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Global asymptotic stability in a class of reaction-diffusion equations with time delay. (English) Zbl 1469.35121

Summary: We study a very general class of delayed reaction-diffusion equations in which the reaction term can be nonmonotone and spatially nonlocal. By using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the positive steady state to the equations subject to the Neumann boundary condition.

MSC:

35K57 Reaction-diffusion equations
35B35 Stability in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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[1] Gourley, S. A.; Wu, J.; Brunner, H.; Zhao, X.-Q.; Zou, X., Delayed non-local diffusive systems in biological invasion and disease spread, Nonlinear Dynamics and Evolution Equations. Nonlinear Dynamics and Evolution Equations, Fields Institute Communications, 48, 137-200 (2006), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 1130.35127
[2] Huang, C.; Yang, Z.; Yi, T.; Zou, X., On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, Journal of Differential Equations, 256, 7, 2101-2114 (2014) · Zbl 1297.34084 · doi:10.1016/j.jde.2013.12.015
[3] Liang, D.; So, J. W.-H.; Zhang, F.; Zou, X., Population dynamic models with nonlocal delay on bounded domains and their numerical computations, Differential Equations and Dynamical Systems, 11, 1-2, 117-139 (2003) · Zbl 1231.35287
[4] So, J. W.-H.; Wu, J.; Yang, Y., Numerical steady state and Hopf bifurcation analysis on the diffusive Nicholson’s blowflies equation, Applied Mathematics and Computation, 111, 1, 33-51 (2000) · Zbl 1023.65108 · doi:10.1016/S0096-3003(99)00047-8
[5] So, J. W.-H.; Wu, J.; Zou, X., A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains, The Royal Society of London Proceedings A: Mathematical, Physical and Engineering Sciences, 457, 2012, 1841-1853 (2001) · Zbl 0999.92029 · doi:10.1098/rspa.2001.0789
[6] Thieme, H. R.; Zhao, X.-Q., Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, Journal of Differential Equations, 195, 2, 430-470 (2003) · Zbl 1045.45009 · doi:10.1016/S0022-0396(03)00175-X
[7] Yi, T.; Zou, X., Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case, Journal of Differential Equations, 245, 11, 3376-3388 (2008) · Zbl 1152.35511 · doi:10.1016/j.jde.2008.03.007
[8] Yi, T.; Zou, X., Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, Journal of Differential Equations, 251, 9, 2598-2611 (2011) · Zbl 1231.35285 · doi:10.1016/j.jde.2011.04.027
[9] Yi, T.; Zou, X., On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, Journal of Dynamics and Differential Equations, 25, 4, 959-979 (2013) · Zbl 1319.35095 · doi:10.1007/s10884-013-9324-3
[10] Zhao, X.-Q., Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Canadian Applied Mathematics Quarterly, 17, 1, 271-281 (2009) · Zbl 1213.35119
[11] Gourley, S. A.; Kuang, Y., Wavefronts and global stability in a time-delayed population model with stage structure, The Royal Society of London Proceedings A: Mathematical, Physical and Engineering Sciences, 459, 2034, 1563-1579 (2003) · Zbl 1047.92037 · doi:10.1098/rspa.2002.1094
[12] Xu, D.; Zhao, X.-Q., A nonlocal reaction-diffusion population model with stage structure, Canadian Applied Mathematics Quarterly, 11, 3, 303-319 (2003) · Zbl 1079.92055
[13] Evans, L. C., Partial Differential Equations. Partial Differential Equations, Graduate Studies in Mathematics, 19 (1998), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0902.35002
[14] Thieme, H. R.; Zhao, X.-Q., A non-local delayed and diffusive predator-prey model, Nonlinear Analysis: Real World Applications, 2, 2, 145-160 (2001) · Zbl 1113.92319 · doi:10.1016/S0362-546X(00)00112-7
[15] Yau, S. T.; Schoen, R., Lectures on Differential Geometry (1994), Beijing, China: Higher Education Press, Beijing, China · Zbl 0830.53001
[16] Smith, H. L., Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41 (1995), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 0821.34003
[17] Martin,, R. H.; Smith, H. L., Abstract functional-differential equations and reaction-diffusion systems, Transactions of the American Mathematical Society, 321, 1, 1-44 (1990) · Zbl 0722.35046 · doi:10.2307/2001590
[18] Wu, J., Theory and Applications of Partial Functional-Differential Equations. Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences, 119 (1996), New York, NY, USA: Springer, New York, NY, USA · Zbl 0870.35116 · doi:10.1007/978-1-4612-4050-1
[19] Thieme, H. R., Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread, Journal of Mathematical Biology, 8, 2, 173-187 (1979) · Zbl 0417.92022 · doi:10.1007/BF00279720
[20] Thieme, H. R., On a class of Hammerstein integral equations, Manuscripta Mathematica, 29, 1, 49-84 (1979) · Zbl 0417.45003 · doi:10.1007/BF01309313
[21] Smith, H., An Introduction to Delay Differential Equations with Applications to the Life Sciences. An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57 (2011), New York, NY, USA: Springer, New York, NY, USA · Zbl 1227.34001 · doi:10.1007/978-1-4419-7646-8
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