Abstract
It is shown that for scalar dissipative delay-diffusion equationsu t−Δu=f(u(t),u(t−τ)) with a small delay, all solutions are asymptotic to the set of equilibria ast tends to infinity.
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References
Fitzgibbon, W. E. (1978). Semilinear functional differential equations in Banach space.J. Diff. Eq. 29, 1–14.
Friesecke, G. (1992). Exponentially growing solutions for a delay-diffusion equation with negative feedback.J. Diff. Eq. 98, 1–18.
Green, D., and Stech, H. W. (1981). Diffusion and hereditary effects in a class of population models.Diff. Eq. Appl. Ecol. Epidem. Popul. Problems, 19–28.
Hale, J. (1988). Asymptotic behavior of dissipative systems.Math. Surv. Monogr. AMS, Vol. 20.
Hale, J. (1977).Theory of Functional Differential Equations, Springer Applied Mathematical Sciences, Vol. 3.
Henry, D. (1981).Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, Vol. 840.
Hutchinson, G. E. (1948). Circular causal systems in ecology.Ann. N. Y. Acad. Sci. 50, 221–246.
Inoue, A., Miyakawa, T., and Yoshida, K. (1977). Some properties of solutions for semilinear heat equations with time lag.J. Diff. Eq. 24, 383–396.
Lighbourne, J. H., and Rankin, S. M. (1981). Global existence for a delay differential equationJ. Diff. Eq. 40, 186–192.
Lions, J. L. (1969).Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod.
Luckhaus, S. (1986). Global boundedness for a delay differential equation.Trans. AMS 294, 767–774.
Paret-Mallet, J. (1988). Morse decompositions for delay differential equations.J. Diff. Eq. 72, 270–315.
Matano, H. (1979). Asymptotic behavior and stability of solutions of semilinear diffusion equations.Publ. Res. Inst. Math. Sci. 15, 401–458.
Memory, M. (1989). Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion.SIAM J. Math. Anal. 20, 533–546.
Memory, M. (1991). Stable and unstable manifolds for partial functional differential equations.Nonlin. Anal. 16, 131–142.
Mielke, A. (1987). Über maximaleL p-Regularität für Differentialgleichungen in Banachund Hubert-Räumen.Math. Ann.,277, 121–133.
Travis, C. C., and Webb, G. F. (1974). Existence and stability for partial functional differential equations.Trans. AMS 200, 395–418.
Yoshida, K. (1982). The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology.Hiroshima Math. J. 12, 321–348.
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Friesecke, G. Convergence to equilibrium for delay-diffusion equations with small delay. J Dyn Diff Equat 5, 89–103 (1993). https://doi.org/10.1007/BF01063736
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DOI: https://doi.org/10.1007/BF01063736