Abstract
Recently, we have investigated the global existence in time and asymptotic profile of solutions of some nonlinear evolution equations with strong dissipation and proliferation arising in mathematical biology. In this chapter, we improve the asymptotic behaviour of the solution to a simpler equation so that its derivative with respect to t converges exponentially to a constant steady state. We apply our result to a chemotaxis model and show the global existence in time and such exponential convergence property of the solution.
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Acknowledgements
We would like to express our sincere gratitude to Professor J.I. Tello for his inestimable suggestions and comments. Also, we are very grateful to thank Professor H. Takamura for his valuable advice. We would like to thank the referee for fruitful suggestions. This work was supported in part by the Grant-in-Aide for Scientific Research (C) 16540176, 19540200, 22540208, 25400148, and 16K05214 from Japan Society for the Promotion of Science.
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Kubo, A., Hoshino, H. (2019). Nonlinear Evolution Equations and Their Application to Chemotaxis Models. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_32
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DOI: https://doi.org/10.1007/978-3-030-04459-6_32
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