LARGE TIME ASYMPTOTIC BEHAVIORS OF TWO TYPES OF FAST DIFFUSION EQUATIONS

Chuqi Cao (1) , Xingyu Li (2)

1 Yau Mathematical Science Center and Beijing Institute of Mathematical Sciences and Applications
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We consider two types of non linear fast diffusion equations in R^N: (1) External drift type equation with general external potential. It is a natural extension of the harmonic potential case, which has been studied in many papers. In this paper we can prove the large time asymptotic behavior to the stationary state by using entropy methods. (2) Mean-field type equation with the convolution term. The stationary solution is the min- imizer of the free energy functional, which has direct relation with reverse Hardy-Littlewood- Sobolev inequalities. In this paper, we prove that for some special cases, it also exists large time asymptotic behavior to the stationary state.

Mots clés: nonlinear diffusion; mean field equations; free energy; large time asymptotics; Hardy-Poincaré inequality AMS subject classifications: 35K55; 35B40; 35P15; nonlinear fast diffusion; Fisher information: large time asymptotic; Hardy-Poincaré inequality; reverse Hardy-Littlewood-Sobolev inequality AMS subject classifications: 35K55

Domaines: Equations aux dérivées partielles [math.AP]

Chuqi Cao, Xingyu Li. LARGE TIME ASYMPTOTIC BEHAVIORS OF TWO TYPES OF FAST DIFFUSION EQUATIONS. 2021. ⟨hal-02987235v3⟩