Waiting time for a non-Newtonian polytropic filtration equation with convection (Q413445)

The mathematical model of filtration of a non-Newtonian polytropic fluid is considered. NEWLINE\[NEWLINE \begin{cases}\frac{\partial u}{\partial t}-\frac{\partial}{\partial x}\left(\left|\frac{\partial\, (u^m)}{\partial x}\right|^{p-2}\;\frac{\partial\, (u^m)}{\partial x}\right)-\lambda\,\frac{\partial (u^q)}{\partial x}=0, &x\in \mathbb{R},\quad t>0,\\ u(x,0)=u_0(x), &x\in \mathbb{R}. \end{cases}NEWLINE\]NEWLINE Here, \(p>1\), \(m(p-1)>1\), \(q>0\), \(\lambda \in \mathbb{R}\), \(u_0\) is a non-negative continuous function with compact support. The sets NEWLINE\[NEWLINE \xi_-(t)=\inf\{x\in \mathbb{R}:\, u(x,t)<0\},\quad \xi_+(t)=\inf\{x\in \mathbb{R}:\, u(x,t)>0\} NEWLINE\]NEWLINE are called the interfaces of solution and a time interval for which these interfaces are stationary is called the waiting time.NEWLINENEWLINEThe authors give some necessary and sufficient conditions on the initial data for the existence of waiting time. These conditions depend on the magnitude of the exponent \(q\).