Asymptotic regularity and attractors of the reaction–diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction–diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term ff is dissipative and the boundary nonlinear term gg is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H1(Ω)H1(Ω) with dissipative internal and boundary nonlinearities.