Quasilinear abstract parabolic evolution equations and exponential attractors

The Exponential attractor, one of notions of limit set in infi nite-dimensional dynamical systems, is known to have strong robustness and is kn own to be constructed under a simple compact smoothing condition. In this paper, w e study a dynamical system determined from the Cauchy problem for a quasilin ear abstract parabolic evolution equation. We give a general strategy for construc ting the exponential attractor and apply the abstract result to a chemotaxis-growt h system in non smooth domain.

[1]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[2]  Hiroki Tanabe,et al.  Functional analytic methods for partial differential equations , 1996 .

[3]  A. Yagi,et al.  Global Stability of Approximation for Exponential Attractors , 2004 .

[4]  Alain Miranville,et al.  The dimension of the global attractor for dissipative reaction-diffusion systems , 2003, Appl. Math. Lett..

[5]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[6]  Atsushi Yagi,et al.  Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems , 2002 .

[7]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[8]  Tohru Tsujikawa,et al.  Exponential attractor for a chemotaxis-growth system of equations , 2002 .

[9]  Atsushi Yagi,et al.  Full discrete approximations by Galerkin method for chemotaxis-growth model , 2001 .

[10]  James C. Robinson.  Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors , 2001 .

[11]  Tomasz W. Dłotko,et al.  Global Attractors in Abstract Parabolic Problems , 2000 .

[12]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[13]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[14]  A. Yagi.  Quasilinear Abstract Parabolic Evolution Equations with Applications , 2002 .

[15]  Sergey Zelik,et al.  Exponential attractors for a nonlinear reaction-diffusion system in ? , 2000 .

[16]  Tohru Tsujikawa,et al.  EXPONENTIAL ATTRACTOR FOR AN ADSORBATE-INDUCED PHASE TRANSITION MODEL , 2002 .

[17]  S. Krein,et al.  Linear Differential Equations in Banach Space , 1972 .

[18]  Alessandra Lunardi,et al.  Abstract quasilinear parabolic equations , 1984 .