Asymptotic behavior of multi-term time fractional nonlocal diffusion equations

We study a class of linear nonlocal diffusion equations with multi-term Caputo time-fractional derivatives in the whole space RN. The model combines temporal memory effects with a convolution-type nonlocal diffusion operator. Using Fourier and Laplace transform methods, we obtain a representation formula for the corresponding Fourier mild solutions in terms of a multi-term relaxation function. We then investigate the large-time behavior of the solutions by comparison with an associated fractional-Laplacian problem. Under suitable assumptions on the low-frequency behavior of the convolution kernel, we show that the difference between the nonlocal solution and its fractional-Laplacian counterpart is of lower order under the natural large-time scaling determined by the smallest fractional order. We also establish convergence for rescaled nonlocal problems and extend the analysis to higher-order nonlocal diffusion operators.