Decay estimates of global solutions for the three-dimensional magnetohydrodynamics equations with horizontal dissipation in a half space

Consider the three-dimensional magnetohydrodynamics (MHD) equations with horizontal dissipation in the upper half-space $\mathbb{R}_{+}^{3}$ . First, under appropriate assumptions on the initial velocity and magnetic field in the new-type space $\mathfrak{L}_{\delta}^{p,p}(\mathbb{R}_{+}^{3})$ , with the help of the integral equation, we prove the global well-posedness of the system for small initial values. Second, through the linear solution formula and using the space $\mathfrak{L}_{\delta}^{p,p}(\mathbb{R}_{+}^{3})$ introduced to overcome the singular integral operators emerging in the solution formula, we obtain the first-order derivatives and the optimal decay rate of the solution in the anisotropic Lebesgue norms. Finally, utilizing the embedding relationship between the $\mathfrak{L}_{1}^{1}(\mathbb{R}_{+}^{3})$ space and the $L_{1}^{1}(\mathbb{R}_{+}^{3})$ space, we obtain the uniform $L_{1}^{1}(\mathbb{R}_{+}^{3})$ -estimate of the solution.