New results for convergence problem of fractional diffusion equations when order approach to $1^-$

This work studies the convergence problem for a class of fractional diffusion equations in which the time-derivative order approaches $1^-$. Up to now, few works have investigated this topic. The purpose of the article consists of three main contents. The first result is related to the convergence of the Caputo derivative and the Mittag-Leffler operators when $\alpha \to 1^-$. The second is to investigate the convergence problem for a linear fractional diffusion equation on $L^p$ spaces. And last result is concerned with the convergence problem for nonlinear fractional diffusion equations. The main analysis and techniques of the paper involve the evaluation related to Riemann-Liouville integration, Caputo derivative and Sobolev embeddings. Our analysis provides a complete and detailed answer to the convergence problem as fractional order tends to $1^-$.