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This open access book deals with a rich variety of taxis-type cross-diffusive equations. Particularly, it intends to show the key role played by quasi-energy inequality in the derivation of some necessary a  priori estimates. This book addresses applied mathematics and all researchers interested in mathematical development of reaction-diffusion theory and its application and can be a basis for a graduate course in applied mathematics.

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Yuanyuan Ke is employed as the professor and doctoral supervisor of School of Mathematics, Renmin University of China. She is mainly engaged in the theoretical research and application of reaction–diffusion equation. She won Beijing Higher Education Young Elite Teacher Project, the Excellent Teacher Award of Baosteel and the Research Award Program for Outstanding Young Teachers in Higher Education Institutions in Beijing. She presided over 8 high-level scientific research projects, 2 educational reform projects and published 50 scientific research papers. Jing Li is employed as the associate professor and master supervisor of College of Science, Minzu University of China. She was also employed as the High-level Young Talent of Minzu University of China from 2019 to 2022. Her research interests are the relevance of chemotaxis in reaction–diffusion processes and mathematical theory of nonlocal problem in biology. She presided over three National Natural Science Foundation projects (NNSF) and published more than 20 scientific research papers. Yifu Wang is employed as the professor of school of Mathematics and Statistics, Beijing Institute of Technology, P.R. of China. His research is  concerned mainly with  qualitative analysis of nonlinear  parabolic equations, such as chemotactic cross-diffusion system in biology and ecology. He has been the supervisor of several master's degree students and Ph.D. students. The research is supported by the National Natural Science Foundation of China (NNSF) and program for new century excellent talents in university (NCET). He has published 80 scientific research papers.

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