Abstract: In this paper, we mainly discuss long time behavior of the strong solution for nonlinearity reaction diffusion equation with fading memory. First of all, by the regularity of solutions and Control convergence principle, we prove that the semigroup of the solutions is a $ H_0^1(\Omega)\times L_\mu^2(\mathbb{R}; D(A))-$ Contractive semigroup. From this, the asymptotic compactness of the semigroup is obtained; Then the existence and regularity of global attractor $\mathcal{A}$ are proved on the product space. It is noteworthy that the nonlinearity $f$ satisfies the polynomial growth of arbitrary order and $\mathcal{A}\subset D(A)\times L_\mu^2(\mathbb{R}; D(A))$.

Key words: Reaction diffusion equation, Control convergence principle, Contractive semigroup, Arbitrary polynomial growth, Global attractor