Properties Of The Global Attractors For The Variable Parameter Diffusion Equation With Fading Memory

In this article, we mainly studied the non-classical diffusion equation with fading memory. The specific study of two questions:i) We study the initial value problems of the non-classical diffusion equation with variable parameters and fading memory: where a≥0. By using semi-group theory, we obtain the continuity of solutions in the weak topological space, when the nonlinearity is critical and the forcing term only belongs to H-1(Q). Therefore, upper semi-continuity of the global attractors for the variable parameter diffusion equation with fading memory is obtained. The results are the improvements and extensions of the results of paper.ii) We study the following a periodic boundary value problem for a non-classical diffusion equation with memory: where uo(x, t)is a known periodic function in x∈R3for t≤0, and we assume L=(L1,L2,L3). By using semi-group theory and fractal dimension theory, we obtain the fractal dimension of the global attractor is finite when the nonlinearity is critical and the forcing term only belongs to C2(R,R). The results are the improvements and extensions of the results of paper.