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J. Korean Math. Soc. 2019; 56(4): 1017-1029

Online first article April 10, 2019      Printed July 1, 2019

https://doi.org/10.4134/JKMS.j180539

Copyright © The Korean Mathematical Society.

Blow-up and global solutions for some parabolic systems under nonlinear boundary conditions

Limin Guo, Lishan Liu, Yonghong Wu, Yumei Zou

Changzhou Institute of Technology; Curtin University; Curtin University; Shandong University of Science and Technology

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In this paper, blows-up and global solutions for a class of nonlinear divergence form parabolic equations with the abstract form of $(\varrho(u))_{t}$ and time dependent coefficients are considered. The conditions are established for the existence of a solution globally and also the conditions are established for the blow up of the solution at some finite time. Moreover, the lower bound and upper bound of the blow-up time are derived if blow-up occurs.

Keywords: blows-up and global solutions, parabolic equations, nonlinear boundary conditions, time dependent coefficients, abstract form of $(\varrho(u))_{t}$

MSC numbers: 35K55, 35K61

Supported by: This research was supported by the National Natural Science Foundation of China (11871302), the Natural Science Foundation of Shandong Province of China (ZR2014AM034), Changzhou institute of technology research fund (YN1775) and Project of Shandong Province Higher Educational Science and Technology Program (J18KA217). The support from the Australian Research council for the research is also acknowledged.

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