In this talk, we mainly study asymptotic behavior of solutions to quasilinear parabolic PDEs near self-similar solutions, when the linearized operators of the self-similar solutions are compact and self-adjoint. Then, we talk about the fully nonlinear parabolic equations. We also consider solutions to elliptic PDEs, whose blow-ups or blow-downs are homogeneous functions. Then, we will discuss how the radila parameter in the elliptic case plays the role of time variable in the parabolic case.