In this article, we are interested in the existence, uniqueness and regularity of solutions of fully nonlinear parabolic equations, with initial data u₀, in the whole space Rⁿ. Our main result is the existence of a strictly subquadratic solution with a local C^1,β regularity with respect to the space variable, assuming C^1,α regularity on u₀ and local uniform ellipticity of the equation. Our proof relies on a result of N. Zhu which shows the local C^1,β regularity of the solution provided it is Lipschitz continuous and Holder continuous in t, with an exponent g / 2 ; we obtain this last property through a continuous dependence result. Then we investigate further regularity for the solution using results of L. Wang.