Well-Posedness of Mild Solutions for Superdiffusion Equations with Spatial Nonlocal Operators

In this paper, we study the well-posedness for a class of semilinear superdiffusion equations with spatial nonlocal operators. We first establish the Gagliardo–Nirenberg inequality in \(\psi \) $\psi$ -Bessel potential spaces. Based on this, the well-posedness results of local and global mild solution for corresponding linear problem are obtained via apriori estimates. We also obtain the well-posedness results for the nonlinear problem under different conditions. These conclusions are mainly based on the Mihlin–Hörmander’s multiplier estimates, embedding theorem and fixed point theory.