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

Part of: Parabolic equations and systems Partial differential equations

Published online by Cambridge University Press:  17 January 2019

Nguyen Anh Dao ,
Jesus Ildefonso Díaz  and
Huynh Van Kha
Nguyen Anh Dao
Affiliation:
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam (daonguyenanh@tdtu.edu.vn)
Jesus Ildefonso Díaz
Affiliation:
Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain (jidiaz@ucm.es)
Huynh Van Kha
Affiliation:
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam (huynhvankha@tdtu.edu.vn)
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Abstract

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.

Keywords

gradient estimatesquenching type of parabolic equationsirregular initial datumfree boundaryinstantaneous shrinking of support

MSC classification

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35K65: Degenerate parabolic equations 35B99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1323 - 1346
Copyright
Copyright © Royal Society of Edinburgh 2019 

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