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We prove the wellposedness of scalar wave equations on spatially flatuniverse as a background with nonminimal coupling with the scalar potentialturned on by introducing the $k$-order linear energy and the correspondingenergy norm. In the local case, we show that both the $k$-order linear energyand the energy norm are bounded for finite time with initial data in$H^{k+1}\times H^{k}$. Whereas in the global case, we have to add threeassumptions related to the nonminimal coupling constant, the scale factor ofspacetimes, and the form of the scalar that has to be a polynomial with a smallpositive parameter. Then, we show that the solution does globally exist with aparticular decay estimate that depends on the scale factor of the spacetimes.Finally, we provide some physical models that support our general setup.

Local and Global Existence of Solutions to Scalar Equations on Spatially Flat Universe as a Background with Non-minimal Coupling