We consider the traveling wave solutions of the degenerate nonlinear parabolic equation ut= up(uxx+ 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 vτ= vp(vxx+ v- v-p+1) related with it. We first give a result on the whole dynamics on the phase space R2 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.
- Asymptotic behavior
- Degenerate nonlinear parabolic equation
- Desingularization of vector fields (blow-up)
- Poincaré compactification
- Weak traveling wave solution with quenching