mu u_{ xx}\) or the hyperbolic \( u_{ tt}+u_t+f(u)_x= u_{ xx}\), in \((\mathbb R^+)^2\) admit solutions that convergent to a stationary wave, provided by some smooth convex function \(f=f(u)\) conveniently chosen, \(\mu>0\) and \(u_b<u_+=\lim_{x\rightarrow \infty}u_0(x)\), with \(u_b\) and \(u_0\) representing respectively a constant boundary value and an initial function. The \(L^p\)-stability analysis is based on the space-time weighted energy method.