On the continuity of solutions to doubly singular parabolic equations

  • Qifan LiEmail author


This paper considers certain quasilinear parabolic equations with one singularity which occurs in the time derivative. A model equation is
$$\begin{aligned} \partial _t\beta (u)-{\text {div}}|Du|^{p-2}Du\ni 0, \end{aligned}$$
where \(\beta :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is monotonically increasing and discontinuous at the origin. We show that bounded weak solutions are locally continuous in the range
$$\begin{aligned} 2-\varepsilon _0\le p<2, \end{aligned}$$
provided that \(\varepsilon _0>0\) is sufficiently small, and the continuity is stable as \(p\rightarrow 2\).


Two-phase Stefan problem Singular parabolic equations Phase transition 

Mathematics Subject Classification

Primary 35R05 35R35 35D30 Secondary 35K59 35K92 



The author wishes to thank Eurica Henriques, Peter Lindqvist, Irina Markina and José Miguel Urbano for the valuable discussions. The author would also like to thank the referees for providing helpful comments to improve the manuscript.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of SciencesWuhan University of TechnologyWuhanPeople’s Republic of China

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