Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semi-group approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation whose potential is divided into a convex part and a non-convex part as well as a semilinear parabolic equation with a nonlinear term involving gradients. Keyword: Doubly nonlinear evolution equation, generalized semiflow, global attractor, reflexive Banach space, generalized Allen-Cahn equation MSC: 37L30, 35B41, 34G25, 35K65 ∗Department of Machinery and Control Systems, School of Systems Engineering, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 3378570 Japan (e-mail: [email protected]). Supported in part by the Shibaura Institute of Technology grant for Project Research (No. 211459 (2006), 211455 (2007)), and the Grant-in-Aid for Young Scientists (B) (No. 19740073), Ministry of Education, Culture, Sports, Science and Technology.