In the current study, we explore asymptotic behaviors for the Cauchy problems of fractional diffusion equations with the Hilfer derivative of order \(0<\alpha <1\) and \(1<\alpha <2\) and of type \(0\le \beta \le 1\) in time, and the integral fractional Laplacian of order \(0<s<1\) in space. By using integral transforms, we derive the solutions in convolutional form for the considered equations and determine their fundamental solutions. Subsequently, the asymptotic behaviors of solutions are studied by means of Young’s inequality for convolution in a general \(L^{p}\) -norm with \(1\le p\le \infty \) . Next, gradient estimates of the solutions are discussed. Finally, we present large time behaviors for the given equations. Especially, we obtain the optimal \(L^{2}\) -decay rate for the case \(0<\alpha <1\) by the Plancherel’s theorem.
