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Asymptotic behavior of a class of reaction--diffusion equations with delays. (English) Zbl 1031.35065
The authors study systems of equations of the form $$\gather {\partial u\over\partial t}= \nabla(G(x, u)\circ\nabla u)- Bu+ F(u_t),\quad x\in\Omega,\quad t\ge t_0,\\ {\partial u\over\partial n}\Biggl|_{\partial\Omega}= 0,\quad t\ge t_0,\\ u(t_0+ s,x)= \phi(s,x)\quad -r\le s\le 0,\quad x\in\Omega,\endgather$$ where $u_t(x)= u(t+ s,x)$, $\phi(s, x)\in C([- r,0]\times \Omega;\bbfR^m)$. By $T(t)\phi= u_t(x)$ the semigroup of operators is defined for $\phi\in C^L= C([- r,0], L^2(\Omega))$. The authors prove that the semigroup $T(t)$ possesses a compact global and connected attractor in $C^L$.

35K57Reaction-diffusion equations
35R10Partial functional-differential equations
35B40Asymptotic behavior of solutions of PDE
35B41Attractors (PDE)
Full Text: DOI
[1] Adams, R. S.: Sobolev space. (1975) · Zbl 0314.46030
[2] Bates, P. W.; Lu, K.; Zeng, Z.: Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. amer. Math. soc., 645 (1998) · Zbl 1023.37013
[3] Carpenter, G. A.: A geometric approach to singular perturbation problems with application to nerve impulse equation. J. differential equations 23, No. 3, 355-367 (1977) · Zbl 0341.35007
[4] Dalmasso, R.: Existence and uniqueness of positive solutions of semilinear elliptic systems. Nonlinear anal. 39, 559-568 (2000) · Zbl 0940.35091
[5] Fabrie, P.; Galusinski, C.: Exponential attractor for partially dissipative reaction system. Asymptot. anal. 12, 295-327 (1996) · Zbl 1028.35026
[6] Galusinski, C.: Existence and continuity of uniform exponential attractor of the singularity perturbed Hodgkin--Huxley system. J. differential equations 44, 99-169 (1998) · Zbl 0912.35030
[7] Hale, J. K.: Asymptotic behavior of dissipative systems. (1988) · Zbl 0642.58013
[8] Idezak, D.: Stability in semilinear problems. J. differential equations 162, 64-90 (2000)
[9] Karachalios, N. I.; Stavrakis, N. M.: Existence of a global attractor for semilinear dissipative wave equations on RN. J. differential equations 157, 183-205 (1999) · Zbl 0932.35030
[10] Lasalle, J. P.: The stability of dynamical system. (1976) · Zbl 0364.93002
[11] Li, D. H.: A course in applied functional analysis. (1999)
[12] Li, S.; Xu, D.; Zhao, H.: Stability region of nonlinear integrodifferential equations. Appl. math. Lett. 13, 77-82 (2000) · Zbl 0974.45006
[13] Liao, X. X.; Fu, Y. L.; Gao, J.; Zhao, X. Q.: Stability of Hopfield neural networks with reaction--diffusion terms. Acta electron. Sinica 28, 78-80 (2000)
[14] Liu, X.; Xu, D. Y.: Uniform asymptotic of abstract functional differential equations. J. math. Anal. appl. 216, 616-624 (1997)
[15] Marion, M.: Attractor for reaction--diffusion equation. Existence and estimate of their dimension. Appl. anal. 25, 101-147 (1987) · Zbl 0609.35009
[16] Martin, R. H.; Smith, H. L.: Abstract functional differential equations and reaction--diffusion system. Trans. amer. Math. soc. 321, 1-44 (1990) · Zbl 0722.35046
[17] Temam, R.: Infinity-dimensional dynamical systems in mechanics and physics. (1997) · Zbl 0871.35001
[18] Travis, C. C.; Webb, G. F.: Existence and stability for partial functional differential equations. Trans. amer. Math. soc. 200, 395-418 (1974) · Zbl 0299.35085
[19] Wu, J.: Theory and application of partial functional differential equations. (1996) · Zbl 0870.35116
[20] Xu, D. Y.: Global attractor of reaction--diffusion equations with delays. Proc. int. Diff. eqs. Comp. simulations, 371-377 (2000) · Zbl 0958.35143
[21] Xu, D. Y.; Xu, A.: Domain of attraction of nonlinear difference systems. Chinese sci. Bull. 44, 121-123 (1999) · Zbl 1006.39007
[22] Zhou, S. F.: Dimension of the global attractor for discretization of damped sine-Gordon equation. Appl. math. Lett. 12, 95-100 (1999) · Zbl 0935.35145