Traveling wave solutions for degenerate nonlinear parabolic equations

A refined asymptotic behavior of traveling wave solutions for degenerate nonlinear parabolic equations

“…In [5], a result on the whole dynamics on the phase space R 2 including infinity generated by the twodimensional ordinary differential equation (ODE for short) (2) is obtained by applying the dynamical system approach and the Poincaré compactification (for instance, see [6] for the details of the Poincaré compactification). Further, connecting orbits on it are focused and several results on the existence of (weak) traveling wave solutions are given.…”

“…Further, connecting orbits on it are focused and several results on the existence of (weak) traveling wave solutions are given. The following theorem is one of main results obtained in [5]. Theorem 1 ([5,Theorem 3]) Assume that p ∈ 2N and δ = 1.…”

“…The following theorem is one of main results obtained in [5]. Theorem 1 ([5,Theorem 3]) Assume that p ∈ 2N and δ = 1. Then, for a given positive constant c, the equation (1) has a family of traveling wave solutions (which correspond to a family of the orbits of (2)).…”

“…1 shows dynamics on the Poincaré disk of (2) (see [6] for the definition of the Poincaré disk). In addition, the asymptotic behavior of the traveling wave solutions (obtained in Theorem 1) for ξ → +∞ is also given in [5], while the asymptotic behavior as ξ → −∞ is not obtained there.…”

“…Our argument here is based on an asymptotic study of solutions in the different form from that provided in e.g. [5,7], which can be applied to asymptotic analysis towards further applications in various phenomena including their numerical calculations.…”