In this paper, a kind of heat conduction problem subject to a moving boundary condition is studied, in which its equation has form of quasilinear parabolic e-quation with vaviable coefficients, under some assumptions, we have defined and provided existence of solution of this problem in domain Ω by singular perturbation methods, where Ω denotes the domain:{σ1(t)≤x≤ σ2(t), 0≤t |