## Abstract

We consider the traveling wave solutions of the degenerate nonlinear parabolic equation u_{t}= u^{p}(u_{xx}+ u) which arises in the model of heat combustion, solar flares in astrophysics, plane curve evolution problems and the resistive diffusion of a force-free magnetic field in a plasma confined between two walls. We also deal with the equation v_{τ}= v^{p}(v_{xx}+ v- v^{-}^{p}^{+}^{1}) related with it. We first give a result on the whole dynamics on the phase space R^{2} with including infinity about two-dimensional ordinary differential equation that introduced the traveling wave coordinates: ξ= x- ct by applying the Poincaré compactification and dynamical system approach. Second, we focus on the connecting orbits on it and give a result on the existence of the weak traveling wave solutions with quenching for c> 0 and p∈ 2 N. Moreover, we give the detailed information about the asymptotic behavior of u(ξ) , u^{′}(ξ) , v(ξ) and v^{′}(ξ) for p∈ 2 N. In the case that p∈ 2 N+ 1 , it is too complicated to determine the dynamics near the singularities on the Poincaré disk, however, we classify the connecting orbits and corresponding traveling wave solutions and obtain their asymptotic behavior.

Original language | English |
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Journal | Journal of Elliptic and Parabolic Equations |

DOIs | |

Publication status | Accepted/In press - 2020 |

## Keywords

- Asymptotic behavior
- Degenerate nonlinear parabolic equation
- Desingularization of vector fields (blow-up)
- Poincaré compactification
- Weak traveling wave solution with quenching