Attractors of the non-autonomous reaction-diffusion equation with nonlinear boundary condition

In this paper, we study the long-time behavior of the non-autonomous reaction-diffusion equation with nonlinear boundary condition and competing nonlinearities. Under the balance conditions between internal and boundary nonlinear terms, which have been proved in Rodríguez-Bernal and Tajdine (2001) [21] to prevent solution from blow-up, we prove the existence of a compact uniform attractor in Lp+1(Ω)Lp+1(Ω) where p>1p>1 is the growing exponent of internal nonlinearity.