A nonlinear Stefan problem with variable exponent and different moving parameters

Abstract

In this paper, we consider a nonlinear diffusion problem with variable exponent, accompanied by double free boundaries possessing different moving parameters, where the variable exponent function m(x) satisfies that m(x) − 1 can change its sign. Local existence and uniqueness of solution are established firstly, and then, some sufficient conditions are achieved for finite time blowup, and as well for global existence. Asymptotic behavior is further investigated for global solution, and existences of fast solution and slow solution are presented by making use of upper-sub solutions, energy and scaling arguments.