Continuity of Weak Solutions for Quasilinear Parabolic Equations with Strong Degeneracy

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form U_t-Δφ(u)=O, whereφ■C~1(R~1)is a strictly monotone increasing function.Clearly,the above equation has strong degeneracy,i.e.,the set of zero points ofφ′(·)is permitted to have zero measure. This is an answer to an open problem in13,p.288].

Continuity of Weak Solutions for Quasilinear Parabolic Equations with Strong Degeneracy

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form u _t − Δ∅(u) = 0, where ∅ ∈ C ¹(ℝ¹) is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of ∅'( · ) is permitted to have zero measure. This is an answer to an open problem in 13, p. 288]. *Project supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE (No. 2000]26), the 973 Project of the Ministry of Science and Technology of China (No. 2006CB805902), the National Natural Science Foundation of China (No. 10571072), the Key Laboratory of Symbolic Computation and Knowledge Engineering of the Ministry of Education of China and the 985 Project of Jilin University.