We introduce a configuration of a q-difference equation and characterize the variants of the q-hypergeometric equation, which were defined by Hatano-Matsunawa-Sato-Takemura, by configurations. We show integral solutions and series solutions for the variants of the q-hypergeometric equation.

In this article we investigate Gevrey regularity of formal power series solutions for a certain class of nonlinear moment partial differential equations, the inhomogeneity of which is of a fixed Gevrey order with respect to the time variable. The results are a generalization of similar ones obtained for standard nonlinear partial differential equations as well as linear moment differential equations.

In this paper, we establish the global well-posedness of a strong solution to the three-dimensional Tropical climate model with fractional dissipation and damping. We prove the global existence and uniqueness of the strong solution under suitable assumptions on the dissipation and damping terms of the barotropic velocity. Our result is based on a priori energy estimates and interpolation inequalities. A key feature of this work is to minimize the required damping exponent when the dissipation is taken to be weaker. In addition, we note that the proof of uniqueness in this framework can also be applied to the Navier-Stokes and MHD equations.