In this paper, we first prove the existence of weak solutions of the initial and boundary value problem for ut=udiv(|∇u|p-2∇u)+γ|∇u|p with p⩾2, γ∈(0,1), and then discuss the asymptotic behavior of weak solutions as γ→0+.

This paper is concerned with a class of quasilinear parabolic equations with singularity and arbitrary degeneracy. The existence and uniqueness of generalized solutions to a kind of boundary value problem is established.

In this paper, we are concerned with the time periodic solutions to the evolutionp-Laplacian equations of the form[formula]and[formula]with the Dirichlet boundary value condition, wherep≥2, ω>0,Ω is a bounded convex domain inRNwith smooth boundary ∂Ω,m(x,t) is continuous onΩ×R, periodic intwith period ω and positive in Ω×R. The existence of nontrivial nonnegative periodic solutions is established provided thatN>1 andp−1<α<p−1+p/N.

: This paper is concerned with a free boundary problem which arises in the study of a class of degenerate parabolic equations. A thorough treatment is given for a special case which can be reduced to a problem in ordinary differential equations upon introduction of the appropriate similarity variable. Beyond its inherent interest, the solvability of the resulting problem establishes that an analysis by Vol'pert and Hudjaev of jump conditions satisfied by solutions of degenerate parabolic equations is not correct in general. (Author)

In this paper,we study the first boundary value problem for quasilinear equations ofthe formUnder certain conditions,the existence of generalized solutions in BV is proved by meansof the method of parabolic regularization.To do this,we need some estimates on the family{u ε }of solutions of regularized problems and the most difficult step is to estimate|grad u ε |L 1+ In addition,some results on the uniqueness and stability of generalized solutions areestablished.