Traveling wave solutions for degenerate nonlinear parabolic equations

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Abstract

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.

Original languageEnglish
JournalJournal of Elliptic and Parabolic Equations
DOIs
Publication statusAccepted/In press - 2020

Keywords

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

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