Communications of the
Korean Mathematical Society CKMS

In this paper we consider a class of nonlinear nonlocal parabolic equations involving $p$-Laplacian operator where the nonlocal quantity is present in the diffusion coefficient which depends on $L^p$-norm of the gradient and the nonlinear term is of polynomial type. We first prove the existence and uniqueness of weak solutions by combining the compactness method and the monotonicity method. Then we study the existence of global attractors in various spaces for the continuous semigroup generated by the problem. Finally, we investigate the existence and exponential stability of weak stationary solutions to the problem.

Keywords: nonlocal parabolic equation, weak solution, global attractor, nonlinearity of polynomial type

MSC numbers: 35B41, 35D30, 35K65