# American Institute of Mathematical Sciences

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July  2018, 17(4): 1613-1632. doi: 10.3934/cpaa.2018077

## $L^∞$-energy method for a parabolic system with convection and hysteresis effect

 1 Hikari Ltd, P.O. Box 85 Ruse 7000, Bulgaria 2 Department of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

* Corresponding author

Received  March 2017 Revised  October 2017 Published  April 2018

Fund Project: The second author is supported by the Grant-in-Aid for Scientific Research #15K13451, the Ministry of Education, Culture, Sports, Science, and Technology, Japan.

The $L^∞$-energy method is developed so as to handle nonlinear parabolic systems with convection and hysteresis effect. The system under consideration originates from a biological model where the hysteresis and convective effects are taken into account in the evolution of species. Some results for the existence of local and global solutions as well as the uniqueness of solution are presented.

Citation: Emil Minchev, Mitsuharu Ôtani. $L^∞$-energy method for a parabolic system with convection and hysteresis effect. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1613-1632. doi: 10.3934/cpaa.2018077
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