Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation. I: The reaction function is a bistable cubic polynomial (Q1627701)

The main purpose of the paper is to study the long-time behavior of weak solutions to the following initial-value problem \[ \begin{cases} u_{\tau \tau}+u_{\tau}=u_{x x}+ \varepsilon\,\big(F(u)+F(u)_{\tau}\big),\quad x\in \mathbb{R}, \,\, \tau>0,\\ u=0\quad \text{for}\quad x>0 \quad \text{and}\quad u=1\quad\text{for}\quad x\leq 0, \end{cases}\tag{1} \] where \(F(u)=u\,(1-u)\,(u-\mu)\), \(\varepsilon >0\) and \(\mu\in (0,1/2]\) are two parameters. More precisely, in dependence of relationship between the parameters \(\varepsilon >0\) and \(\mu\in (0,1/2)\), the authors establish the existence of so called \textit{permanent-form traveling wave} (PFTW). A PFTW is a positive solution to the problem (1) which depends only on the single variable \(z=x-V\,t\), where \(V>0\) is the constant propagation speed. Also is proved that if \(\mu=1/2\) then the system evolves to the steady state.