Global existence and blow-up for the fast diffusion equation with a memory boundary condition
Authors:
Keng Deng and Qian Wang
Journal:
Quart. Appl. Math. 74 (2016), 189-199
MSC (2010):
Primary 35A01, 35B44, 35K59
DOI:
https://doi.org/10.1090/qam/1425
Published electronically:
December 7, 2015
MathSciNet review:
3472525
Full-text PDF Free Access
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References |
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Additional Information
Abstract: In this paper, we study the long-time behavior of solutions to the fast diffusion equation with a memory boundary condition. The problem corresponds to a model introduced in previous studies of tumor-induced angiogenesis. We establish global existence and finite time blow-up results for the problem.
References
- Jeffrey R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), no. 1, 105–143. MR 1096835, DOI 10.1080/03605309108820753
- Jeffrey R. Anderson and Keng Deng, Global solvability for the porous medium equation with boundary flux governed by nonlinear memory, J. Math. Anal. Appl. 423 (2015), no. 2, 1183–1202. MR 3278193, DOI 10.1016/j.jmaa.2014.10.041
- Jeffrey R. Anderson, Keng Deng, and Qian Wang, Global behavior of solutions to the fast diffusion equation with boundary flux governed by memory, preprint.
- Ján Filo, Diffusivity versus absorption through the boundary, J. Differential Equations 99 (1992), no. 2, 281–305. MR 1184057, DOI 10.1016/0022-0396(92)90024-H
- Howard A. Levine, Serdal Pamuk, Brian D. Sleeman, and Marit Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma, Bull. Math. Biol. 63 (2001), 801-863.
- Ming Xin Wang, The long-time behavior of solutions to a class of quasilinear parabolic equations with nonlinear boundary conditions, Acta Math. Sinica (Chin. Ser.) 39 (1996), no. 1, 118–124 (Chinese, with English and Chinese summaries). MR 1412913
- Noemí Wolanski, Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), no. 2, 317–326. MR 1205529, DOI 10.1137/0524021
- Yonghui Wu, Existence and behavior of solutions of nonlinear evolution equations, Doctoral Dissertation, Institute of System, Chinese Academy of Sciences, 1993.
References
- Jeffrey R. Anderson, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), no. 1, 105–143. MR 1096835 (92d:35163), DOI 10.1080/03605309108820753
- Jeffrey R. Anderson and Keng Deng, Global solvability for the porous medium equation with boundary flux governed by nonlinear memory, J. Math. Anal. Appl. 423 (2015), no. 2, 1183–1202. MR 3278193, DOI 10.1016/j.jmaa.2014.10.041
- Jeffrey R. Anderson, Keng Deng, and Qian Wang, Global behavior of solutions to the fast diffusion equation with boundary flux governed by memory, preprint.
- Ján Filo, Diffusivity versus absorption through the boundary, J. Differential Equations 99 (1992), no. 2, 281–305. MR 1184057 (94d:35083), DOI 10.1016/0022-0396(92)90024-H
- Howard A. Levine, Serdal Pamuk, Brian D. Sleeman, and Marit Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma, Bull. Math. Biol. 63 (2001), 801-863.
- Mingxin Wang, The long-time behavior of solutions to a class of quasilinear parabolic equations with nonlinear boundary conditions, Acta Math. Sinica (Chin. Ser.) 39 (1996), no. 1, 118–124 (Chinese, with English and Chinese summaries). MR 1412913 (97h:35118)
- Noemí Wolanski, Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), no. 2, 317–326. MR 1205529 (93j:35023), DOI 10.1137/0524021
- Yonghui Wu, Existence and behavior of solutions of nonlinear evolution equations, Doctoral Dissertation, Institute of System, Chinese Academy of Sciences, 1993.
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Additional Information
Keng Deng
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Louisiana 70504
MR Author ID:
225222
Email:
deng@louisiana.edu
Qian Wang
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Louisiana 70504
Email:
qxw3519@louisiana.edu
Keywords:
Fast diffusion equation,
global existence,
finite time blow-up,
memory boundary condition.
Received by editor(s):
November 12, 2014
Published electronically:
December 7, 2015
Article copyright:
© Copyright 2015
Brown University
