Reaction diffusion equation with spatio-temporal delay |
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| Affiliation: | 1. School of Mathematics Physics, University of Science and Technology Beijing, Beijing 100083, PR China;2. Shanxi Agricultural University, Taigu, Shanxi 030801, PR China;1. Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, PR China;2. School of Automation, Southeast University, Nanjing, 210096, PR China;3. Institute of Information and Control Engineering Technology, Nanjing Normal University, Nanjing, Jiangsu 210042, PR China;4. College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing, Jiangsu 210042, PR China;1. College of Science, Shandong University of Science and Technology, Qingdao 266590, PR China;2. Shandong Key Laboratory of Robotics and Intelligent Technology, Qingdao 266590, PR China;1. MOE Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China;2. College of Information Engineering, Tarim University, Alar, Xinjiang 843300, China;3. School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi 710048, China;1. School of Mathematical Sciences and Physics, North China Electric Power University, Beijing 102206, China;2. School of Economics and Management, North China Electric Power University, Beijing 102206, China;1. Université d’Avignon et des Pays de Vaucluse, UMR 1114 EMMAH, F-84018 Avignon Cedex, France;2. UMR 1114 EMMAH, Centre INRA d’Avignon, F-84914 Avignon Cedex 9, France;1. Department of Automatization, Wuhan University, Wuhan 430072, PR China;2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China;3. College of Electrical and Information Engineering, Hunan University, Changsha 410082, PR China |
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| Abstract: | We investigate reaction–diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction–diffusion equation with spatio-temporal delay. Applying this theory to Lotka–Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem’s steady-state solution. |
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| Keywords: | Reaction–diffusion equation Spatio-temporal delay Global solution Upper and lower solution Steady-state solution |
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