On the large time behavior of solutions for some degenerate quasilinear parabolic systems (Q2367030)

We consider the large time behavior of the solutions for the following Cauchy problem: \[ u_ t= (u^ m)_{xx}- v^ n u^ n, \quad v_ t= (v^ m)_{xx}- u^ n v^ n \qquad \text{in} \quad \mathbb{R}\times (0,\infty) \tag{1} \] with initial conditions (2) \(u(\cdot, 0)= u_ 0\) and \(v(\cdot, 0)= v_ 0\) on \(\mathbb{R}\). Here, \(m>1\) and \(n\geq 1\) are real numbers. If \(u_ 0\equiv v_ 0\) on \(\mathbb{R}\), the solutions \(u\) and \(v\) of (1) and (2) would coincide in \(\mathbb{R}\times [0,\infty)\) and satisfy the following Cauchy problem with \(p=2n\): \[ (3) \quad u_ t= (u^ m)_{xx} -u^ p \text{ in } \mathbb{R}\times (0,\infty), \qquad (4) \quad u(\cdot, 0)= u_ 0 \text{ on } \mathbb{R}. \] The purpose of this paper is to investigate whether the large time behavior of the solutions for the system (1) differs from the behavior of the solutions for the equation (3).