On $ L^{\infty } $-estimates and the structure of the global attractor for weak solutions of reaction-diffusion equations

In this paper, we study the structure of the global attractor for weak and regular solutions of a problem governed by a scalar semilinear reaction-diffusion equation with a non-regular nonlinearity, such that uniqueness of solutions can fail to happen. First, using the Moser–Alikakos iterations we obtain the estimates of the weak solutions in the space $ L^{\infty }(\Omega) $. After that, using these estimates we improve the existing results on the structure of the attractor. Finally, estimates of the Hausdorff and fractal dimension of the attractor are obtained.