Global existence and blow-up for the fast diffusion equation with a memory boundary condition. (English) Zbl 1336.35080
Summary: 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.
MSC:
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35K59 | Quasilinear parabolic equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
Keywords:
fast diffusion equation; global existence; finite time blow-up; memory boundary condition; tumor-induced angiogenesisReferences:
| [1] | Anderson, Jeffrey R., Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations, 16, 1, 105-143 (1991) · Zbl 0738.35033 · doi:10.1080/03605309108820753 |
| [2] | Anderson, Jeffrey R.; Deng, Keng, Global solvability for the porous medium equation with boundary flux governed by nonlinear memory, J. Math. Anal. Appl., 423, 2, 1183-1202 (2015) · Zbl 1382.35140 · doi:10.1016/j.jmaa.2014.10.041 |
| [3] | [[ADW]] Jeffrey R. Anderson, Keng Deng, and Qian Wang, Global behavior of solutions to the fast diffusion equation with boundary flux governed by memory, preprint. · Zbl 1355.35109 |
| [4] | Filo, J{\'a}n, Diffusivity versus absorption through the boundary, J. Differential Equations, 99, 2, 281-305 (1992) · Zbl 0761.35048 · doi:10.1016/0022-0396(92)90024-H |
| [5] | [[Levine]] 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. · Zbl 1323.92029 |
| [6] | Wang, Mingxin, The long-time behavior of solutions to a class of quasilinear parabolic equations with nonlinear boundary conditions, Acta Math. Sinica (Chin. Ser.), 39, 1, 118-124 (1996) · Zbl 0880.35059 |
| [7] | Wolanski, Noem{\'{\i }}, Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal., 24, 2, 317-326 (1993) · Zbl 0778.35047 · doi:10.1137/0524021 |
| [8] | [[Wu]] Yonghui Wu, Existence and behavior of solutions of nonlinear evolution equations, Doctoral Dissertation, Institute of System, Chinese Academy of Sciences, 1993. \endbiblist |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
