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Korean Mathematical Society JKMS

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J. Korean Math. Soc. 2018; 55(3): 531-551

Online first article February 9, 2018      Printed May 1, 2018

https://doi.org/10.4134/JKMS.j170233

Copyright © The Korean Mathematical Society.

Global attractors for nonlocal parabolic equations with a new class of nonlinearities

Cung The Anh, Le Tran Tinh, Vu Manh Toi

Hanoi National University of Education, Hong Duc University, Thuyloi University

Abstract

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In this paper we consider a class of nonlocal parabolic equations in bounded domains with Dirichlet boundary conditions and a new class of nonlinearities. We first prove the existence and uniqueness of weak solutions by using the compactness method. Then we study the existence and fractal dimension estimates of the global attractor for the continuous semigroup generated by the problem. We also prove the existence of stationary solutions and give a sufficient condition for the uniqueness and global exponential stability of the stationary solution. The main novelty of the obtained results is that no restriction is imposed on the upper growth of the nonlinearities.

Keywords: nonlocal parabolic equation, weak solution, global attractor, fractal dimension, stability, exponential nonlinearity

MSC numbers: 35B41, 35D30, 35K65

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