Complete quenching phenomenon and instantaneous shrinking of support of solutions of degenerate parabolic equations with nonlinear singular absorption

This paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as...

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Bibliographic Details
Published inProceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 149; no. 5; pp. 1323 - 1346
Main Authors Dao, Nguyen Anh, Díaz, Jesus Ildefonso, Kha, Huynh Van
Format Journal Article
LanguageEnglish
Published Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.10.2019
Cambridge University Press
Subjects
Boundary conditions
Cauchy problems
Dirichlet problem
Mathematical analysis
Nonlinear equations
Quenching
Primary 35K55
free boundary
instantaneous shrinking of support
quenching type of parabolic equations
35B99
gradient estimates
irregular initial datum
35K65
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Summary:This paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2018.68