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The nonhomogeneous incompressible Magnetohydrodynamic Equations withdensity-dependent viscosity is studied in three-dimensional (3D) exteriordomains with slip boundary conditions. The key is the constraint of anadditional initial value condition $B_0\in L^p (1\leqslant p<12/7)$, whichincrease decay-in-time rates of the solutions, thus we obtain the globalexistence of strong solutions provided the gradient of the initial velocity andinitial magnetic field is suitably small. In particular, the initial density isallowed to contain vacuum states and large oscillations. Moreover, thelarge-time behavior of the solution is also shown.

Global Strong Solutions to the incompressible Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum in 3D Exterior Domains