Uniform quasi-differentiability of semigroup to nonlinear reaction-diffusion equations with supercritical exponent

A new approach is established to show that the semigroup {S(t)}t≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in Lq (Ω) (2 ≤ q < ∞) with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.