On quasilinear bidegenerate parabolic equations (Q1909362)

Consider the following initial value problem: \[ \partial/\partial t(|u|^{p- 1} u)= \partial/\partial x(|u_x|^{p- 1} u_x)+ \lambda |u|^{r- 1} u, (t, x)\in [0, \infty)\times \mathbb{R}, u(0, x)= u_0(x), x\in \mathbb{R}, \] where \(p\), \(q\) and \(r\) are natural numbers and \(\lambda\) is a real number. This problem is the mathematical model of a diffusion process in a medium in which the density \(u= u(x, t)\) depends on the power of \(u\) and \(u_x\). The authors construct explicit solutions to this quasilinear bidegenerate parabolic equation, characterize the behavior as the time increases and give an observation of general solutions for the initial value problem.