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Attractors in $L^2(\bbfR^N)$ for a class of reaction-diffusion equations. (English) Zbl 1221.35203
The authors study the class of nonlinear reaction-diffusion equations: $$ \align &\frac{\partial u}{\partial t} = b\Delta u - cu - f(u) - a(x)h(u) + g(x), \quad x \in \Bbb R^N,\\ &u(x,0) = u_0(x). \endalign $$ They prove the existence of a global attractor $A$ in $L^2(\Bbb R^N)$ for the semigroup associated to this problem; $A$ is compact, invariant and attracts every bounded subsets of $L^2(\Bbb R^N)$. The proof relies on results of {\it Q.-F. Ma, S.-H. Wang} and {\it C.-K. Zhong} [Indiana Univ. Math. J. 51, No.,6, 1542--1558 (2002; Zbl 1028.37047)].

35K57Reaction-diffusion equations
35B41Attractors (PDE)
47H20Semigroups of nonlinear operators
34C20Transformation and reduction of ODE and systems, normal forms
34D20Stability of ODE
Full Text: DOI
[1] Babin, A. V.; Vishik, M. I.: Attractors of partial differential equations in an unbounded domain. Proc. roy. Soc. Edinburgh 116A, 221-243 (1990) · Zbl 0721.35029
[2] Efendiev, M. A.; Zelik, S. V.: The attractor for a nonlinear reaction--diffusion system in an unbounded domain. Commun. pure appl. Math. 54, 0625-0688 (2001)
[3] . Physica D 128, 41-52 (1999)
[4] Robinson, C.: Infinite-dimensional dynamical systems: an introduction to dissipative parabolic pdes and the theory of global attractors. (2001) · Zbl 0980.35001
[5] Ma, Q.; Wang, S.; Zhong, C.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana math. J. 5, No. 6, 1542-1558 (2002)
[6] Zhong, C. K.; Yang, M. H.; Sun, C. Y.: The existence of global attractors for the norm-to-weak continuous semigroup and applications to the nonlinear reaction--diffusion equatoins. J. differential equations 223, 367-399 (2006) · Zbl 1101.35022
[7] Zhang, Yanhong; Mu, Lugui: Existence of the solution to a class of nonlinear reaction--diffusion equation in RN. J. gansu normal college 12, No. 2, 1-4 (2007)